UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de Matemática, Estatística e Computação Científica ABNER DE MATTOS BRITO Concerning Formal Concept Analysis on Complete Residuated Lattices Sobre Análise de Conceitos Formais em Reticulados Residuados Completos Campinas 2019 Abner de Mattos Brito Concerning Formal Concept Analysis on Complete Residuated Lattices Sobre Análise de Conceitos Formais em Reticulados Residuados Completos Dissertação apresentada ao Instituto de Matemática, Estatística e Computação Científica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestre em Matemática Aplicada. Dissertation presented to the Institute of Mathematics, Statistics and Scientific Computing of the University of Campinas in partial fulfillment of the requirements for the degree of Master in Applied Mathematics. Supervisor: Laécio Carvalho de Barros Co-supervisor: Marcelo Esteban Coniglio Este exemplar corresponde à versão final da Dissertação defendida pelo aluno Abner de Mattos Brito e orientada pelo Prof. Dr. Laécio Carvalho de Barros. Campinas 2019 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado CRB 8/5467 Brito, Abner de Mattos, 1991- B777c BriConcerning formal concept analysis on complete residuated lattices / Abner de Mattos Brito. – Campinas, SP : [s.n.], 2019. BriOrientador: Laécio Carvalho de Barros. BriCoorientador: Marcelo Esteban Coniglio. BriDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. Bri1. Teoria dos reticulados. 2. Lógica simbólica e matemática. 3. Lógica fuzzy. 4. Teoria dos conjuntos. 5. Conjuntos fuzzy. I. Barros, Laécio Carvalho de, 1954-. II. Coniglio, Marcelo Esteban, 1963-. III. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. IV. Título. Informações para Biblioteca Digital Título em outro idioma: Sobre análise de conceitos formais em residuados reticulados completos Palavras-chave em inglês: Lattice theory Symbolic and mathematical logic Fuzzy logic Set theory Fuzzy sets Área de concentração: Matemática Aplicada Titulação: Mestre em Matemática Aplicada Banca examinadora: Laécio Carvalho de Barros [Orientador] Walter Alexandre Carnielli Renata Wassermann Data de defesa: 28-03-2019 Programa de Pós-Graduação: Matemática Aplicada Identificação e informações acadêmicas do(a) aluno(a) ORCID do autor: https://orcid.org/0000-0002-0920-6801 Currículo Lattes do autor: http://lattes.cnpq.br/4462666377529458 Powered by TCPDF (www.tcpdf.org) Dissertação de Mestrado defendida em 28 de março de 2019 e aprovada pela banca examinadora composta pelos Profs. Drs. Prof(a). Dr(a). LAÉCIO CARVALHO DE BARROS Prof(a). Dr(a). WALTER ALEXANDRE CARNIELLI Prof(a). Dr(a). RENATA WASSERMANN A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica. To a certain stray dog I never got to know, but without which this work would not be. Acknowledgements This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil (CAPES) Finance Code 001. In the development of this work, parts of the material here presented - particularly Section 3.1, together with other portions of text here exposed - have been already published (BRITO et al., 2018) . I am very grateful to Laécio Barros and Esteban Coniglio, who have been my supervisors in the development of this work, and who have shared with me not only their mathematical knowledge but their experience in the form of wise advice. Furthermore, I would like to thank Estevão Laureano who has provided many valuable insights that allowed the development of this work. I am indebted to Fábio Bertato, who has encouraged me to enter the Masters in Applied Mathematics programme at Unicamp together with Masters in Philosophy programme, which I am currently developing at the Centre for Logic, Epistemology and the History of Science (CLE-Unicamp), where Fábio is my co-supervisor. He is the one who has proposed the development of a work related to Formal Concept Analysis. Over the years Fábio has provided me with guidance not only with in academic matters but in other subjects of life. There are many people whom I met through my studies and who have helped me in one way or another, but I would especially like to thank some friends, namely Laryssa Abdalla, Melissa Carvalho and Joyce Clímaco (IMECC), as well as João Toniolo and Nicola Salvatore (CLE). Although we didn't spend so much time together as I would like over these last two years, I consider each of you a friend. It is always great to hear that you are well in spite of all difficulties that life brings upon us. I hope that I can aid each of you in difficult times. I am verily thankful to Marcelo Cabral, Priscila Akemi, Altamiro Menezes and Ricardo Silva for your friendship and spiritual guidance. The Lord has used each of you to relief my burden in times of distress, and to make my joy complete when life is merry. I hope I can ever repay you for what you did for me, but I am sure that in the right time the Lord will. There are so many people from CEU who have been by my side through this time that I am afraid I will forget some names, but I will give it a try: Alexandre, "tia" Alice, Aline Muzio, Amadeo, Bia, Bruna, Felipe Macedo, Fernando (aka Ferpa), Laura, Maira, Marília, Marina, Murilo, Natã, Neusa, Quesia, Romulo, Simone. Thank you all for being at my side and remembering me in your prayers. Among all these friends, I specially thank Gesiel and Mateus. Now we share a roof, but you shared your lives with me prior to that. Together we have thouhgt and contemplated, walked and sat down, traveled and stayed, talked and made silence, laughed and cried, listened and prayed. As Lewis writes in The Weight of Glory, you are signs of Grace to my senses for in you "Christ vere latitat [...] Glory Himself is truly hidden." Much less important than this, thanks Mateus for helping me to prove a result I was stuck on. Thank you so much Letícia for being at my side over the last few months. You have supported me, allowed me to be myself, taken care of me when my health was poor, cheered me up when I didn't have heart to proceed. It feels like every meal at your side tastes better. Thank you for praying with/for me. The Lord knows our hearts and He knows everything else. Surely He will guide us according to His will in all things. I thank my father Edney, my mother Cláudia and my brother Willyam, who have always supported me in so many ways: financially, in conversations and through your prayers. Without you I wouldn't be who I am or where I am. Thank you so much for your unconditional love. Finally, I praise my Lord Christ Jesus, together with the Father and the Holy Spirit, One True God. Unto Him be all the glory now and evermore! Resumo Análise de Conceitos Formais (FCA) é uma teoria matemática elegante, intimamente ligada à teoria de reticulados completos. Conforme exibido no presente trabalho, a teoria pode ser generalizada, permitindo-nos trabalhar com contextos formais nos quais objetos e atributos estão relacionados entre si apenas parcialmente e, de fato, o "grau"de tal relação pode ser extraído de reticulados residuados completos arbitrários. O presente texto propõe-se a ser uma introdução concisa, intuitiva e autocontida à FCA Fuzzy. Para tanto, o leitor é introduzido às ideias de ordem, reticulados (completos) e operadores de fecho em tais reticulados. Estas - juntamente com noções de teoria ingênua dos conjuntos - formam a base necessária para a compreensão de FCA clássica. Posteriormente introduzimos a teoria de conjuntos fuzzy que, juntamente com a teoria de conectivos fuzzy (em particular t-normas e seus resíduos), provê as ferramentas necessárias para se estender a teoria clássica. Acreditamos que os exemplos apresentados ao longo do texto compõem um guia esclarecedor para a intuição. Palavras-chave: Análise de conceitos formais. Reticulado de conceitos. Análise de conceitos formais fuzzy. Conceitos formais fuzzy. Reticulado de conceitos fuzzy. Abstract Formal Concept Analysis (FCA) is an elegant mathematical theory closely related to the theory of complete lattices. As shown in this work the theory can be generalized, allowing us to work with formal contexts in which objects and attributes are only partially related to each other and, in fact, the "degree" of such a relationship may be drawn from arbitrary complete residuated lattices. The present text aims to be a concise, intuitive and self-contained introduction to fuzzy FCA. In order to reach this objective, the reader is introduced to the ideas of orders, (complete) lattices and closure operators on such lattices. That - together with notions of naïve set theory - is the necessary basis for an understanding of classical FCA. Afterwards, we introduce the theory of fuzzy sets which, together with fuzzy connectives (particularly tnorms and their residua), provides us with the tools necessary to extend the classical theory. It is our belief that the examples provided throughout the text have been illuminating as a guide for intuition. Keywords: Formal concept analysis. Concept lattice. Fuzzy formal concept analysis. Fuzzy formal concepts. Fuzzy concept lattice. List of Figures Figure 1 – Isomorphic Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 2 – Concept Lattice of Animals . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 3 – Complexity of the Algorithms . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 4 – Membership to the Fuzzy Set of Young People . . . . . . . . . . . . . . 51 Figure 5 – Operations with Fuzzy Subsets . . . . . . . . . . . . . . . . . . . . . . 53 Figure 6 – Triangular norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 7 – Triangular conorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Figure 8 – Fuzzy negations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 9 – Fuzzy implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 10 – Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Figure 11 – Output (left) and Lattice Diagram (right) of Alg. 5 applied to Table 3. 94 Figure 12 – Medical Diagnosis: Gödel's implication . . . . . . . . . . . . . . . . . . 97 Figure 13 – Number of concepts with multivalued Łukasiewicz implications . . . . . 98 Figure 15 – Medical Diagnosis: 5-valued Łukasiewicz implication . . . . . . . . . . 100 List of Tables Table 1 – A formal context of animals . . . . . . . . . . . . . . . . . . . . . . . . . 38 Table 2 – Truth Tables of Classical Connectives . . . . . . . . . . . . . . . . . . . 57 Table 3 – Revisiting example 4.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Table 4 – Clinical signs observed for diagnosis of pneumonia . . . . . . . . . . . . 95 Table 5 – Fuzzy relation of clinical signs and diagnoses . . . . . . . . . . . . . . . 96 Table 6 – Relation of clinical signs and diagnoses, with values in L5 . . . . . . . . 97 List of abbreviations and acronyms iff If and only if List of symbols χ A Characteristic function of the set A μ A Membership function of the fuzzy set A - formally μ A is the set A 2Y Collection of functions on Y mapped to t0, 1u LY Collection of functions on Y mapped to L PpY q Collection of the subsets of Y - bijective with 2Y FpY q Collection of the fuzzy subsets of Y - formally the same as LY AB Cartesian product of A and B A1  ... An Cartesian product of the sets A1, ..., An A  B A is a subset of B, possibly equal to B A  B A is a proper subset of B, i. e., A  B but A  B© S Infimum (greatest lower bound) of the set S, also denoted by inf Sa S Supremum (least upper bound) of the set S, also denoted by supS ra, bs Closed interval of the elements a ¤ x ¤ b sa, br Open interval of the elements a x b ra, br Half-open interval of the elements a ¤ x b sa, bs Half-open interval of the elements a x ¤ b xA,By Ordered pair with first coordinate A and second coordinate B xA,B,Cy Ordered triple with first coordinate A, second coordinate B and third coordinate C xAαyαPJ A sequence indexed by J pa, bq Ordered pair, where we explicitly treat a, b as elements of given sets A,B paαqαPJ A sequence indexed by J , where we explicitly treat aα as an element Aα x ÞÑ y An unnamed map is presented, with the rule that maps each x to a y φ ÝÑ ψ Classical (material) implication of φ and ψ xñ y Fuzzy implication of x and y φ Classical negation of φ νpxq Fuzzy negation of x φ^ ψ Classical conjunction ("and") of φ and ψ x M y Triangular norm (or t-norm, or fuzzy conjunction) of x and y φ_ ψ Classical disjunction ("or") of φ and ψ xOy Triangular conorm (or t-conorm, or fuzzy disjunction) of x and y xO,A, Iy Formal context with object set O, attribute set A and relation I, usually denoted by C xO,A, Ify Fuzzy formal context with object set O, attribute set A and fuzzy relation I, usually denoted by Cf BpCq Collection of formal concepts on the formal context C BfpCfq Collection of fuzzy formal concepts on the fuzzy formal context Cf C pY q Collection of closed (perhaps fuzzy) subsets of Y with respect to a given closure operator List of Algorithms Algorithm 1 – Standard Computation of Intents . . . . . . . . . . . . . . . . . . . 43 Algorithm 2 – Next Intent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Algorithm 3 – UpperNeighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Algorithm 4 – GenerateFrom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Algorithm 5 – Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Contents Topics Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 0 ORDER THEORY AND LATTICES . . . . . . . . . . . . . . . . . . 21 0.1 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 0.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1 FORMAL CONCEPT ANALYSIS . . . . . . . . . . . . . . . . . . . 37 1.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.2 Computing the concept lattice . . . . . . . . . . . . . . . . . . . . . . 43 2 FUZZY SETS AND FUZZY CONNECTIVES . . . . . . . . . . . . . 49 2.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Fuzzy relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 Extending classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.1 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.2 Predicates and quantification . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.4 Continuity and semicontinuity . . . . . . . . . . . . . . . . . . . . . . 67 2.5 Residuated Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3 FUZZY FORMAL CONCEPT ANALYSIS . . . . . . . . . . . . . . . 79 3.1 Formal concept analysis in the fuzzy setting . . . . . . . . . . . . . . 79 3.2 Formal concept analysis on complete residuated lattices . . . . . . . 85 3.2.1 The approach of Burusco and Fuentes-Gonzáles . . . . . . . . . . . . . . . 85 3.2.2 The approach of Pollandt and Bělohlávek . . . . . . . . . . . . . . . . . . 86 4 COMPUTING THE FUZZY CONCEPT LATTICE . . . . . . . . . . 89 4.1 Computational algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 An application of fuzzy FCA . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.1 Posing the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.2 Computing the fuzzy concept lattice . . . . . . . . . . . . . . . . . . . . . 96 4.2.3 Interpreting the diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2.4 Final considerations concerning applications . . . . . . . . . . . . . . . . . 101 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 18 Subjects flowchart Chapter 2 Chapter 3 Partial_Orders, Lattices Chapter 0 Fuzzy Sets L-fuzzy_Sets, Residuated_Lattices Formal Concept Analysis Chapter 1 Fuzzy FCA FCA on Complete Residuated Lattices Algorithms Chapter 4 19 Introduction According to (JOSEPH, 2002), a concept is an abstraction produced by the intellect (e.g. the concept of tree abstracts what is common to several trees observed). A concept may then be communicated by a term, i.e., a symbol - a sound, a sequence of characters, an image etc. with a meaning imposed to it by convention - which expresses the concept (e.g., an image of a skull which stands for danger, or the sequence of characters T-R-E-E which stands for the notion of tree). Also according to JOSEPH, a term has both extension and intension, the former corresponding to the set of all objects to which the term corresponds (e.g., the set of all trees), and the later, to "the sum of the essential characteristics that the term implies." Moreover, "As a term increases in intension, it decreases in extension." Given two concepts C1 and C2, if the extension of C1 is a part of the extension of C2 - i.e., if every object corresponding to C1 also corresponds to C2 - we say that C1 is a subconcept of C2. Formal Concept Analysis (FCA), a mathematization of the philosophical idea of concept (GANTER; WILLE, 1999) , based on the idea - slightly different from JOSEPH's - that the concept that has extension and intension. In this text, we present an extension of FCA by grounding the theory on fuzzy sets. On the fuzzy setting computational algorithms are presented, which are applied to explore an example of medical diagnosis. Because there is a hierarchy of concepts, in Chapter 0 basic notions of order theory are presented. We define orders, partially ordered sets, lattices covering relations and describe how a diagram may be drawn out of a lattice (or poset) in such a manner that the lattice (resp. poset) can be read back from the diagram. Furthermore, some basic notions concerning closure operators and Galois connections are presented. In Chapter 1 the classical theory of FCA is introduced. Given a set of objects O, a set of attributes A and a binary relation I on O  A - together they constitute a formal context -, we define two maps , ^, which are used to define formal concepts, which are composed of an extent and an intent. It turns out that these maps constitute a Galois connection between xO,y and xA,y, and so ^ and ^ are closure operators. As a consequence, the collection of formal concepts of a given formal context is a complete lattice. An algorithm for computing every formal concept of a formal context is presented, and the idea of another algorithm, for computing the concepts together with the lattice structure is mentioned. In order to extend FCA, fuzzy sets are introduced in Chapter 2, as well as fuzzy set operations (union, intersection and complementation). By means of fuzzy sets, we extend the classical connectives for conjunction (^), disjunction (_), implication (ÝÑ) CONTENTS 20 and negation ( ) respectively to t-norms (M), t-conorms (O), fuzzy implications (ñ) and fuzzy negations (ν). In particular, we consider ñ as the residuum of a t-norm. We prove some properties concerning lower semicontinuous t-norms. Some basic notions of residuated lattices are also presented. Chapter 3 deals with the extension of FCA to fuzzy sets. In Section 3.1 show that, if we use lower semicontinuous t-norms and their residua in order to define maps  and ^, all the basic results of the classical theory as presented in Ch. 1 are preserved. The results have been found independently by us, but it turns out that they fall within the scope of a more general theory of FCA on residuated lattices. The idea of this more general approach is presented, alongside with another approach - pioneer in generalizing FCA to (L-)fuzzy sets, but without some useful properties. Finally, Chapter 4 deals with the problem of computing the fuzzy concept lattice when we consider the set L of truth values as a finite subset of r0, 1s - actually, the algorithm presented applies to any finite linearly ordered set. The algorithm is used to explore an example of the relation between clinical signs and pneumonia vs. non-pneumonia diseases in children. We have attempted to write a self-contained text, accessible to people who understand mathematical reasoning and who have an intuitive knowledge of sets, functions, real numbers etc., as well as the notion of what is an algorithm. No rigorous, axiomatic, knowledge of such notions is required. Nonetheless, it is certainly helpful that the reader understands what are the supremum and infimum of (limited nonempty) subsets of R, and why these may not exist for (limited nonempty) subsets of Q. Also useful - particularly for some proofs in the auxiliary Section 2.4 is basic knowledge related to sequences. 21 0 Order Theory and Lattices The idea of concept shall be (briefly) presented later in this text, but we mention from the start that there is a hierarchy of concepts. For example, the concept of animal is broader than that of human in the sense that the former includes all individuals of the later - and in fact other, non-human animals. On the other hand, although the concept of dog is narrower than that of animal it is neither broader nor narrower than the concept of human. In terms of order theory, we say that this is "hierarchy" is a nonlinear order. Orders will be extensively used in this text, and for this reason we present some basic notions of order theory. The content of the present chapter has been drawn mostly from (GRÄTZER, 1971) and (BIRKHOFF, 1948) , to which we refer the interested reader. Some of the content of this chapter (and others) corresponds to set theory1, and can be found in any good textbook on the field, such as (HRBÁČEK; JECH, 1999) . 0.1 Orders Definition 0.1.1. A n-place relation on a Cartesian product of sets A1  ...  An is a subset R  A1  ... An. If A : A1  ...  An we say that R is a n-place relation on A. In particular, a 2-place relation is called a binary relation. It is common practice to use infix notation for binary relations, i.e., pa1, a2q P R is expressed as a1Ra2. Given S2  S1 and a binary relation R1 on S1, the restriction of R1 to S2 is the subset R2 of R1 defined as the intersection of R1 and S2  S2, that is, R2  tpx, yq P R1 : x, y P S2u . Example 0.1.2. Given a set S, an equivalence relation on S is a set R  S2 that is reflexive (aRa), symmetric (aRb implies bRa) and transitive (aRb and bRc imply aRc). For instance, we say that a, b P Z are congruent modulo n, denoted by a  b mod n, if a b  kn for some k P Z. Congruence modulo n is an equivalence relation on Z for all n P Z. A very special class of binary relations on a given set is that of order relations. Definition 0.1.3. A partial order, or simply order , on a set X is a binary relation ¤ on X that satisfies, for all x, y, z P X: 1 We assume the ZFC axioms as presented by (HRBÁČEK; JECH, 1999). Chapter 0. Order Theory and Lattices 22 1. x ¤ x (reflexivity) 2. if x ¤ y and y ¤ x then x  y (antisymmetry) 3. if x ¤ y and y ¤ z then x ¤ z (transitivity) An order is called linear or total if it additionally satisfies 4. x ¤ y or y ¤ x (linearity) A set P equipped with a partial order ¤, denoted by xP,¤y, is called a partially ordered set, or poset. If the order is linear, we say to have a linearly ordered set, a totally ordered set or a chain. We may refer to xP,¤y simply as P if there is no risk of confusion concerning the order ¤. Two elements a, b P P are said to be comparable if either a ¤ b or b ¤ a. Otherwise, a and b are incomparable. Clearly there are incomparable elements in a poset iff it is not a chain. The binary relation defined as a b iff a ¤ b and a  b (0.1.4) is called a strict order. Strict orders can be defined on their own rights rather than being defined in terms of partial orders. Proposition 0.1.5. A binary relation on X is a strict order iff, for all x, y, z P X, 1. x x does not hold (irreflexivity) 2. if x y then y x does not hold (asymmetry) 3. if x y and y z then x z (transitivity) Proof. We first prove that strict orders are irreflexive, asymmetric and transitive. 1. If we had x x then we would also have x  x by (0.1.4), which is a contradiction. Thus, x x does not hold. 2. Suppose for the sake of contradiction that both x y and y x. Then by (0.1.4) we would have x ¤ y and y ¤ x, implying (by antisymmetry of ¤) that x  y, which would contradict x y. Thus, x y and y x cannot both hold. 3. Suppose that x y and y z. Then x ¤ y and y ¤ z, whence x ¤ z. Furthermore, were x  z we would have x y and y x, contradicting asymmetry. Thus, x z. On the other hand, suppose that is an irreflexive, asymmetric and transitive relation. Define   by x   y iff x y or x  y . (0.1.6) Chapter 0. Order Theory and Lattices 23 Then x y iff x   y and x  y2. Now it is clear from (0.1.4) that and ¤ are the same relation. To finish the discussion concerning strict orders, we highlight the fact that proof of Proposition 0.1.5 tells us that there is a one-one correspondence between partial orders and strict orders, in such a way that one uniquely defines the other, i.e., a partial order ¤ defines a strict order by means of (0.1.4), a strict order defines a partial order as in (0.1.6) and ¤, are the same. The dual relation of ¤ is ¥ defined as x ¤ y iff y ¥ x, which is easily seen to be an order3. The reader should see that the dual of ¥ is ¤. Correspondingly, the dual of is ¡, the strict order defined from ¥. The dual of a statement4 about posets is the statement we get by replacing every occurrence of an order (¤ or ¥) by its dual (¥ or ¤, respectively). Thus we are able to state a nice, work saving principle. Duality Principle. If a statement is true in all posets, then its dual is also true in all posets. Indeed, consider one such statement Φ, and let xP,¤y be any poset. Because Φ is true in all posets, it is also true in xP,¥y. Replacing each occurrence of ¥ by ¤ yields the dual Φd of Φ, and Φd is true in xP,¤y. Example 0.1.7. Consider the following. 1. The sets N,Q and R, equipped with their respective usual orders, are each a linearly ordered set with their respective usual orders. 2. Let S be a non-empty set and let PpSq be the set of its subsets (informally, the "power set" of S). Then xPpSq,y is a poset, where  is set inclusion5. Moreover, if S has at least two elements then PpSq is nonlinear, as tau, tbu P PpSq are incomparable with respect to set inclusion whenever a  b. 2 If x y then it is true that x y or x  y, whence x   y. Conversely, if x   y then x y or x  y whence, if additionally we have x  y, we must have x y. 3 Check for yourself that ¥ is reflexive, antisymmetric and transitive, and that if ¤ is linear so is ¥. 4 A statement can be given a precise, rigorous meaning which forms the base of both Statement and First Order Predicate Calculi. These are used to work out the foundations of mathematics in logical terms. For more on the Statement Calculus and First Order Predicate Calculus, See (MENDELSON, 2009) . 5 Because an order on a set S is a subset of S2, whenever we consider set inclusion as an order we restrict ourselves to the order PpSq on PpSq, i.e.: A PpSq B iff A  S and B  S and A  B . But whenever we use PpSq, the power set of S is already implicit in the context, so that we write  for brevity. Chapter 0. Order Theory and Lattices 24 3. Let S  ta, b, cu. Let ¤1 tpa, aq, pa, cq, pb, bq, pb, cq, pc, cqu6 and let ¤2 be the dual order of ¤1 (i.e., replace pa, cq by pc, aq and pb, cq by pc, bq). Then both xS,¤1y xS,¤2y are posets. 4. Consider @ R2,¤p D , where px, yq ¤p pz, wq iff x ¤R z. It is reflexive, transitive but is not antisymmetric, and thus is not a poset. In fact, p0, 0q ¤p p0, 1q and p0, 1q ¤p p0, 0q, but of course, p0, 0q  p0, 1q. 5. "Parenthood" on the set of human beings is (vacuously7) antisymmetric, but is irreflexive (no person is its own parent) and non-transitive (strictly speaking, a grandparent is not a parent). 0.2 Lattices We now proceed to define the suprema and infima of subsets of a given set. Let P be a poset and let H  P . An element a P P is an upper bound of H if h ¤ a for all h P H. A least upper bound, or supremum, of H is an upper bound s of H such that s ¤ a for every upper bound a of H. Antisymmetry of ¤ implies that the supremum s of H is unique, and we shall denote it as either of the following. supH , a H ,a hPH h , na i1 hi , where the last notation applies only if H  th1, ..., hnu. Notice that the supremum of H may or may not be an element of H. For instance, in R the supremum of both A s0, 1r and B  r0, 1s is 1, which belongs to B but not to A. The definition of a lower bound is dual to the definition of an upper bound, and the definition of the infimum of a set is dual to that of the supremum of a set. Notice that the infimum of a set is unique by the duality principle. Dual to the notations for the 6 This means that a ¤1 a, a ¤1 c, b ¤1 b, b ¤1 c, c ¤1 c and nothing else satisfies ¤1. 7 Given some assertion about something, if one is to say that such an assertion is false one needs to present a counterexample. For instance, the statement "3 is the greatest natural number" is false because a counterexample may be presented, such as 4, a natural number greater than 3. Now, consider the statement "every negative natural number is greater than 8". Because there are no negative natural numbers, we cannot present one such number which is not greater than 8. Thus, we accept such a ridiculous statement as true - a rather vague truth - simply because it cannot be false. That is the reason why we accept as true every assertion about elements of the empty set, simply because there are no counterexamples. Such assertions are said to be vacuously true. Chapter 0. Order Theory and Lattices 25 supremum, the infimum of H is denoted as either of infH , © H ,© hPH h , n© i1 hi . Given a, b P P , we shall denote supta, bu and infta, bu by a _ b and a ^ b respectively. In order theory it is usual to call ^ the meet and _ the join. Let xL,¤y be a poset. If for all a, b P L both a_ b and â b exist, then xL,¤y is called a lattice. The induction principle implies that this is equivalent to the existence of both infimum and supremum of arbitrary finite subsets of of L. Definition 0.2.1. A poset xL,¤y is a lattice iff both a H and © H exist for any nonempty finite H  L. If a H and © H exist for arbitrary (finite or infinite) nonempty H  L, we say that the lattice is complete. This lattice is said to be finite or infinite depending on whether L is finite or infinite. Notation. If a L exists (which is always the case for complete lattices), then it is denoted by 1 and it is the greatest element of L. Dually, if L has a smallest element © L, it is denoted by 0. In order to explicit the smallest and greatest elements of a complete lattice, it may be denoted as xL,¤, 0, 1y. Notice that every element of L is (vacuously) both an upper bound and a lower bound of the empty set H. Thus, in the lattice L, if 0 exists then a H  0, and dually if 1 exists then © H  1. Example 0.2.2. 1. Every finite lattice is complete by definition. 2. Given a nonempty set S, xPpSq,y is a complete lattice with infimum given by set intersection and supremum given by set union. 3. Any closed interval of R is a complete lattice under its usual order, where infimum and supremum are defined as usual. 4. R is not a complete lattice because R itself has neither supremum nor infimum. However, if we consider R : RY t8, 8u  r8, 8s , then we have a complete lattice. 5. Let IQ  tx P Q : 0 ¤ x ¤ 1u Chapter 0. Order Theory and Lattices 26 be ordered by the restriction ¤IQ of the usual order ¤Q of Q to IQ. Then IQ is a lattice, but not a complete lattice. In fact, the set S  tr P IQ : 2r2 ¤IQ 2u has no supremum in IQ8. In fact, no interval I  Q is a complete lattice - although I is always a lattice under the restriction of ¤Q to I - as we can always find a subinterval of I whose supremum is "absent", i.e., is not rational. The definition of R via Dedekind cuts9 is made in order to literally fill the gaps that prevent intervals of Q from being complete lattices. 6. The posets in item 3. of Example 0.1.7 are not lattices. In fact, ta, bu has no infimum with respect to ¤1 and no supremum with respect to ¤2. Lattices can also be defined algebraically in terms of the meet and the join of elements of L. Although we shall not present the definition here, we draw the reader's attention to the fact that a ¤ b iff â b  a (0.2.3) iff a_ b  b . (0.2.4) Also, both _ and ^, considered as binary operations, are idempotent, commutative, associative and satisfy the following absorption identities: â pa_ bq  a , a_ pâ bq  a . These properties suffice in order to define a lattice xL,_,^y in a way that is equivalent to xL,¤y. Now we turn to the question of whether it is possible to find a binary subrelation of ¤ such that all the information in ¤ can be recovered from ¤. As we shall see, the answer is "yes", and this will allow us to represent (finite) posets and lattices graphically. A binary relation R on a set S can always be made reflexive by adding the diagonal of S2 to it, that is, by taking the set Rref  R Y tps, sq P S 2 : s P Su . 8 Indeed, the supremum should be ? 2{2 R Q. 9 See (HRBÁČEK; JECH, 1999) . Chapter 0. Order Theory and Lattices 27 Similarly, Rtrans  R Y tpx, zq P S 2 : xRy and yRz for some y P Su is a transitive binary relation. Thus, a relation such as that of "Parenthood" (see Example 0.1.7 item 6.), which is antisymmetric but is neither reflexive nor transitive, could be made into an order ord by first extending it to a reflexive relation, then to a transitive relation10. However, we would have a great difficulty to make an antisymmetric relation out of a relation which is presents symmetries, because if there are two distinct x, y P S such that xRy and yRx , we would need to exclude either px, yq or py, xq from R, in which case information from the original relation would be lost11. Fortunately orders are already antisymmetric and, because symmetries do not arise by excluding pairs from a binary relation, we do not need to concern ourselves about such a process. As seen above, a binary relation can always be made reflexive and transitive, so we now create a sub-relation of ¤ by excluding all the information about reflexivity and transitivity. Definition 0.2.5. Let xP,¤y be a finite poset. We define for each a, b P P by a b iff $& %a b , andEc P P such that a c b . (0.2.6) The relation is called a covering relation on P (with respect to ¤). If a b we say that a is covered by b, or that a is a lower neighbour of b. Conversely, we say that b covers a, or that b is an upper neighbour of a. Here, of course, a c b is short for a c and c b. We know that can be made into an order relation ord, but we need to check that ord really is the same as ¤. Suppose that a b. If P is finite there exists a maximal chain a  c0 c1 ... cn1  b, that is, there is no c P P such that ci c ci 1 for i  0, ..., n 2. In fact, of all the (finitely many) subsets of P we may take all the (finite) chains H with smallest element a and greatest element b (at least one such H exists, namely ta, bu). From the finitely many such H we choose one of larger size and let n be the number of elements of 10 We would reach the same order if the "Parenthood" relation were extended first transitively and only then reflexively, because transitive extensions do not add new information for diagonal elements. In fact, for x  y we have, in the extension equation above, that px, zq P Rtrans iff py, zq P R. 11 After this process, we would have no straightforward criteria for deciding which elements of the reduced relation we should need to make symmetric again. Chapter 0. Order Theory and Lattices 28 H. We can now assume that H  tc0, c1, ..., cn1u is such that a  c0 c1 ... cn1  b. Now by maximality of H, there is no c P P such that ci c ci 1, for i  0, ..., n  2 (otherwise there would be a chain larger than H with smallest and greatest elements a and b respectively). This proves the following: Lemma 0.2.7. Let xP,¤y be a finite poset. Then a ¤ b iff a  b or there exists a finite sequence c0, ..., cn1 of elements of P such that a  c0 c1 ... cn1  b , where is as defined in 0.2.6. Given a poset xP,¤y, we say that a covers b (or b is covered by a) if a b. For this reason, the relation is called the covering relation (or simply covering) corresponding to ¤. This lemma, holds true of lattices, which are a special case of posets. Example 0.2.8. Consider the Example 0.1.7 item 3. The orders ¤1 and ¤2 have the following coverings respectively: 1  tpa, cq, pb, cqu , 2  tpc, aq, pc, bqu . The covering relation allows us to draw a diagram of a finite lattice (and of a finite poset in general). This diagram is such that there is a unique circle, called a node, corresponding to each element of L and, for each a, b P L, a b iff the circle of a is below the circle of b and there is a line connecting them. Example 0.2.9. 1. Consider the set S  ta, bu. Then PpSq ordered by set inclusion  and has the following diagram: H tau tbu ta, bu Nodes tau and tbu are incomparable. H is smaller than ta, bu according to  because there exists an upwards path connecting nodes H and ta, bu, namely the one passing through tau. It has the covering  tpH, tauq, pH, tbuq, ptau, ta, buq, ptbu, ta, buqu . Chapter 0. Order Theory and Lattices 29 2. Consider the set S  ta, b, cu with orders ¤1 and ¤2 given in Example 0.1.7 item 3. They have the following diagrams: c a b (a) xS,¤1y c a b (b) xS,¤2y The coverings 1 and 2 have been given in Example 0.2.8. 3. Consider the lattice which corresponds to the following diagram: a b c d e f Notice that, although node c is higher than node d, they are not comparable because it is not possible to, starting from node d, reach node c going strictly upwards on the diagram. 4. Using the same idea as in previous items, N with its usual order could be represented as 0 1 2 3 giving the additional information that the diagram continues upward with the elements given natural succession. 5. Neither Q nor R is well-ordered12, and so they do not have covering relations. Nonetheless, they can be represented as usual with axes in which we point out the zero and other elements of interest. 12 Intuitively, a set is well-ordered if, for each element, we know what is the "next" element. Because both Q and R are dense sets (i.e., given any two elements, there is another element between them), it is impossible to determine the "next" element to any rational or real number. For more on well-orders, See (HRBÁČEK; JECH, 1999) . Chapter 0. Order Theory and Lattices 30 In many cases it happens that certain elements can be represented as the supremum of certain elements, or as their infimum. (GANTER; WILLE, 1999) present the following definitions. Definition 0.2.10. Given a complete lattice xL,¤, y, consider for each x P L the elements x#  a ty P L : y xu , x#  © ty P L : x yu . Then x is called supremum-irreducible, and denoted as a -irreducible, if x  x#. Dually, if x  x# it is said infimum-irreducible, and denoted as © -irreducible. The sets of alla , © -irreducible elements of L are denoted JpLq and MpLq13, respectively, or simply J,M if there is no risk of confusion concerning L . Definition 0.2.11. Sets X, Y  L are respectively called supremum-dense and infimumdense iff for all z P L we have respectively z  a tx P X : x ¤ zu , z  © ty P Y : z ¤ yu . It is clear that L is both a supremum-dense and an infimum-dense set. Nonetheless, in the finite case it is possible to find smaller sets which satisfy these properties, as stated in Prop. 0.2.12 proved by (GANTER; WILLE, 1999). Proposition 0.2.12. Let xL,¤y be a finite lattice. An element x P L is a -irreducible (resp. © -irreducible) iff it has exactly one lower neighbour (resp. upper neighbour). The sets J , M are respectively supremum-irreducible and infimum-irreducible and, if X, Y  L are respectively a supremum-irreducible and an infimum-irreducible set, then J  X and M  Y . Hence, J , M are minimal supremum-irreducible and infimum-irreducible sets, respectively. Example 0.2.13. Consider the following lattice. 0 a b c 1 13 As in join and meet. Chapter 0. Order Theory and Lattices 31 0 a b 1 0 α β 1 Figure 1 – Isomorphic Lattices Nodes a, b, 1 have exactly one lower neighbour, and nodes a, b, c have exactly one upper neighbour. Hence, J  ta, b, 1u , M  ta, b, cu . Furthermore, 0  a tj P J : j ¤ 0u  a H , 0  © tm PM : 0 ¤ mu  © M , a  a tj P J : j ¤ au  a tau , a  © tm PM : a ¤ mu  © ta, cu , b  a tj P J : j ¤ bu  a tbu , b  © tm PM : b ¤ mu  © tb, cu , c  a tj P J : j ¤ cu  a ta, bu , c  © tm PM : c ¤ mu  © tcu , 1  a tj P J : j ¤ 1u  a J , 1  © tm PM : 1 ¤ mu  © H . Notice that it is also possible that JpLq MpLq, depending on the structure of L14. There are certain circumstances under which two lattices may be considered to be "the same", i.e., they may have different elements, but have the same structure. That is what happens, for instance, with the lattices corresponding to the diagrams in Fig. 1. Definition 0.2.14. Let L1  xL1,¤1y, L2  xL2,¤2y be lattices. We say that L1 is isomorphic to L2 iff there exists a bijective function f : L1 Ñ L2 such that, for all x, y P L1, x ¤1 y iff fpxq ¤2 fpyq . In this case, f is called an isomorphism. Clearly L1 is isomorphic to L2 iff L2 is isomorphic to L115, so that we often say that L1 and L2 are isomorphic. Example 0.2.15. 1. The lattices xL1,¤1y, xL2,¤2y, with L1  ta, bu , ¤1  tpa, aq, pa, bq, pb, bqu L2  tα, βu , ¤2  tpα, αq, pα, βq, pβ, βqu 14 Consider Fig. 1. 15 With isomorphism f1. Chapter 0. Order Theory and Lattices 32 are isomorphic with isomorphism f : L1 Ñ L2 given by fpaq  α, fpbq  β. 2. Let S  tξ, ζu. Then the lattice xPpSq,y is isomorphic to, say, the left lattice of Fig. 1 with wither of the following isomorphisms defined on PpSq. fpHq  0 , gpHq  0 , fptξuq  a , gptξuq  b , fptζuq  b , gptζuq  a , fpSq  1 , fpSq  1 . 3. Consider the following result16. Let pP, q, pQ, q be countable dense linearly ordered sets without endpoints17. Then pP, q and pQ, q are isomorphic. This implies that xQ,¤Qy is isomorphic to xIQ,¤IQy, where IQ  tx P Q : 0 x 1u , and ¤IQ is the restriction of ¤Q to IQ. When working with Formal Concept Analysis in later chapters of this text, special attention shall be given to the set inclusion order . For the moment, we shall use a slightly different approach for working with members of PpY q. A set-theoretic approach to number theory defines 2 : t0, 1u18. Let us denote by 2Y the set of all functions Y Ñ t0, 1u. Given a function χ P 2Y , we can define a set Aχ  Y as Aχ  ty P Y : χpyq  1u . Conversely, given A  Y , there is a unique χ A P 2Y such that χ A pyq  $& %1, if y P A0, if y R A Furthermore, it can be proven that χ Aχ  A , Aχ A  A , 16 The proof can be found in (HRBÁČEK; JECH, 1999, Ch. 4). 17 A set S is countable if there exists a bijection between it and N; it is dense if it has at least two elements and for all a, b P S, a b implies a c b for some c P S; and it has no endpoints if there are no x, y P S such that x a y for all a P S. 18 In fact, writing n for the set-theoretic number n, we have 0 : H , n  1 : t0, ..., nu . Thus, n has n members according to our intuition of natural numbers. For all purposes of this text we do not need to distinguish between the n and n, hence we shall always write n. For more on set-theoretic arithmetic, see (HRBÁČEK; JECH, 1999) . Chapter 0. Order Theory and Lattices 33 so that we have a natural bijection between 2Y and PpY q. Given A  Y , we call χ A its characteristic function. It is clear that, given A,B  Y , we have A  B iff χ A pyq ¤ χ B pyq for all y P Y . This can be used in order to extend the notion of subsetness as follows. In the remainder of this text, BA denotes the set of all functions AÑ B. Proposition 0.2.16. Let Y be a nonempty set and xL,¤Ly a poset. Then the binary relation ¤ on LY defined for each f, g P LY as f ¤ g iff fpyq ¤L gpyq for all y P Y is an order. If additionally xL,¤Ly is a (complete) lattice, so is @ LY ,¤ D . Proof. We first prove that ¤ is an order on LY . Let f, g, h P LY . Then 1. (Reflexivity). fpyq ¤L fpyq for each y P Y , and so f ¤ f . 2. (Antisymmetry). Suppose that f ¤ g and g ¤ f . Then for each y P Y , fpyq ¤L gpyq and gpyq ¤L fpyq, so that fpyq  gpyq. Hence, f  g. 3. (Transitivity). Suppose that f ¤ g and g ¤ h. Let y P Y . Then fpyq ¤L gpyq and gpyq ¤L hpyq, so that fpyq ¤L hpyq. Hence, f ¤ h. Now, let us suppose that xL,¤Ly is a (complete) lattice. Then, given a finite (arbitrary) set F  LY , define F , F P LY respectively, for each y P Y , by F pyq  a FPF F pyq , F pyq  © FPF F pyq . Let f, g be respectively an upper bound and a lower bound of F . Let y P Y be fixed. Then, for each F P F we have fpyq ¥ F pyq , gpyq ¤ F pyq , whence fpyq ¥ a FPF F pyq gpyq ¤ © FPF F pyq  F pyq ,  F pyq , so that F  a F , F  © F . Chapter 0. Order Theory and Lattices 34 In particular, with this order on 2Y , xPpY q,y and @ 2Y ,¤ D are isomorphic with isomorphism fpAq  χ A . There is an interesting procedure for finding complete lattices ordered by set inclusion which uses the idea of closure operators defined below. Definition 0.2.17 and Theorem 0.2.20 below generalize results19 by E. H. Moore, which are presented (in a form closer to contemporary mathematical practice) by BIRKHOFF (1948, p. 49). Definition 0.2.17. Let Y be a nonempty set, xP,¤P y a poset and ¤ the order on P Y we defined in Proposition 0.2.16. A closure operator on xP Y ,¤y is a map Cl : P Y Ñ P Y such that, for all X,Z P P Y : 1. X ¤ ClpXq (extensivity) 2. ClpClpXqq  ClpXq (idempotency) 3. If X ¤ Z then ClpXq ¤ ClpZq (monotonicity) If the closure operator on a context is clear we may denote ClpXq as X. If X  X we say that X is closed (with respect to the closure operator). Notice that, because of extensivity, X is closed iff X ¤ X. Example 0.2.18. An example of closure operator is mapping a subset of a topological space S to its topological closure. 1. The closure of s0, 1r on R is r0, 1s. 2. Consider R2 with its usual metric. For each r ¡ 0 the closure of tpx, yq P R2 : x2   y2 ru is tpx, yq P R2 : x2   y2 ¤ ru . Example 0.2.19. Let us consider a classical deductive system xF , |ùy, in which F is a set of formulas20 and, given two formulas φ, ψ P F , let φ |ù ψ mean that ψ is deductible from φ. If Γ,∆  F , we write Γ |ù ∆ meaning that, assuming the formulas in Γ we derive each formula in ∆. Then xF , |ùy is a poset in which the deductive closure of a set of formulas Γ is the set of all the theorems deductible from it. 19 By extending such results from xPpY q,y, or equivalently from @ 2Y ,¤2Y D , to @ LY ,¤LY D . 20 Think of a formula as a truth statement. For example, "2   2  4," or "3 ¤ 5," or "under normal temperature and pressure conditions water is liquid." Chapter 0. Order Theory and Lattices 35 As an example, the deductive closure of the set of logical axioms of the Statement Calculus21 (SC) is the set of theorems of SC. Theorem 0.2.20. Let Y be a nonempty set, L a complete lattice and LY be ordered as in Proposition 0.2.16. The family C pLY q of closed elements of LY (according to any given closure operator) is a complete lattice. Given a collection xAjyjPJ of elements of C pLY q, we have bxAjyjPJ  © jPJ Aj , (0.2.21) `xAjyjPJ  a jPJ Aj , (0.2.22) where © , a are the meet and join of LY , and b,` are the meet and join of C pLY q. Proof. Let C  xAjyjPJ be any nonempty collection of closed elements of LY , and let B  © jPJ Aj , C  a jPJ Aj . Then monotonicity yields22 B ¤ Aj  Aj for each j P J , and so B is a lower bound of C. Thus, B ¤ B , i.e., B P C pLY q. Now, since B  © jPJ Aj P C pLY q , we conclude that B  bxAjyjPJ . Conversely, for each j P J we have Aj ¤ a jPJ Aj  C. Now, if D P C pLY q is an upper bound of C on C pLY q, then by monotonicity of the closure we have C ¤ D  D , 21 The Statement Calculus is a formal deductive system which considers formulas as constants and uses the logical connectives ,ÝÑ (logical negation and implication, respectively), from which connectives of conjunction ("and") and disjunction ("or") are defined. We shall explore a little of the Statement Calculus on Sec. 2.3. The interested reader is referred to (MENDELSON, 2009), (MARGARIS, 1990). 22 Recall that Xj is closed. Chapter 0. Order Theory and Lattices 36 hence C  `xAjyjPJ . Example 0.2.23. In Example 0.2.19 we stated that the topological closure on a topological space is an instance of a closure operator, as well as the deductive closure on a (classical) deductive system. Therefore, it follows from Theorem 0.2.20 that both the set of closed sets on a topological space and the set of deductive closures on a (classical) deductive system constitute complete lattices. Just before following to the next chapter, we state the following definition, in accordance with (FALMAGNE; DOIGNON, 2011) . Definition 0.2.24. Let xP,¤y, xQ, y be ordered sets. A Galois connection between xP,¤y and xQ, y is a pair pf, gq of functions f : P Ñ Q, g : Q Ñ P such that, for all p, p1, p2 P P and for all q, q1, q2 P Q, 1. p1 ¤ p2 implies fpp2q   fpp1q 1'. q1   q2 implies gpq2q ¤ gpq1q 2. p ¤ pg  fqppq 2'. q   pf  gqpqq Clearly pf, gq is a Galois connection between xP,¤y and xQ, y iff pg, fq is a Galois connection between xQ, y and xP,¤y. It can be proved23 that f  g, g  f are closure operators on Q and P respectively. In fact, in the remainder of this text the most important24 closure operators we shall work with are Galois connections. Now the reader has all the tools necessary to understand the ideas concerning orders and lattices on this text. We are ready to introduce the classical theory of Formal Concept Analysis. 23 See (FALMAGNE; DOIGNON, 2011) . 24 But we shall not work exclusively with them. In fact, in Sec. 4.1 a computational algorithm is presented which applies to general closure operators on finite sets. 37 1 Formal Concept Analysis According to (JOSEPH, 2002), a concept is an abstraction produced by the intellect (e.g. the concept of tree abstracts what is common to several trees one has contact with). A symbol is a sign (e.g., a sound, a sequence of letters, an image) with a meaning imposed to it by convention (e.g., an image of a skull which stands for danger, or the sequence of characters T-R-E-E which stands for the notion of tree). A term is a concept communicated by a symbol (e.g., "tree" is a term which conveys the idea of the concept tree). Thus, she distinguishes concepts and terms by stating that a concept is an idea (in the mind) which represents a reality, whereas a term is that idea in transit, being communicated. Also according to JOSEPH, a term has both extension and intension, the former corresponding to the set of all objects to which the term corresponds (e.g., the set of all trees), and the later, to "the sum of the essential characteristics that the term implies." Moreover, "As a term increases in intension, it decreases in extension." Formal Concept Analysis (FCA) is to be regarded as a mathematization of the philosophical idea of concept (GANTER; WILLE, 1999) , based on the slightly different idea that it is the concept that has extension and intension. 1.1 Definitions and properties The definitions and theorems presented in this section follow those presented by (GANTER; WILLE, 1999) with a different notation and slightly different proofs. Definition 1.1.1. A formal context is an ordered triple C : xO,A, Iy, in which O and A are non-empty sets, and I  O A is a binary relation. The elements of O are called objects, and the elements of A attributes. We say that, in the context C, an object o has an attribute a if and only if po, aq P I. We shall write oIa for po, aq P I. Example 1.1.2. One may represent a finite formal context in an easy manner by using a table in which rows are indexed by objects and columns by attributes. A cell in row o and column a is marked if and only if oIa. As an example, Table 1 represents a formal context of animals. Notice that in this example there is no distinction between swans and geese. We should add more attributes if we wanted to distinguish between them. On the other hand, because in this context the attribute for having wings gives us no new information Chapter 1. Formal Concept Analysis 38 O A Vertebrate Lay eggs Carnivorous Has wings Flies Quadruped Crawls Eagle X X X X X Snake X X X X Goose X X X X Swan X X X X Lion X X X Table 1 – A formal context of animals (every winged animal in this context flies) we could rule this attribute out. However it would be an important attribute if we had for instance the object "Chicken." This example illustrates the fact that formal contexts are often narrower than the real world. Nonetheless, it is a way of representing information that can be made very powerful as we shall see further ahead. Definition 1.1.3. Let C  xO,A, Iy be a formal context. We define two maps,  : 2O Ñ 2A , ^ : 2A Ñ 2O , given for each O  O and A  A by the relations O : ta P A : oIa for all o P Ou (1.1.4) Â : to P O : oIa for all a P Au , (1.1.5) where write O for pOq and Â for ^pAq. According to this definition, O is the set of all attributes common to every object in O, and Â is the set of all objects having every attribute in A. We can now define the central object of FCA: Definition 1.1.6. Let C  xO,A, Iy be a formal context. A formal concept (often referred to as concept) of C is an ordered pair C  xO,Ay such that O  O, A  A, O  A and Â  O. The sets O and A are called the extent and intent of the concept C respectively. In other words, C  xO,Ay is a concept if the following conditions hold: 1. A is precisely the set of all attributes common to every object of O; and 2. no object of OzO has every attribute of A. Example 1.1.7. The following are formal concepts of the context presented in Table 1: CLion  xtLionu, tQuadruped, Carnivorous, Vertebrateuy , (1.1.8) CCarnivorous  xtEagle, Snake, Lionu, tCarnivorous, Vertebrateuy . (1.1.9) Chapter 1. Formal Concept Analysis 39 Notice that there is an inverse relation between the numbers of elements in the intent and the extent of a concept. If we increase the number of elements in the extent (we added "Eagle" and "Snake" to it), the number of elements in the intent is reduced (eagles are not quadrupeds and neither are snakes)1. In fact, the following useful properties hold: Theorem 1.1.10. Let O,O1, O2  O and A,A1, A2  A. Then 1. If O1  O2 then O2  O1 1'. If A1  A2 then Â2  Â1 2. O  O^ 2'. A  Â 3. O  O^ 3'. Â  Â^ 4. O  Â iff A  O iff O  A  I Proof. We shall prove items 1., 2., 3. and 4. Items with a prime can be proved analogously. 1. Let a P O2 . Then oIa for all o P O2. In particular, oIa for all o P O1. Thus, a P O1 . 2. Let o P O. Then oIa for all a P O, by definition of O. Thus, by definition of O^, we have o P O^. 3. From item 2'. with A  O we already know that O  O^. Let a P O^. Then piq oIa for all o P O^, by definition of O^. Now let õ P O^ be fixed. For all ã P A, piiq if õIã then ã P O, by definition of O^. From piq we have õIa. Thus, using piiq we conclude that a P O. 4. Suppose O  Â. By item 1., Â  O. Using item 2'. and transitivity of , we have A  O. Now suppose A  O. By items 1'. and 2., we have O  O^  Â. Assuming A  O, let o P O and a P A. By definition of O, oIã for all ã P O. By hypothesis, a P A  O. Thus, oIa. Since o P O and a P A are arbitrary, O  A  I. Hence, A  O ñ O  A  I. Finally, suppose O A  I. Let o P O. By hypothesis, for all a P A we have oIa. By definition of O, if oIa then a P O. Thus if a P A then a P O. This completes the proof. Items 1., 1'., 2., 2'. of Theorem 1.1.10 yield the following. 1 Notice how this, together with items 2., 2'. of Theorem 1.1.10, resembles JOSEPH's statement that "As a term increases in intension, it decreases in extension" (2002). Chapter 1. Formal Concept Analysis 40 Corollary 1.1.11. Let C  xO,A, Iy be a formal context. Then p,^ q is a Galois connection2 between xO,y and xA,y, hence ^ and ^ are both closure operators3. From properties 3. and 3'. of Theorem 1.1.10 we see that, given O  O and A  A, xO^, Oy and xÂ, Ây are concepts. On the other hand if C  xO,Ay is a concept then by definition C  xO,Oy  xÂ, Ay . This proves the following. Lemma 1.1.12. Let C  xO,A, Iy be a formal context. Then C is a formal concept of C iff there exist O P O and A P A such that C  xO,Oy  xÂ, Ay  xO,Ay . In particular, for all O  O, A  A, the following are formal concepts: xO^, Oy , xÂ, Ây . Lemma 1.1.12 gives us a procedure for finding concepts4. For example, if we want to find the concept of "Carnivorous" we presented in Example 1.1.7 - that is, the concept with the smallest intent such that "Carnivorous" is an attribute -, start with the set tCarnivorousu and then apply respectively the maps ^ and  to it: tCarnivorousû  tEagle, Snake,Lionu  tVertebrate,Carnivorousu (1.1.13) From (1.1.13), using Theorem 1.1.10 we have the concept CCarnivorous  xtEagle, Snake,Lionu, tVertebrate,Carnivorousuy (1.1.14) as presented in (1.1.9). Now in Table 1, consider the concept of eagle: CEagle  xtEagleu, tFlies, Has Wings, Lay eggs, Carnivorous, Vertebrateuy . (1.1.15) Comparing (1.1.15) and (1.1.14), we see that as we intersect the intents of the concepts of lion and eagle we get the intent of another concept: that of carnivorous. In fact, an arbitrary intersection of intents is an intent which is, of course, smaller than the original one. This suggests the existence of an order between intents - and, in fact, between attributes. 2 See Def. 0.2.24. 3 See Def. 0.2.17. 4 Computational algorithms for this process shall be explored in Sec. 1.2. Chapter 1. Formal Concept Analysis 41 Definition 1.1.16. Let C  xO,A, Iy be a formal context, and denote by BpCq the set of all formal concepts on C. Define an order ¤C on BpCq for each xO1, A1y, xO2, A2y by xO1, A1y ¤C xO2, A2y iff O1  O2 , (1.1.17) iff A2  A1 . Given two concepts C1, C2 P BpCq, we say that C1 is a subconcept of C2 (or that C2 is a superconcept of C1) iff C1 ¤C C2. Interestingly, xBpCq,¤Cy is not simply a poset. It can be shown that, with the order introduced in Def. 1.1.16, the set of all formal concepts of any context constitutes a complete lattice. Theorem 1.1.18 (The Basic Theorem on Concept Lattices). Let C be a formal context. Then LC : xBpCq,¤Cy is a complete lattice, called the concept lattice of C. If K is an index set and Cκ  xOκ, Aκy P BpCq for each κ P K then inf κPK Cκ  C£ κPK Oκ, ¤ κPK Aκ ^G , (1.1.19) sup κPK Cκ  C¤ κPK Oκ ^ , £ κPK Aκ G . (1.1.20) Furthermore, a complete lattice xL,¤Ly is isomorphic to BpCq iff there are mappings ω : O Ñ L and α : A Ñ L such that ωpOq is supremum-dense in L, αpAq is infimum-dense in L and, for all o P O, a P A, oIa iff ωpoq ¤ αpaq . Proof. We prove that xBpCq,¤Cy is a complete lattice, and that (1.1.19) and (1.1.20) hold. The remainder of the proof - i.e., the part concerning isomorphisms and the maps ω, α - is left undone in this text, and the proof can be found in (GANTER; WILLE, 1999) . Let us consider the concepts Cκ indexed by K. As we have already noticed in Cor. 1.1.11, the maps ^ and ^ are closure operators on O,A respectively, whose closed elements are respectively extents and intents. Thus, Theorem 0.2.20 implies5 that the collections C pOq,C pAq of closed subsets of O,A with respect to ^, ^ are complete lattices6 and, in particular, © κPK Oκ  £ κPK Oκ , a κPK Oκ  ¤ κPK Oκ ^ , pon xC pOq,yq © κPK Aκ  £ κPK Aκ , a κPK Aκ  ¤ κPK Aκ ^ . pon xC pAq,yq 5 Here L  t0, 1u, hence by isomorphism Theorem 0.2.20 applies to Y  O,A ordered by set inclusion. 6 Ordered by , according to Theorem 1.1.10, with infimum £ and supremum ¤ . Chapter 1. Formal Concept Analysis 42 C1 a C2 b C3 c C4 d,e 3,4 C5 C6 1 C7 g 2 C8 f 5 C9 Figure 2 – Concept Lattice of Animals The equations in the first row imply7 that BpCq is a complete lattice, as well as the formulae for extents of inf Cκ and supCκ. Now, the restriction of  to C pOq establishes an isomorphism between xC pOq,y and xC pAq,y8, so that the second row of equations above impose respectively formulae for supremum and infimum of intents on xBpCq,¥y, so that by duality we get the intents of (1.1.19) and (1.1.20). Theorem 1.1.18 allows us to use lattice theory for finding out many properties that come from a formal context. In particular, a finite concept lattice has an easy visual representation (see Example 1.1.21 below). In order to interpret the concept lattice from the diagram, one may write, for each concept on the diagram, the elements of its intent and extent. However, from the order ¤ of the concept lattice a tidier manner of presenting the diagram can be devised: for a given concept, instead of writing every element of its extent (resp. intent), we write only those objects (resp. attributes) that did not appear below (resp. above) in the concept lattice. This is possible because of the Basic Theorem. Example 1.1.21. The concept lattice of the context presented in Table 1 is shown in Fig. 2. Each concept is represented by a circle. Here animals are represented by numbers 1-5 in the order they appear in Table 1. Attributes are represented by letters a-g, also in the order they appear in the table. The extent (intent) of a given concept C has an object (attribute) iff that object (attribute) appears near a concept C such that there is a descending (ascending) path from C to C. 7 By definition of the order on BpCq. 8 By Lemma 1.1.12, elements of C pOq have the form Â, and elements of C pAq have the form O. To see that  is one-to-one, let A1, A2 be such that A^1  A^2 . Then A^1  A^2 , whence Â1  Â2 by Theorem 1.1.10 item 3'. Also,  is onto, because given A P C pAq, A  O for some O P O, whence A  pO^q by Theorem 1.1.10 item 3. Finally,  is order-preserving by item 1. of the same theorem. Chapter 1. Formal Concept Analysis 43 Take, for instance, the concept C3. It has ascending paths to concepts C1 and C3, and descending paths to concepts C8, C5, C6, C7 and C9. On the other hand, C3 has no (strictly ascending or descending) paths to C2 or C4. Thus, C3  xt1, 2, 5u, ta, cuy. Concept lattices are visual tools that allow us to find out relations on attributes and objects (for example, any animal with property e also has property b). However, limitations may arise on the theory, as, for example, the definition of formal context allows us only to work only with precise relations. For example, a chicken can fly for short distances, but this information could not be expressed on a formal context as defined earlier. In the next section we generalize these ideas to allow fuzzy objects, attributes and relations. 1.2 Computing the concept lattice For purposes of computing the concept lattice of a given formal context C  xO,A, Iy, let us consider that O,A are finite. Recall that according to Lemma 1.1.12, given any A  A the set Â is the intent of a concept9, and from an intent the extent can be uniquely determined10. Thus, we shall consider that, in order to compute the formal concepts of C, it is sufficient to evaluate closures of sets of attributes. Thus, we shall write A for Â. The idea of Lemma 1.1.12 that every intent is the closure of a set of attributes provides us with an algorithm for computing all the intents of C and thus, all of its concepts. Algorithm 1 – Standard Computation of Intents input :Formal context C. output :The set Intents of all intents of C. 1 IntentsÐH; 2 foreach A P PpAq do 3 if A R Intents then 4 append A to Intents; 5 end 6 end 9 Analogously, given O  O, the set O^ is the extent of a concept. In fact, all the ideas presented in this section for subsets of A and intents are true (with analogous proofs) if we consider subsets of O and extents instead, and then interchange the operators ^ and . 10 In reality we have more than that: in the process of computing Â we do compute Â which can be stored in the process. Nonetheless in the algorithms presented in this section, in order to find all the concepts we only need Â to compute Â (and, of course, we store Â), and so we shall consider that only Â needs to be computed. Chapter 1. Formal Concept Analysis 44 For running the loop started in row 2 we assume a linear order on PpAq. Algorithm 1 has a critical computational issue: it has exponential computational complexity. More precisely, if A has n elements this algorithm has to consider 2n subsets A of A and evaluate A for each of them11. We are provided in (GANTER; WILLE, 1999) with an algorithm which is not necessarily of exponential complexity. This algorithm is based on the idea of finding a lectic order on PpAq which is a linear strict order12 on this set, and then restricting this order to a linear strict order on the set BpCq of formal concepts of C. In what follows, we assume that A  t1, 2, ..., nu is linearly ordered by the usual order of natural numbers13. Definition 1.2.1. Let A,B  A. Then A is said lectically smaller than B, and this is denoted as A B, iff There exists i P BzA such that AX t1, ..., i 1u  B X t1, ..., i 1u , where t1, ..., 0u : H . It can be shown that the lectic order is a linear strict order on PpAq. Example 1.2.2. The lectic order is not related with the number of members of a set, but with the smallest element distinguishing them. 1. If A  B, then A B. In fact, let i be the smallest element of BzA. Then AX t1, ..., i 1u  B X t1, ..., i 1u . In particular, if H  A  A then H A A. 2. If A  t1, 2u and B  t3u, then B A. In fact, the smallest element i  1 of A is an element of AzB, and B X t1, ..., i 1u  B XH  H  AXH  AX t1, ..., i 1u . Due to monotonicity of the closure operator, from item 1. of Example 1.2.2 we see that H is the lectically smallest intent of C. If a procedure can be found so that, given an intent, the next intent according to the lectic order can be found, we shall have all that is necessary for finding all the intents of C up to A. 11 We can save several computations here if instead of evaluating Â in row 3 we verified in that row whether A has already been computed and only then (i. e., only if we have a newly found extent) we compute Â and append it to the set of intents. Nonetheless, the number of evaluations of A would remain 2n. 12 See Prop. 0.1.5. 13 If A  ta1, a2, ...anu, we only need to order A according to the indices of its elements. Chapter 1. Formal Concept Analysis 45 Define, for each A  A and each i P A the following: À i : pAX t1, ..., i 1uq Y tiu . Because À i is the closure of a set of objects it is an intent. Furthermore, if i R A, then AX t1, ..., i 1u  pAX t1, ..., i 1uq Y tiu , and so A pAX t1, ..., i 1uq Y tiu ¤ À i , implying that A À i. Is it possible to determine an i P A so that À i is the smallest intent greater than A? Ganter and Wille define in (GANTER; WILLE, 1999) a binary relation i by A i B iff i P BzA and AX t1, ..., i 1u  B X t1, ..., i 1u . Notice that A B iff there exists a unique i P A such that A i B. Then, the authors present us with the following. Theorem 1.2.3. The smallest intent greater than A  A with respect to the lectic order is À i , where i is the greatest element of A such that A i À i. The result is the following algorithm. Algorithm 2 – Next Intent input :Formal context C. output :The set Intents of all intents of C. 1 B = H; 2 append B to Intents; 3 while B  A do 4 A = B; 5 for i = MaxpAzAq:1: 1 do 6 B = À i; 7 if A i B then Break; 8 end 9 append B to Intents; 10 end The notation in row 5 means that the variable i assumes values from maxpAzAq down to 1 decreasing 1 unit each loop. Algorithm 2 tends to be much faster than the Chapter 1. Formal Concept Analysis 46 0 10 20 30 40 50 Number of Concepts (×103) 0 25 50 75 100 125 150 175 200 Ap pl ica tio n of th e Cl os ur e Op er at or (× 10 4 ) Upper Neighbour Next Intent Figure 3 – Complexity of the Algorithms standard algorithm (Alg. 1) because the number of applications of the closure operator is in general not much larger than the number of concepts. For some computational experiments, see Fig. 3. Example 1.2.4. The complexity of the algorithm depends on the order of the rows of the formal context. In fact, consider the following formal context: O A 1 2 3 a X X b X c X By applying Alg. 2, we have the following sequence of events. Step Evaluation Comparison Action 1 H  H Store H 2 H ` 3  t2, 3u H ¢3 t2, 3u Try ` 2 3 H ` 2  t2u H 2 t2u Store t2u 4 t2u ` 3  t2, 3u t2u 3 t2, 3u Store t2, 3u 5 t2, 3u ` 1  t1u t2, 3u 1 t1u Store t1u 6 t1u ` 3  t1, 2, 3u t1u ¢3 t1, 2, 3u Try ` 2 7 t1u ` 2  t1, 2, 3u t1u 2 t1, 2, 3u Store t1, 2, 3u If we interchange columns 1 and 3 we get the following context. Chapter 1. Formal Concept Analysis 47 O A 11 21 31 a X X b X c X This gives us the following sequence of steps. Step Evaluation Comparison Action 1 H  H Store H 2 H ` 31  t31u H 31 t31u Store t31u 3 t31u ` 21  t21u t31u 21 t21u Store t21u 4 t21u ` 31  t11, 21, 31u t21u ¢31 t11, 21, 31u Try ` 11 5 t21u ` 11  t11, 21u t21u 11 t11, 21u Store t11, 21u 6 t11, 21u ` 31  t11, 21, 31u t11, 21u 31 t11, 21, 31u Store t11, 21, 31u Now if we recall that 11  3 , 21  2 , 31  1 , we see that we get the same results, but with in different sequences and in different numbers of steps. Nonetheless, both executions apply the closure operator a smaller number of times than the standard algorithm, for which necessarily 23  8 applications are made. Algorithm 2 presents itself as a (comparably) fast algorithm for finding every formal concept of a given formal concept. However, it lacks a procedure for evaluating the structure of the concept lattice. After computing the concepts, another algorithm must be used for computing their lattice structure. LINDIG proposed (2000) another algorithm for computing the concepts together with its lattice structure. LINDIG's Upper Neighbour algorithm shall not be presented here. Rather, we shall present an extended version of it later in this text14. We do nonetheless present the basic idea of the algorithm. Recall that according to Theorem 0.2.20, given a closure operator, the collection of closed sets with respect to it constitutes a complete lattice. In particular, the operator ^ is a closure operator, and so the set of intents constitutes a complete lattice on A, which is isomorphic to xBpCq,¥y15. LINDIG's Upper Neighbour16 algorithm is based on the idea of first evaluating the smallest element of this lattice, which is the closure of the emptyset, and then recursively, 14 See Ch. 4. 15 That is the dual of the concept lattice. See (1.1.17). 16 See Def. 0.2.5 Chapter 1. Formal Concept Analysis 48 for each closed set A, to compute the set of upper neighbours of A (and dually attributing A as a lower neighbour of each if its upper neighbours). Figure 3 presents the number of applications of the closure operator of Alg. 2 ("Next Intent") and LINDIG's "Upper Neighbour" algorithm in terms of the number of concepts of the context. Our computational experiments were made with randomly generated m n formal contexts, with m  5, 10, ..., 40 and n  5, 10, ..., 50. Our goal in the remainder of this text is to present the reader with an extension of the theory of Formal Concept Analysis which allows us to work with graded values of relationships between objects and attributes, i. e., we intend to allow other possibilities than "object o has attribute a," by replacing this statement with (possibly) more flexible statements of the form "object o has attribute a to a certain degree d". This goal may be achieved by replacing the classical notion of set with a more flexible, fuzzy notion. 49 2 Fuzzy Sets and Fuzzy Connectives Fuzzy sets extend classical set theory in order to deal with uncertainty. Whereas classical mathematics works on the assumption that we have absolute precision concerning data (we know whether or not an element belongs to a given set, or we know exactly how a function behaves within some neighbourhood of a point), fuzzy logic deals with imprecisions. There are of course social conventions which behave classically, such as the age from which someone is allowed to drive, or to drink alcoholic beverages. Nonetheless, considerations on how such legislations vary across different countries indicates that these conventions do not correspond precisely to reality, i.e., in reality people are sufficiently mature to drink or drive at different ages, but because maturity is a subjective matter and the legislation should (in theory at least) treat individuals objectively, it establishes a number with some degree of arbitrariness1. In the present chapter we present the required theory of fuzzy sets and fuzzy connectives as it shall be necessary in the following chapters. 2.1 Fuzzy sets In this section we define what we mean by a fuzzy set. In order to avoid misleading the reader with seemingly nonsensical definitions, we start by considering the classical theory of sets. Then, by extending the classical definitions, we shall define fuzzy sets. We hope that our approach will help the reader to build an intuition on fuzzy sets and, at the same time, to understand what the definitions mean. Nonetheless, we shall not present here a detailed discussion on axiomatic set theory. We refer the reader interested in the axiomatic approach to (HRBÁČEK; JECH, 1999). Classical sets are usually conceived, at least intuitively, as collections of objects. We may speak of the collection of glasses in the cupboard, or the cutlery in the drawer. In both situations we know implicitly that we are talking about objects in a kitchen. In some cases however, ambiguity may arise unless we state explicitly what set bounds our speech. For example, depending on the context the term number may be interpreted as a natural number (with or without zero), an integer or a real number. Hence it is often a good starting place to state precisely the set U of all the objects of which we are speaking, 1 As an example, the minimum legal drinking age in Austria is 16 years old, whereas in the USA it is 21 years old. For information on countries around the globe, see <https://www.who.int/substance_ abuse/publications/global_alcohol_report/profiles/en/> Chapter 2. Fuzzy Sets and Fuzzy Connectives 50 which is often called the universe of discourse2. Recall that subsets of U may be represented by their characteristic functions, i.e., for each A P PpUq we have χ A P 2Y given by χ A pxq  $& %1, if x P A0, if x R A . Fuzzy sets are defined by extending 2  t0, 1u to I  r0, 1s  R, thus enabling us to work with uncertain degrees of membership, i.e., rather than stating something or its negation (1 and 0, respectively), we are allowed to say something in between. Definition 2.1.1. Let U be a (classical) set. A fuzzy subset F of U is a function3 μ F : U Ñ I . The set of all fuzzy subsets of U is denoted by IU . Henceforth we shall frequently refer to fuzzy subsets as fuzzy sets, as long as there is no risk of confusion concerning the set U . It is a common practice to state that F is characterized by the function μ F , which is called the membership function of F . In order to unify this practice and our definition, henceforth we shall consider a fuzzy set and its membership function to be the same entity, and at times we may say that a fuzzy set is characterized by its membership function. We may also denote the set of fuzzy subsets of U by FpUq4. A fuzzy set μ such that μpuq P t0, 1u for every u P U is called a crisp set. If the set U  tu1, ..., unu is finite and A is a fuzzy subset of U we may write A  μ A pu1q u1   ...  μ A punq un . Example 2.1.2. What does it mean to be young? Individuals a and b, aged 20 and 80 respectively, may have very different opinions about individual c, aged 40, being young. 2 By speaking about a "universe of discourse" we do not state that there is a "universal set", i. e., a set which contains all sets as elements of itself as one such set would be subject to Russell's paradox (HRBÁČEK; JECH, 1999) . On the contrary, the universe of discourse is a set conceived in order to avoid paradoxes. 3 Some clarification may be useful here. Many authors (such as (ZADEH, 1965) and (BARROS; BASSANEZI; LODWICK, 2017)) define F as being characterized by a function μ F . We do understand that this is a good idea in terms of intuition, and we shall use this idea later for practical purposes. Nonetheless, this idea presents a philosophical difficulty: if F is characterized by μ F , then what is F itself? We then decide to define a fuzzy set to be the function μ F . Additionally, this definition is in accordance with the definition of L-fuzzy sets stated later in this chapter. 4 This is merely a notation. Although for classical sets there is a distinction between 2U and PpUq and they are related by a bijection, in the fuzzy case IU and FpUq are one and the same set. Chapter 2. Fuzzy Sets and Fuzzy Connectives 51 Nonetheless they are all likely to agree that Mr. a is young, whereas Mr. b is not. Having this in mind, we may define μ Y , the membership to the set of young people, as follows5: Let U  tn P N : n ¤ 120u. Define6 μ Y pxq  $'''& '''% 1, if 0 ¤ x ¤ 20 60 x 40 , if 20 x ¤ 60 0, if 60 x ¤ 120 . 0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4 – Membership to the Fuzzy Set of Young People Given two classical sets A,B  U , their intersection (AXB) and union (AYB) are expressed by the following characteristic functions: χ AXB pxq  $& %1, if x P A and x P B0, if x R A or x R B (or both) , χ AYB pxq  $& %0, if x R A and x R B1, if x P A or x P B (or both) . The complement of A (that is, the set UzA) denoted by A1, has the following characteristic function: 5 We are aware that age alone is no sufficient criteria for defining someone as young or old. As sang by the children in the Mexican TV show El Chavo del Ocho, "There are young people in their eighties and there are old people who are 16 years old." Nonetheless we do choose to work only with age for simplicity. 6 Notice that the points at x  20 and x  60 give us constraints. If we want to make the membership function (when extended to r0, 120s) of class Cn, each of these points give us two additional constraints, meaning that it could be achieved by fitting a 2n 1 degree polynomial to the 2n constraints. However, high degree polynomials constrained to a limited interval may become wavy. If we used a function of the form 1{1 epr sxq we would have a smooth function that approaches but never reaches 0 nor 1. Chapter 2. Fuzzy Sets and Fuzzy Connectives 52 χ A1 pxq  $& %0, if x P A1, if x R A . Notice that for any x P U we have the following7: χ AXB pxq  χ A pxq ^ χ B pxq , χ AYB pxq  χ A pxq _ χ B pxq , χ A1 pxq  1 χ A pxq . Now one sees that extending the classical definitions is an easy matter. The definitions presented below are called the standard fuzzy set operations. Definition 2.1.3. Let A,B be fuzzy subsets of U . Then the fuzzy subsets A X B and AYB, called the intersection and union of A and B respectively, and the set A1, called the complement of A, have the following membership functions: μ AXB pxq  μ A pxq ^ μ B pxq , μ AYB pxq  μ A pxq _ μ B pxq , μ A1 pxq  1 μ A pxq . When working with fuzzy sets, properties distinct from those of classical sets often arise. Example 2.1.4. From the set Y of young people defined in Example 2.1.2, we shall define the set O of old people as the complement Y 1 of Y (Fig. 5a), that is: μ O pxq  1 μ Y pxq  $'''& '''% 0, if 0 ¤ x ¤ 20 x 20 40 , if 20 x ¤ 60 1, if 60 x ¤ 120 Thus, we get the following results for Y YO and Y XO (Fig. 5b): μ YYO pxq  $''''''& ''''''% 1, if 0 ¤ x ¤ 20 60 x 40 , if 20 x ¤ 40 x 20 40 , if 40 x ¤ 60 1, if 60 x ¤ 120 μ YXO pxq  $''''''& ''''''% 1, if 0 ¤ x ¤ 20 60 x 40 , if 20 x ¤ 40 x 20 40 , if 40 x ¤ 60 1, if 60 x ¤ 120 7 Recall that ^ and _ are the meet (infimum) and join (supremum), defined in Sec. 0.2, on the lattice 2  t0, 1u with the usual order of natural numbers. Chapter 2. Fuzzy Sets and Fuzzy Connectives 53 0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 0.8 1.0 (a) Fuzzy Set O of Old People ■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ 0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 0.8 1.0 (b) Y YO (top) and Y XO (bottom) Figure 5 – Operations with Fuzzy Subsets Notice that differently from the classical case, the union and intersection of a fuzzy set with its complement may not be the full set U or the empty set. For example, μ YYO p40q  0.5  μ YXO p40q. As we have seen in Sec. 0.28, given A,B  U , A  B iff χ A ¤ χ B , where χ A ¤ χ B expresses χ A puq ¤ χ B puq for all u P U . We extend this criteria of subsetness to fuzzy sets. Definition 2.1.5. Let A,B be fuzzy sets of U . We say that A is a fuzzy subset of B, denoted by A  B, iff μ A ¤ μ B , that is, iff for all x P U , μ A pxq ¤ μ B pxq9. Considering that we want to have, for every fuzzy subset A of U , the relations H  A  U , it follows that the membership functions of U and H are pointwise given by μ U pxq  1 and μ H pxq  0, respectively. 2.2 Fuzzy relations Recall that a classical n-place relation on a product V1  ... Vn is defined as a set R  V1  ...Vn (Def. 0.1.1). The notion of relation is present in the definition of a formal context (Def. 1.1.1), where I is a binary relation on AO. If we are to present a fuzzy theory of FCA, then it is clear that we shall need fuzzy relations. 8 See Prop. 0.2.16. 9 Recall from Prop. 0.2.16 that with this order @ IU ,¤ D is a complete lattice in which, for every (classical) subset M of IU we havea μPM μ pyq  a μPM μpyq , © μPM μ pyq  © μPM μpyq , where the right-hand side of each equation corresponds to inf, sup in r0, 1s. Chapter 2. Fuzzy Sets and Fuzzy Connectives 54 Definition 2.2.1. Let U1, ..., Un be (classical) sets. A n-place fuzzy relation (or simply fuzzy relation) on U  U1  ... Un is a fuzzy subset of U . Example 2.2.2. Consider the idea of people belonging to different generations. It is no problem to think that a parent and a child belong to different generations, but we may say that in a sense two siblings born 15 years apart do not belong to the same generation. Their ages are (almost certainly) closer than a parent and a child's. Still, they are further apart than cousins born only 2 or 3 years apart One way to describe different generations is as follows. Let A be a set of people and f : AÑ Y be a function that maps a person a to the year fpaq when a was born. Define the fuzzy subset G of A A by μ G pa1, a2q  a" 0, 1 |fpa1q  fpa2q|25 * . Then G is a fuzzy (binary) relation which accounts for membership of two people in A to the same generation. Here we consider that two people are considered to belong to the same generation if they are born in the same year, whereas they are of different generations if they are born 25 (or more) years apart Recall that we defined an order (Def. 0.1.3) as a reflexive, transitive, antisymmetric relation. These properties (as well as other classical properties) can be extended to fuzzy sets. Definition 2.2.3. Let U be a (classical) set and A be a binary fuzzy relation on U , i.e., a fuzzy subset of U  U . We say that A is: 1. reflexive if μ A px, xq  1 for all x P U ; 2. symmetric if μ A px, yq  μ A py, xq for all x, y P U ; 3. transitive if rμ A px, yq ^ μ A py, zqs ¤ μ A px, zq for all x, y, z P U ; 4. antisymmetric if 0 μ A px, yq and 0 μ A py, xq imply x  y for all x, y P U . Example 2.2.4. Consider the fuzzy relation G of generations in Example 2.2.2. G is reflexive and symmetric. It is neither transitive nor antisymmetric. Proof. 1. (Reflexivity) Every person a is born in the same year as itself, and so μ G pa, aq  a" 0, 1 |fpaq  fpaq|25 *  a t0, 1 0u  a t0, 1u  1 . 2. (Symmetry) For all a1, a2 P A we have μ G pa1, a2q  a" 0, 1 |fpa1q  fpa2q|25 *  a" 0, 1 |fpa2q  fpa1q|25 *  μ G pa2, a1q . Chapter 2. Fuzzy Sets and Fuzzy Connectives 55 3. (No transitivity) Let c1, c2 be siblings born in 1990 and 1985 respectively. Let p be c1 and c2's parent, born in 1965. Then μ G pc1, c2q  4 5 μGpc2, pq  1 5 μ G pc1, pq  0 μ G pc1, c2q ^ μGpc2, pq  1 5 μ G pc1, pq  0 rμ G pc1, c2q ^ μGpc2, pqs ¡ μGpc1, pq 4. (No antisymmetry) Consider c1 and c2 of item 3. Clearly only diagonal binary fuzzy relations10 can be simultaneously symmetric and antisymmetric. In fact, if A is both symmetric and antisymmetric, let x, y P U be such that 0 μ A px, yq. By symmetry, 0 μ A py, xq, so that antisymmetry gives x  y. Although a study of fuzzy orders an interesting topic11, for our purposes crisp orders on sets of fuzzy functions suffice. It is worth mentioning that an important family of subsets of a Cartesian product (thus a family of relations) is that of Cartesian products of subsets. For instance, item 4. of Theorem 1.1.10 informs us that, given O  O, A  A, the objects in O share all attributes in A - expressed as O  Â - iff O  A  I. Now, if Si  Vi for each i  1, ..., n, the relation S  S1 ...Sn on V1 ...Vn satisfies, for each n-tuple px1, ..., xnq P V1  ... Vn, the equation χ S px1, ..., xnq  n© i1 χ Si pxiq , which motivates the following definition. Definition 2.2.5. Let U1, ..., Un be (classical) sets and let Ai be a fuzzy subset of Ui for each i  1, ..., n. The fuzzy Cartesian product A  A1 ...An of the Ai is a fuzzy relation with membership function given by: μ A px1, ..., xnq  n© i1 μ Ai pxiq . Example 2.2.6. Let C  tc1, c2, c3u be a set of cars. Let the fuzzy subsets E,F of U , describing respectively expensive cars and fast cars, be given by the following: μ E pc1q  0.1 , μEpc2q  0.4 , μEpc3q  0.9 , μ F pc1q  0.3 , μF pc2q  0.5 , μF pc3q  0.6 . 10 That is, fuzzy relations on U2 such that if x  y then μ A px, yq  0. 11 See (ŠEŠELJA; TEPAVČEVIĆ, 2007) . Chapter 2. Fuzzy Sets and Fuzzy Connectives 56 Then the fuzzy Cartesian product E  F is as follows: μ EF pc1, c1q  0.1 , μEF pc1, c2q  0.1 , μEF pc1, c3q  0.1 , μ EF pc2, c1q  0.3 , μEF pc2, c2q  0.4 , μEF pc2, c3q  0.4 , μ EF pc3, c1q  0.3 , μEF pc3, c2q  0.5 , μEF pc3, c3q  0.6 . This Cartesian product satisfies none of the properties in Def. 2.2.3. 2.3 Extending classical logic This section is divided in three parts. In the first part we introduce the Statement Calculus (SC), which deals with statements (sentences that are either true or false) and connectives that "paste" statements together creating new statements. Then we extend the classical connectives to fuzzy connectives. In the second part we consider what is the meaning of using the universal and existential quantifiers (@ and D, respectively), and then we present their fuzzy correspondents. We refer the interested reader to (MARGARIS, 1990) and (MENDELSON, 2009) . These references further develop SC as well as the so called First Order Predicate Calculus, which builds up on SC and includes the universal and existential quantifiers. For the fuzzy counterparts of the logical connectives the reader is referred to (BARROS; BASSANEZI; LODWICK, 2017) and (KLEMENT et al., 2000) . Finally, we state definitions of continuity and semi-continuity of functions, and provide some basic results concerning continuous and semi-continuous t-norms and implications. These results will be crucial in the next chapter in order to prove theorems concerning an extended version of FCA. 2.3.1 Logical connectives According to MARGARIS, "A statement is a declarative sentence that is either true or false (but not both)" (1990, p. 1). For example, the sentence All swans are white is a statement, but the sentence When did you arrive here? is not. Statements can be connected with one another in ways that produce new statements. Given two statements P and Q, the ways in which statements are connected in classical logic are the following: Chapter 2. Fuzzy Sets and Fuzzy Connectives 57 P Q P ^Q 0 0 0 1 0 0 0 1 0 1 1 1 (a) Conjunction P Q P _Q 0 0 0 1 0 1 0 1 1 1 1 1 (b) Disjunction P Q P ÝÑ Q 1 1 1 1 0 0 0 1 1 0 0 1 (c) Implication P P 0 1 1 0 0 0 (d) Negation Table 2 – Truth Tables of Classical Connectives 1. Conjunction: is the statement "P and Q", which is true iff both P and Q are true. It is denoted by P ^Q. 2. Disjunction: is the statement "P orQ", which is true iff at least one of the statements P and Q is true. It is denoted by P _Q. 3. Implication: is the statement "If P then Q", which is devised to preserve truth: if P is true, so is Q. It is denoted by P ÝÑ Q. And finally, "connecting" one single statement P , we have: 4. Negation: is the statement "Not P ", which is true iff P is false. It is denoted by P . The attentive reader will have noticed that the symbols for conjunction and disjunction, which are usual in formal (mathematical) logic, are the same as those we used earlier for denoting the meet and join, related to lattices. This is no mere coincidence. Once we assume the values 1 for truth and 0 for falsehood, conjunction and disjunction have the truth tables presented in Tables 2a and 2b. The last two rows of Table 2c may seem difficult to understand. A statement of the form P ÝÑ Q with P false is said to be vacuously true12 (MARGARIS, 1990, p. 45). This principle lies behind the mathematical practice of assuming the truth of any statement concerning elements of the empty set: "if a P H then Rpaq for any property R", simply because a P H is always false. These rows entail traditional principle known as ex falso quodlibet (Latin for "anything follows from falsehood") or the principle of explosion: if something is false ( P ), then it implies anything (P ÝÑ Q), i.e., P ÝÑ pP ÝÑ Qq . We now extend these connectives. Triangular norms, which extends conjunction, and triangular conorms, which extends disjunction, are defined in accordance with (KLEMENT et al., 2000) . The examples of triangular norms and triangular conorms presented in this section can be found in the same reference. 12 This idea was already presented in Footnote 7 in Sec. 0.1. Chapter 2. Fuzzy Sets and Fuzzy Connectives 58 (a) Minimum (b) Product (c) Łukasiewicz (d) Drastic product Figure 6 – Triangular norms Definition 2.3.1. A triangular norm (or t-norm) is a map M: r0, 1s2 Ñ r0, 1s satisfying, for all x, y, z P r0, 1s: 1. xM y  yMx (commutativity) 2. xM pyM zq  pxM yqM z (associativity) 3. If y ¤ z then xM y ¤ xM z (monotonicity) 4. xM 1  x (boundary condition) Example 2.3.2. The mapsMM,ML,MP,MD: r0, 1s2 Ñ r0, 1s defined below are t-norms. See Figure 6 for graphical representations of each of them. 1. xMM y  mintx, yu (minimum) 2. xMP y  xy (product) 3. xML y  maxtx  y  1, 0u (Łukasiewicz t-norm) 4. xMD y  $'''& '''% 0, if px, yq P r0, 1r2 x, if y  1 y, if x  1 (drastic product) Every t-norm is defined on the whole boundary of r0, 1s2. In fact, for each x P r0, 1s we have 1Mx  xM 1  x , Chapter 2. Fuzzy Sets and Fuzzy Connectives 59 and 0 ¤ xM 0  0Mx ¤ 0M 1  0, whence 0Mx  0  xM 0 . Furthermore, if x1 ¤ x2 and y1 ¤ y2 then x1 M y1 ¤ x1 M y2  y2 Mx1 ¤ y2 Mx2  x2 M y2 . Definition 2.3.3. Let M1 ,M2 be t-norms. If for all x, y P r0, 1s we have M1¤M2 then we say that M1 is weaker than M2 (or equivalently M2 is stronger than M1), and we write M1¤M2 . If moreover M1M2 , we write M1 M2 . Proposition 2.3.4. The t-norms of Example 2.3.2 have the following order: MD ML MP MM . Moreover, MD and MM are respectively the weakest and strongest t-norms. Proof. Since all t-norms are equal on the boundary of r0, 1s2 and they only assume values on r0, 1s it is immediate that MD is the weakest t-norm. To show that MM is the strongest, consider a t-norm M. Monotonicity yields xM y ¤ xM 1  x xM y ¤ 1M y  y , so that xM y ¤ min tx, yu  xMM y . Thus, MM is the strongest t-norm. Now, for all x, y Ps0, 1r we have xp1 yq p1 yq iff x xy 1 y iff x  y  1 xy and 0 xy. Hence we have xML y xMP y . Therefore, MD ML MP MM . We proceed with the extension of disjunction (maximum). Definition 2.3.5. A triangular conorm (or t-conorm) is a map O : r0, 1s2 Ñ r0, 1s satisfying, for all x, y, z P r0, 1s: Chapter 2. Fuzzy Sets and Fuzzy Connectives 60 (a) Maximum (b) Probabilistic sum (c) Łukasiewicz (d) Drastic sum Figure 7 – Triangular conorms 1. xOy  yOx (commutativity) 2. xOpyOzq  pxOyqOz (associativity) 3. xOy ¤ xOz, if y ¤ z (monotonicity) 4. xO0  x (boundary condition) Example 2.3.6. The maps OM , OL , OP , OD : r0, 1s2 Ñ r0, 1s defined below are t-conorms. See Figure 7 for graphical representations of each of them. 1. xOM y  maxtx, yu (maximum) 2. xOP y  x  y  xy (probabilistic sum) 3. xOL y  mintx  y, 1u (Łukasiewicz t-conorm) 4. xOD y  $'& '% 1, if px, yq Ps0, 1s2 x, if y  0 (drastic sum) y, if x  0 Notice that for each j P tM,P,L,Du, we have xO j y  1  p1 xqM j p1 yq  (2.3.7) for all x, y P r0, 1s. In fact, it can be shown that a map O : r0, 1s2 Ñ r0, 1s is a t-conorm iff there exists a t-norm M such that for all x, y P r0, 1s xOy  1 rp1 xqM p1 yqs . (2.3.8) Chapter 2. Fuzzy Sets and Fuzzy Connectives 61 This property, together with the boundary conditions for t-norms, tells us that every t-conorm assumes the same values on the boundary of r0, 1s. Moreover, if we define order relations for t-conorms as we have defined them for t-norms, that is O1 ¤ O2 iff xO1 y ¤ xO2 y for all x, y P r0, 1s , and if moreover O1  O2 then O1 O2 , the following result holds. Proposition 2.3.9. The t-conorms of Example 2.3.6 have the following order: OM OP OL OD . Moreover, OM and OD are respectively the weakest and strongest t-conorms. Proof. Let O1 , O2 be t-conorms and let M1 ,M2 respectively be their dual t-norms, that satisfy (2.3.8). Suppose that M1¤M2 . Then for all x, y P r0, 1s 1 xO1 y  p1 xqM1 p1 yq ¤ p1 xqM2 p1 yq  1 xO2 y , so that O2 ¤ O1 . The result follows from this, together with Proposition 2.3.4 and (2.3.7). For more details on t-norms and t-conorms, see (KLEMENT et al., 2000) . Definition 2.3.10. A fuzzy negation (or simply negation) is a map ν : r0, 1s Ñ r0, 1s such that for all x, y P r0, 1s: 1. νp0q  1 and νp1q  0 (boundary conditions) 2. νpνpxqq  x (involution) 3. If x ¤ y then νpyq ¤ νpxq (monotonicity) Example 2.3.11. The following maps are negations (see Figure 8): 1. ν1pxq  1 x 2. ν2pxq  1 x 1  x Notice that (2.3.8) expresses that there exists a one-one correspondence between t-norms and t-conorms that satisfy, for all x, y P r0, 1s, the identities ν1 pxM yq  ν1pxqOν1pyq , ν1 pxOyq  ν1pxqM ν1pyq , Chapter 2. Fuzzy Sets and Fuzzy Connectives 62 (a) ν1pxq  1 x (b) ν2pxq  1 x 1  x Figure 8 – Fuzzy negations which extend the De Morgan laws. Finally we define an extended implication. Definition 2.3.12. A fuzzy implication is a map ñ: r0, 1s2 Ñ r0, 1s such that for all x, y, z P r0, 1s: 1. p0 ñ 0q  1, p1 ñ 0q  0, (boundary conditions) p0 ñ 1q  1 and p1 ñ 1q  1 2. If y ¤ x then pxñ zq ¤ py ñ zq (monotonicity in the first component) 3. If x ¤ y then pz ñ xq ¤ pz ñ yq (monotonicity in the second component) In classical logic it is usual to define some connectives in terms of others. For example, the following three relations hold: 1. P ÝÑ Q  p P q _Q 2. P ÝÑ Q  maxtx P t0, 1u : P ^ x ¤ Qu 3. P ÝÑ Q  p P q _ pP ^Qq . To see that equalities of items 1. and 3. hold, see the following truth table. P Q P p P q _Q P ^Q p P q _ pP ^Qq P ÝÑ Q 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 Now let us check that the equality of item 2. holds. If P ¤ Q it is clear that P ^ x ¤ Q for x P t0, 1u. Thus, if Q  1 or if P  0  Q we have max tx P t0, 1u : P ^ x ¤ Qu  1 . Chapter 2. Fuzzy Sets and Fuzzy Connectives 63 On the other hand, if P  1 and Q  0 then P ^0  0 ¤ Q, but P ^1  1 ¦ Q. Thus, max tx P t0, 1u : P ^ x ¤ Qu  0 . Hence, for all P,Q P t0, 1u we have max tx P t0, 1u : P ^ x ¤ Qu  P ÝÑ Q . If on the right side of the first two equalities we replace classical connectives by their corresponding fuzzy connectives, with supremum rather than maximum, we get fuzzy implications. The third of these equalities, however, does not always give rise to a fuzzy implication as the following example shows. Example 2.3.13. Let νpxq  1 x and consider the minimum t-norm ^ and maximum t-conorm _. Let P1  0.9, P2  1 and Q  1. Then νpP1q _ pP1 ^Qq  νp0.9q _ p0.9^ 1q  0.1_ 0.9  0.9 ; νpP2q _ pP2 ^Qq  νp1q _ p1^ 1q  0_ 1  1 . Definition 2.3.14. A S-implication is an implication ñS defined by pxñS yq  νpxqOy , where ν is a fuzzy negation and O is a t-conorm. A R-implication (or residual implication, or simply residuum) is an implication ñR defined by pxñR yq  suptz P r0, 1s : xM z ¤ yu , where M is a t-norm. We say that ñ is the residuum of M. Notice that for the residuum ñR of M, if x ¤ y then xM 1  x ¤ y, and so pxñR yq  1. Example 2.3.15. In the following examples, the first two items correspond to S-implications. The other items are R-implications.Figure 9 illustrates the fact that, increasing y from the diagonal x  y, the value of a residuum always equals 1, but this must not be true for S-implications. Chapter 2. Fuzzy Sets and Fuzzy Connectives 64 1. The map ñkd defined as pxñkd yq  p1 xqOM y is a S-implication, called the Kleene-Dienes implication, with the negation νpxq  1 x and the maximum t-conorm. 2. If we consider the negation ν2 of Example 2.3.11 and the Łukasiewicz t-conorm, we define the S-implication pxñS yq   1 x 1  x   y

^ 1 3. The map ñg, defined as pxñg yq  $& %1, if x ¤ yy, if x ¡ y and called Gödel implication is the residuum of the minimum t-norm. 4. The map ñgn, defined as pxñgn yq  $& %1, if x ¤ yy{x, if x ¡ y and called Goguen implication is the residuum of the product t-norm. 5. The map ñł defined as pxñł yq  p1 x  yq ^ 1 is the residuum of the Łukasiewicz t-norm, called the Łukasiewicz implication. 6. The map ñd defined as pxñd yq  $& %1, if x 1y, if x  1 is the residuum of the drastic product t-norm. Now that we have extended the classical connectives, we can also extend the notion of Cartesian product, introduced in Definition 2.2.5. Definition 2.3.16. Let U1, U2 be (classical) sets, and let A1, A2 be a fuzzy subsets of U1, U2 respectively. Let M be a t-norm. Then the Cartesian product of A1, A2 induced by M is the fuzzy set A1 M A2 that has membership function μ A1MA2 px1, x2q  μA1 px1qMμA2 px2q . If A1, ..., An, An 1 are fuzzy subsets of the classical sets U1, ..., Un, Un 1 respectively, and A1 M ...M An has already been defined, then A1 M ...M An M An 1 : pA1 M ...M Anq M An 1 . Chapter 2. Fuzzy Sets and Fuzzy Connectives 65 (a) Kleene-Dienes (b) A S-implication (c) Gödel (d) Goguen (e) Łukasiewicz (f) Drastic product Figure 9 – Fuzzy implications Having extended the classical connectives to a fuzzy setting, we make a brief discussion on how to "extend" the quantifiers (@ and D), which shall be necessary when we develop fuzzy FCA. For a deeper investigation of D and @, and the so called First Order Predicate Calculus, See (MARGARIS, 1990) and (MENDELSON, 2009). 2.3.2 Predicates and quantification According to ANGIONI (2006, pp. 17–18), By predication it is understood the statement that (i) has the form "S is P" or some equivalent form reducible to that, and (ii) intends to report given facts in the world [...] The basic structure of predication, as proposed by Aristotle, is constituted of a minimum of three elements: two terms (one of which is the subject and the other the predicate) and the copulative operator.13 The statements The snow is white and 2  2  4 13 "Por predicação, entende-se o enunciado que (i) possui a forma 'S é P' ou alguma forma equivalente e redutível àquela, (ii) pretende repostar-se a fatos dados no mundo [...] A estrutura básica da predicação, tal como proposta por Aristóteles, constitui-se de três elementos mínimos: dois termos (sendo um deles o sujeito e o outro o predicado) e o operador copulativo." Chapter 2. Fuzzy Sets and Fuzzy Connectives 66 are examples of predication (in the second case, the sentence can be read as "2+2 is equal to 4"). In symbolic (mathematical) logic it is usual to consider the whole predicative sentence as a predicate. In this case, each term is an element of a given set, and a predicate is the characteristic function of a relation on the sets under consideration. When speaking about predicates, we shall denote the characteristic function of n-place relation P applied to the point px1, ..., xnq by P px1, ..., xnq. For instance, consider the sets W : t"Things made of water"u , C : t"Colours"u . Let P : tpx, yq P W  C : x has colour yu . Then the predicate The snow is white is true as P psnow,whiteq holds. At times we may want to state that non-specified elements of the set satisfy certain predicate. For instance, we may want to say that everything that is made of water has a colour, i.e. @xDyP px, yq . The operators for universality ("everything", "each", expressed symbolically as @) and existence ("some", "at least one", written as D) are called quantifiers ("universal quantifier" and "existential quantifier," respectively). Although P px, yq is not a statement, since the elements x and y have not been specified, it does become a statement when each of its variables is either replaced by an element of the corresponding set (e.g., P psnow,whiteq) or is under the scope of a quantifier (e.g., DyP psnow, yq, i.e., "snow has a colour"), and so all the theory of the statement calculus apply to it. As we have already seen, a fuzzy relation is a fuzzy subset of the Cartesian product of the sets under consideration. Thus, we shall express fuzzy predicates by membership functions. But how do we extend the use of quantifiers? Remember that we use real numbers, 0 and 1, to express truth and falsehood. Now a statement @xQpxq is true iff Qpaq holds for each a in the set. If Qpa0q is false for at least one a0 P U , then the statement @xQpxq is false. Thus, classically we have @xQpxq  min aPU Qpaq . Chapter 2. Fuzzy Sets and Fuzzy Connectives 67 A similar argument leads us to the classical expression DxQpxq  max aPU Qpaq . Thus, the way in which we extend the universal and existential quantifiers is by taking, respectively, the infimum and supremum of the membership function over its domain, as inf extends min and sup extends max. As we shall see in Ch. 3, what we need to preserve the theorems of FCA presented in Ch. 1 are the residua of lower semicontinuous t-norms. The next section is dedicated to define continuity, semicontinuity and prove some results that will be used in Ch. 3. 2.4 Continuity and semicontinuity In our development of a fuzzy extension of FCA we shall need some ideas concerning continuity and semicontinuity of t-norms and implications. In this section we shall define continuity, semicontinuity and find how semicontinuity of a t-norm relates to its residuum. Our definitions of semicontinuity are in accordance with (BOURBAKI, 1966). Definition 2.4.1. Let p P N. The distance between two points x  px1, ..., xpq, y  py1, ..., ypq P Rp is defined as dppx, yq  d p i1 pxi  yiq2 . In particular, d0px, yq  0. Definition 2.4.2. Given a point x P Rp and r ¡ 0, the open ball of centre x and radius r is the set Brpxq : ty P Rp : dppx, yq ru. A set S  Rp is open iff it is the (arbitrary) union of open balls. A neighbourhood of a set A  Rp is a set V such that A  S  V , such that S is an open set. In particular, S is a neighbourhood of A. A neighbourhood of a point a is a neighbourhood of A  tau. In other words, an open ball of centre x is the collection of those points close to x (according to the distance dp), and a neighbourhood of a set (or a point) A corresponds to the "surroundings" of A. Definition 2.4.3. A sequence on a set X is a map N Ñ X. If for each n P N we have n ÞÑ xn we write pxnq or pxnqnPN. Now, suppose that X is an ordered set. If for all n P N we have xn ¤ xn 1 we say that the sequence is increasing. If it additionally happens for all n P N that xn  xn 1 - i.e., xn xn 1 - the we say that the sequence is strictly increasing. Chapter 2. Fuzzy Sets and Fuzzy Connectives 68 Decreasing and strictly decreasing sequences are defined dually. Definition 2.4.4. We say that a sequence pxnq on Rp converges to x if for every open ball B containing x, there exists n0 P N such that xn P B for all n ¥ n0, and we write xn Ñ x or lim nÑ8 xn  x. If p  1 so that Rp is (linearly) ordered and pxnq is decreasing (resp. increasing) such that xn Ñ x, we may write xn × x (resp. xn Õ x). Example 2.4.5. Consider the following sequences on R. 1. The sequence (see Fig. 10a)  1 n 

is a strictly decreasing sequence that converges to 0, i.e., 1{pn 1q× 0. 2. The sequence (see Fig. 10b) plnpn  1qq is an increasing diverging14 sequence, where ln is the natural logarithm. 3. The sequence (see Fig. 10c)  ņ k0 p1qk{k 1 nPN is neither increasing nor decreasing, yet it converges to ln 2. Definition 2.4.6. Let X  Rp and Y  Rq be non-empty (classical) sets. Let f : X Ñ Y be a map. We say that f is continuous at x P X if, for all ε ¡ 0, there exists a δ ¡ 0 such that for all y P X, dppx, yq δ implies dq pfpxq, fpyqq ε . In other words, f is continuous at x if, for y P X sufficiently close to x - distance less than δ -, the points fpxq, fpyq are close in Y - less than ε apart. If f is continuous at every x P X it is said to be continuous. In our investigation based on t-norms, t-conorms and implications, we have X  r0, 1s2 and Y  r0, 1s. We shall denote d2 by d and d1px, yq by |x y|. Example 2.4.7. The maximum t-conorm OM is continuous. Consequently, the minimum t-norm MM is also continuous. In fact, let x  px1, x2q P r0, 1s2, ε ¡ 0 and take δ  ε. Let 14 It is unbounded above, i.e., for all k P N there is an n P N such that lnpn   1q ¡ k. A sequence pxnq is called unbounded below iff pxnq is unbounded above. A sequence is unbounded iff it is either unbounded above or below (or both). Chapter 2. Fuzzy Sets and Fuzzy Connectives 69 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 (a) Strictly Decreasing 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (b) Logarithm 0 5 10 15 20 25 30 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 (c) Partial Sum Figure 10 – Sequences y  py1, y2q P r0, 1s2 be such that dpx, yq δ. Define mx : x1OM x2 and my : y1OM y2. Since mx,my ¥ 0 we have x1y1 ¤ mxy1 ¤ mxmy . Similarly, x2y2 ¤ mxmy, and so adding and multiplying by -2 we have 4mxmy ¤ p2x1y1   2x2y2q . Thus, p|mx my|q 2  pmx myq 2 ¤ 2pmx myq2  2m2x   2m2y  4mxmy ¤ px21   x 2 2q   py 2 1   y 2 2q  p2x1y1   2x2y2q  px21   y 2 1  2x1y1q   px22   y22  2x2y2q  px1  y1q 2   px2  y2q 2  dpx, yq2 δ2  ε2 . Chapter 2. Fuzzy Sets and Fuzzy Connectives 70 Since ε ¡ 0, taking the square root on both sides we get |mx my| ε , and so OM is continuous at x. But x is arbitrary, hence OM is continuous. Now, according to (2.3.7), xOM y  1 rp1 xqMM p1 yqs , that is, MM is a composition of continuous functions (the continuous OM and differences), hence it is continuous. Continuity of a t-norm does not imply continuity of its residuum - for example, the residuum of the continuous minimum t-norm is the discontinuous Gödel implication15. Nonetheless, we can say something about such implications. The following definition is made by (BOURBAKI, 1966). Definition 2.4.8. Let X  Rn, Y  R and f : X Ñ Y . We say that f is lower semicontinuous at x0 P X if for h such that h f px0q there is a neighbourhood V of x0 such that h fpxq for all x P V . If f is upper semicontinuous at each x0 P X we say that it is upper semicontinuous. The notion of upper semicontinuity (either at a point, or of the whole function) is defined dually16. Notice that f is lower semicontinuous iff f is upper semicontinuous17. Furthermore, see that a map is continuous at x0 iff it is both lower and upper semicontinuous at x0. BOURBAKI also proves18 the following. Proposition 2.4.9. A function f : X  Rn Ñ R is lower semicontinuous at x0 P X iff lim inf nÑ8 fpxnq ¥ fpx0q 15 Indeed, ñg is discontinuous at px, xq for all x P r0, 1r. In fact, let ε  1 x. Then for all δ ¡ 0, px, x δ{2q P Bδpx, xq, butpxñg xq   xñg  x δ

 x  ¡ 1 x  ε . 16 I.e., replace each instance of by ¡. 17 Where f is the map x ÞÑ fpxq. 18 In fact, Bourbaki's result is more general, but Prop. 2.4.9 follows when considering the Fréchet filter on N, i.e., the complements of finite subsets of N. Chapter 2. Fuzzy Sets and Fuzzy Connectives 71 for each sequence pxnqnPN converging to x0, where lim inf nÑ8 kn : inf n¡0  sup m¥n km

for any sequence pknqnPN. Because of the duality between lower and upper semicontinuousness19, we have the following. Corollary 2.4.10. A function f : X  Rn Ñ R is upper semicontinuous at x0 P X iff lim sup nÑ8 fpxnq ¤ fpx0q for each sequence pxnqnPN converging to x0, where lim sup nÑ8 kn : sup n¡0  inf m¥n km for any sequence pknqnPN. We now derive a useful expression concerning lower semicontinuous t-norms in Lemma 2.4.14, and for its proof we use Proposition 2.4.11. Proposition 2.4.11. Let M be a lower semicontinuous t-norm and x, y P r0, 1s. Let A  tz P r0, 1s : xM z ¤ yu . Then supA P A. Proof. Let z0  supA and pznq be a strictly increasing sequence on r0, 1s that converges to z0. By definition of z0 we have xM zm ¤ y for all m ¡ 0 , as zm z0 for each m ¡ 0. Now, for each n ¡ 0 we have inf m¥n pxM zmq ¤ xM zn ¤ y , so that sup n¡0  inf m¥n pxM znq ¤ y . By lower semicontinuity of M, xM z0 ¤ lim inf nÑ8 pxM znq  sup n¡0  inf m¥n pxM zmq ¤ y . Therefore, z0 P A. 19 In the sense that f is lower semicontinuous iff f is upper semicontinuous. Chapter 2. Fuzzy Sets and Fuzzy Connectives 72 Corollary 2.4.12 and Lemma 2.4.14 are based on (HÁJEK, 1998) . Corollary 2.4.12. Let M be a lower semicontinuous t-norm and let ñ be its residuum. Then for all x, y, z P r0, 1s we have xM z ¤ y iff z ¤ pxñ yq . (2.4.13) Proof. Suppose that xM z ¤ y. Then z P tz0 P r0, 1s : xM z0 ¤ yu , whence z ¤ suptz0 P r0, 1s : xM z0 ¤ yu  pxñ yq . Conversely, if z ¤ pxñ yq then either 1. z pxñ yq, that is z suptz0 P r0, 1s : xM z0 ¤ yu , whence z Ptz0 P r0, 1s : xM z0 ¤ yu ; or 2. z  pxñ yq, so that z P tz P r0, 1s : xM z ¤ yu by Proposition 2.4.11. In both cases, xM z ¤ y. The maps M and ñ satisfying (2.4.13) are called an adjoint pair. Lemma 2.4.14. Let M: r0, 1s2 Ñ r0, 1s be a lower semicontinuous t-norm and let ñ be its residuum. Then, for all x, y P r0, 1s, x ¤ rpxñ yq ñ ys . Proof. Let x, y P r0, 1s. Notice that px ñ yq  supA in Proposition 2.4.11, and so, xM pxñ yq ¤ y. Using commutativity of M, it is clear that x P B : tz P r0, 1s : pxñ yqM z ¤ yu . Hence, x ¤ supB  rpxñ yq ñ ys . Lower semicontinuity of M is a necessary hypothesis for Lemma 2.4.14 as the following example shows. Chapter 2. Fuzzy Sets and Fuzzy Connectives 73 Example 2.4.15. Let ñd be the residuum of the drastic product t-norm, as defined in item 6 of Example 2.3.15. Let x  23 and let y  1 3 . Then pxñd yq   2 3 ñd 1

 1 , as x ¡ y. Thus, rpxñd yq ñd ys   1 ñd 1 3   1 3 2 3  x , and ñd does not satisfy Lemma 2.4.14. Lemma 2.4.16 and Prop. 2.4.17 are proved by (BRITO et al., 2018). Lemma 2.4.16. Let X, Y  R, f : X Ñ Y be an increasing function, pxnq be a sequence on X such that xn × x P X and pynq be a sequence on Y . If for all n ¡ 0, fpxnq ¤ yn, then fpxq ¤ lim inf nÑ8 yn ¤ lim sup nÑ8 yn . Proof. Suppose that fpxnq ¤ yn for all n ¡ 0. Since pxnq is decreasing, for each n ¡ 0 we have sup j¥n xj  xn , so that inf k¡0  sup j¥k xj

¤ xn , so that we have x  lim sup jÑ8 xj ¤ xn . as pxnq Thus, for each n ¡ 0, fpxq ¤ fpxnq ¤ yn . Now, taking the limit inferior on the right-hand side, we conclude the proof. Proposition 2.4.17. Let M be a lower semicontinuous t-norm, and let ñ be the residuum of M. Then for each x0, y0 P r0, 1s the maps x ÞÑ px ñ y0q and y ÞÑ px0 ñ yq are upper semicontinuous. Proof. Let x1, y1 P r0, 1s. Let pxnq, pynq be sequences on r0, 1s converging respectively to x1 and y1. For each n ¡ 0, consider the following definitions: Chapter 2. Fuzzy Sets and Fuzzy Connectives 74 zp1qn  pxn ñ y0q ; zp2qn  px0 ñ ynq ; xp1q  x1 ; xp2q  x0 ; xp1qn  infm¥nxm ; x p2q n  x0 ; ỹp1q  y0 ; ỹp2q  y1 ; ỹp1qn  y0 ; ỹp2qn  sup m¥n ym ; zp1qn    xp1qn ñ ỹ p1q n  ; zp2qn    xp2qn ñ ỹ p2q n  . Notice the following: 1. The sequence pxp1qn q is constructed as the infima of the decreasing sets An  txm : m ¥ nu. Thus, pxp1qn q is increasing. Furthermore as n increases the elements left in each An get always closer to x1. Thus, xp1qn Õ x1. In general, xpiqn Õ xpiq, for i  1, 2. 2. By an argument analogous to that of item 1., ỹpiqn × ỹpiq, for i  1, 2. 3. For each i  1, 2, considering monotonicity of ñ in each component, one sees that for all n ¡ 0, zpiqn ¤ zpiqn . Thus, lim sup nÑ8 zpiqn ¤ lim sup nÑ8 zpiqn . 4. The sequence pzpiqn q is decreasing, for i  1, 2. In the following, i  1, 2. By Proposition 2.4.11, for each n ¡ 0, xpiqn M z piq n ¤ ỹ piq n as M is lower semicontinuous. For n0 ¡ 0 fixed and by monotonicity of M we have, for each n ¡ n0, xpiqn0 M z piq n ¤ ỹ piq n . Thus, an application of Lemma 2.4.16 yields xpiqn0 M  lim sup nÑ8 zpiqn

¤ lim inf nÑ8 ỹpiqn  ỹ piq . But n0 ¡ 0 is arbitrary, and so lim inf nÑ8  xpiqn M  lim sup nÑ8 zpiqn

 ¤ ỹpiq . Remember that M is lower semicontinuous and so, since pxpiqn q converges to xpiq, we see that xpiqM  lim sup nÑ8 zpiqn

¤ ỹpiq . Chapter 2. Fuzzy Sets and Fuzzy Connectives 75 Hence we have lim sup nÑ8 zpiqn ¤ suptz P r0, 1s : xpiqM z ¤ ỹpiqu    xpiq ñ ỹpiq  . Thus, lim sup nÑ8 zpiq ¤   xpiq ñ ỹpiq  . For i  1 this means x ÞÑ px ñ y0q is upper semicontinuous at x1, and for i  2, y ÞÑ px0 ñ yq is upper semicontinuous at y1. Since x1, y1 P r0, 1s are arbitrary, both the maps x ÞÑ pxñ y0q and y ÞÑ px0 ñ yq are upper semicontinuous. Being able to extend classical methods to fuzzy ones is already a very powerful tool. But what would happen if what we have at hand consists in a finite subset of r0, 1s, or even if the truth-values at hand are not linearly ordered? These questions are dealt with in the next section. 2.5 Residuated Lattices Definition 2.5.1. Let U be a (classical) set, and let xL,¤y be a complete lattice, where L  H. A L-fuzzy subset20 F of U is a function μ F : U Ñ L . The set of all L-fuzzy subsets of U is denoted by LU . Notice that, we have special cases of L-fuzzy sets when L  t0, 1u (in which case we have characteristic functions of classical subsets of U) and when L  r0, 1s (which, under the usual order, corresponds to standard fuzzy sets). Now recall from Definitions 2.3.1, 2.3.5 and 2.3.12 that t-norms, t-conorms and fuzzy implications respectively were defined solely in terms the (standard) order of r0, 1s and of equality. Thus, each of these definitions can be generalized here. Definition 2.5.2. Let L be a complete lattice. Then a triangular norm and a triangular conorm on L are maps M,O : L2 Ñ L respectively, which satisfy the following for all 20 The definition of L-fuzzy sets was first introduced in 1967 by Goguen in (GOGUEN, 1967), as a generalization of the idea of fuzzy sets introduced two years earlier by Zadeh in (ZADEH, 1965). Chapter 2. Fuzzy Sets and Fuzzy Connectives 76 x, y, z P L the following: 1. xM y  yMx , 1'. xOy  yOx , 2. pxM yqM z  xM pyM zq , 2'. pxOyqOz  xOpyM zq , 3. If y ¤ z then xM y ¤ xM z , 3'. If y ¤ z then xOy ¤ xOz , 4. xM 1  x , 4'. xO0  x . Example 2.5.3. Clearly if L  r0, 1s then t-norms and t-conorms have the meaning that has been presented earlier. Let us now consider some other examples. 1. For each n ¡ 0, let Ln 1  tk{n : k  0, ..., nu . The respective restrictions M Ln 1 ,O Ln 1 of a t-norm and a t-conorm M,O : r0, 1s2 Ñ r0, 1s to Ln 1 are a t-norm and a t-conorm on Ln 1 respectively. In fact, for each x, y P Ln 1, xM Ln 1 y  xM y , xO Ln 1 y  xOy . 2. If L is an arbitrary finite linearly ordered set with n 1 elements, then it is isomorphic to Ln 1 in the sense that there exists a bijection f : L Ñ Ln 1 such that, for all x, y P L, x ¤L y iff fpxq ¤Ln 1 fpyq . Thus, each t-norm and t-conorm M,O on Ln 1 induces a t-norm and a t-conorm M f ,O f on L, defined respectively, for all x, y P L, by xM f y  fpxqM fpyq , xO f y  fpxqOfpyq . 3. Let L be a (complete) lattice. Then the meet and join21 operators on L are respectively a t-norm and a t-conorm on L. For instance, the (finite, hence) complete lattice defined by 0 a b 1 21 By meet and join of X  tx, yu  L we mean respectively the greatest lower bound and the least upper bound of X on L. The meet and join binary operators are respectively the maps L2 Ñ L that map every pair px, yq to their meet and join. Chapter 2. Fuzzy Sets and Fuzzy Connectives 77 has respective meet and join given by22 ^ 0 a b 1 0 0 0 0 0 a 0 a 0 a b 0 0 b b 1 0 a b 1 _ 0 a b 1 0 0 a b 1 a a a 1 1 b b 1 b 1 1 1 1 1 1 We could now define L-fuzzy implications by generalizing the definition of a fuzzy implication, but for our purposes it is sufficient to generalize the idea of residuum of a t-norm. Recall that Corollary 2.4.12 stated that, given a left semicontinuous t-norm M and its residuum they satisfy, for all x, y, z P r0, 1s, z ¤ pxñ yq iff xM z ¤ y . (2.5.4) That's the property which is generalized. Definition 2.5.5. A residuated lattice is a tuple xL,_,^, ,ñ, 0, 1y such that: 1. xL,_,^, 0, 1y is a lattice23 with least element 0 and greatest element 1; 2.  is a binary operation on L that is associative, commutative and such that 1  x  x for all x P L; 3.  and ñ form an adjoint pair, i. e., for all x, y, z P L we have z ¤ pxñ yq iff x  z ¤ y . (2.5.6) A residuated lattice is called complete iff the lattice in item 1. is complete. Notice that a t-norm always satisfies item 2., with the additional property that t-norms are monotonic increasing maps. Example 2.5.7. It follows from (2.5.4) that if M is a lower semicontinuous t-norm on L  r0, 1s and ñ is its residuum, then xL, sup, inf,M,ñ, 0, 1y is a residuated lattice. Now, consider the following. 1. Let L be a finite linearly ordered set with smallest and greatest elements 0, 1 respectively, and let M be a t-norm on it. Define ñ for all x, y P L as follows. pxñ yq  max tz P L : xM z ¤ yu . 22 For any finite (and sufficiently small) set S, it is usual to present a binary operator S2 Ñ S as a table in which rows and columns are "indexed" by the elements of S and, if px, yq ÞÑ z, then the element of row x and column y is z. 23 Recall that (0.2.3, 0.2.4) uniquely define the order relation in terms of the join and meet binary operators. Chapter 2. Fuzzy Sets and Fuzzy Connectives 78 Then xL,_,^,M,ñ, 0, 1y is a residuated lattice, where _,^ are respectively the join and meet operators on L. It is sufficient to prove that (2.5.6) holds. Let x, y, z P L. Suppose that z ¤ pxñ yq. Since L is finite and linearly ordered, we have z P tz0 P L : xM z0 ¤ yu , whence xM z ¤ y. Conversely, if xM z ¤ y, then z ¤ max tz0 P L : xM z0 ¤ yu  pxñ yq . 2. The following lattice, t-norm and residuum constitute a residuated lattice. 0 a b 1 ^ 0 a b 1 0 0 0 0 0 a 0 a 0 a b 0 0 b b 1 0 a b 1 ñ 0 a b 1 0 1 1 1 1 a b 1 b 1 b a a 1 1 1 0 a b 1 Again, we only need to prove that (2.5.6) holds. If z  0 then both z ¤ pxñ yq and 0  x^ z ¤ y are true for all x, y, P L. If z  1, then (2.5.6) reduces to pxñ yq  1 iff x ¤ y, which can be seen by inspection. Let us consider 0 z 1. Then either z  a or z  b. Suppose WLOG that z  a - the proof for z  b is analogous. Then both x^ a ¤ y , a ¤ pxñ yq are true iff at least one of the following holds. x  0 , x  b , y  a , y  1 . Therefore, (2.5.6) is always true. We now have developed the tools necessary to extend/generalize any expression of classical (first order) logic. We are finally in position to go back to FCA and generalize it. 79 3 Fuzzy Formal Concept Analysis We now introduce the reader to some developments on fuzzy extensions of FCA, here referred to as Fuzzy Formal Concept Analysis. In Sec. 3.1 we shall presented the ideas and results by using standard fuzzy sets (i. e., with L  t0, 1u). The contents in this section have been developed independently by the authors of this text1 and later found to fall within the scope of a broader theory, which we shall briefly discuss in Sec. 3.2. 3.1 Formal concept analysis in the fuzzy setting Recall that the theory of FCA as presented in Ch. 1 is based on a given formal context which consists of sets of objects and attributes, and a binary relation on them. Because a fuzzy set is defined on a classical universal set U , we continue to consider the universal sets of objects and attributes as classical sets, but we are now allowed to work with fuzzy subsets of objects and attributes2. Furthermore, we consider the binary relation to be fuzzy. Definition 3.1.1. A fuzzy formal context is an ordered triple Cf  xO,A, Ify, in which O and A are non-empty (classical) sets, and If  O A is a fuzzy binary relation. The fact that we have a fuzzy relation enables us to express to which degree the object o has the attribute a, rather than the classical "either oIa or po, aq R I". Recall that for the crisp relation I we defined in (1.1.4) and (1.1.5) the maps  : 2O Ñ 2A and ^ : 2A Ñ 2O by O : ta P A : oIa for all o P Ou Â : to P O : oIa for all a P Au . Consider for instance the first of these equations. In order to decide whether a given a P A is a member of O, it suffices to test whether oIa holds for each o P O. We need not concern ourselves with o R O. That is exactly the mathematical practice regarding the (classical) implication. When we write down Φ ÝÑ Ψ, we mean that whenever Φ is 1 See (BRITO et al., 2018) . 2 When applying the theory there are several situations in which it makes no sense to work with fuzzy subsets of objects, but there are other situations in which it does make sense. For instance, a patient is not 2{3 of a person, but a "partial" symptom (noise when breathing to a degree 2{3) may be related to a "partial" disease (a not fully developed pneumonia, with degree 3{4). Chapter 3. Fuzzy Formal Concept Analysis 80 true, so must be Ψ, but in case Φ is false we have no interest in Ψ3. Hence we may say that a P O iff o P O ÝÑ oIa. By using an analogous argument for Â we see that the expressions for O and Â can be translated respectively as χOpãq  @o P Opo P O ÝÑ oIãq , χÂpõq  @a P Apa P A ÝÑ õIaq . Now recall that in Section 2.3.2 we argued that the universal quantifier can be extended by taking the infimum. As a result, we may define the following. Definition 3.1.2. Let Cf  xO,A, Ify be a fuzzy formal context and let ñ be a fuzzy implication. Then given fuzzy subsets O  O and A  A we define fuzzy subsets O  A and Â  O respectively by their membership functions: μOpaq  inf oPO rμOpoq ñ μIf po, aqs , (3.1.3) μÂpoq  inf aPA rμApaq ñ μIf po, aqs . (3.1.4) A fuzzy formal concept is a pair xO,Ay P OA such that O  A and Â  O. In order to extend results achieved working out the classical theory, we need to find out what fuzzy connectives preserve the classical properties we want to keep. For instance, remember that item 2. of Theorem 1.1.10 stated that given a formal context C  xO,A, Iy and given O  O, we have O  O^ . Considering how the fuzzy sets O and Â are defined, we need to do some investigation concerning fuzzy implications. We start by asking ourselves: does continuity preserve the properties we want? Example 3.1.5. Let Cf  xO,A, Ify be the fuzzy context defined so that O  to1, o2u, A  ta1, a2u and If is expressed by a1 a2 o1 0.2 0.7 o2 0.4 0.6 3 We say that the classical implication is vacuously true, that is, an expression Φ ÝÑ Ψ is true whenever Φ is false. It is also true, of course, when both Φ and Ψ are true, and false only in case Φ is true and Ψ is false. Chapter 3. Fuzzy Formal Concept Analysis 81 Let px ñ yq : p1  xq _ y for each x, y P r0, 1s (this is the Kleene-Dienes implication). Consider the fuzzy set O  0.5 o1   0.8 o2 . Then μOpa1q  inf oPO rμOpoq ñ μIf po, a1qs  rp1 0.5q _ 0.2s © rp1 0.8q _ 0.4s  0.5^ 0.4  0.4 and μOpa2q  inf oPO rμOpoq ñ μIf po, a2qs  rp1 0.5q _ 0.7s © rp1 0.8q _ 0.6s  0.7^ 0.6  0.6 , so that O  0.4 a1   0.6 a2 . Now, we have μO^po2q  inf aPA rμOpaq ñ μIf po2, aqs  rp1 0.4q _ 0.4s © rp1 0.6q _ 0.6s  0.6^ 0.6  0.6 . Since μO^po2q  0.6 0.8  μOpo2q, we conclude that O  O^. The implication used in this example is continuous as it is the composition of continuous functions (maximum and difference). Thus continuity is not a sufficient condition on a fuzzy implication for us to extend Theorem 1.1.10. As we shall see, continuity is not a necessary condition as well. Since item 4. of the aforementioned theorem uses a Cartesian product we are led to think that t-norms may be involved and that we must use their residua. However, not any residuum solves the problem we are currently tackling. For example, remember that the item 4. of Theorem 1.1.10 states that if C  xO,A, Iy is a formal context, then given O  O and A  A have A  O iff O  A  I . Example 3.1.6. Let Cf  xtou, tau, Ify be a fuzzy formal context in which μIf po, aq  0.4. Let O  0.5{o and let A  1{a. Let M be the drastic product t-norm4 and ñ the residuum 4 As introduced in Example 2.3.2, xM D y  $'& '% 0, if px_ yq 1 x, if y  1 y, if x  1 . Chapter 3. Fuzzy Formal Concept Analysis 82 of M. Then pμOpoq ñ μIf po, aqq  p0.5 ñ 0.4q  suptz P r0, 1s : 0.5M z ¤ 0.4u  1 as 0.5M z  0 for all z P r0, 1r. Thus, O  1{a, and we have A  O. Nevertheless, μOpoqMμApaq  0.5M 1  0.5 ¡ 0.4  μIf po, aq . Therefore, O M A  If . We see that O M A  If results from the discontinuity of M. Nonetheless, it turns out that, for many of the results we wish to prove, something weaker than continuity of the t-norm is required: all we need are lower semicontinuous t-norms5. Theorem 3.1.7. Let C  xO,A, Ify be a fuzzy context. Let O,O1, O2 be fuzzy subsets of O and A,A1, A2 be fuzzy subsets of A. Let M be a lower semicontinuous t-norm, and let ñ be the residuum of M. Then: 1. If O1  O2 then O2  O1 1'. If A1  A2 then Â2  Â1 2. O  O^ 2'. A  Â 3. O  O^ 3'. Â  Â^ 4. O  Â iff A  O iff O M A  If Proof. We shall prove items 1., 2., 3. and 4. Items with a prime can be proved analogously. 1. Suppose that O1  O2. Let a P A. By hypothesis, μO1poq ¤ μO2poq for each o P O. Since ñ is decreasing in its first component we have pμO2poq ñ μIf po, aqq ¤ pμO1poq ñ μIf po, aqq . Taking the infimum over o on both sides, we have (by definition of O1 and O2 ) μO2 paq ¤ μO1 paq. But a P A is arbitrary. Thus O  2  O  1 . 2. Let o P O. Remember that ñ is decreasing in its first component. Thus μOpoq ¤ rpμOpoq ñ μIf po, aqq ñ μIf po, aqs Lemma 2.4.14 ¤  inf õPO pμOpõq ñ μIf põ, aqq ñ μIf po, aq  Monotonicity of ñ  rμOpaq ñ μIf po, aqs . 5 Lower semicontinuous functions were introduced in Def. 2.4.8. Chapter 3. Fuzzy Formal Concept Analysis 83 Taking the infimum over a on the right side, we get μOpoq ¤ μO^poq. Since o P O is arbitrary, O  O^. 3. From item 2., we have O  O^. Thus, using 1., O^  O. On the other hand, using A  O in 2'., we have O  O^. Hence, O  O^. 4. We first prove that O  Â iff A  O. Suppose that O  Â. By 1., Â  O. Using 2'. and transitivity of , we have A  O. The proof that A  O implies O  Â is analogous. Now we prove that O  Â iff OM A  If . In fact, O  Â iff for all o P O and for all a P A μOpoq ¤ μÂpoq  inf ãPA rμApãq ñ μIf po, ãqqs ¤ rμApaq ñ μIf po, aqs , By Corollary 2.4.12 (and applying commutativity of M) this holds iff μOpoqMμApaq ¤ μIf po, aq . Thus O  Â iff O M A  If . Notice that, as in the classical case, p,^ q is a Galois connection between xFpOq,y and xFpAq,y, so that ^ and ^ are closure operators. Another interesting matter is that not only the set BfpCfq of fuzzy concepts on Cf is a complete lattice (under an order similar to that of classical concepts): the formulae for finding infima and suprema on this lattice are similar to those of the classical case. Proposition 3.1.8. Let Cf  xO,A, Ify be a fuzzy context. Let M be a lower semicontinuous t-norm, and let ñ be its residuum. Let J be an index set and, for each α P J , let Oα  O and Aα  A. Then £ αPJ Oα  ¤ αPJ Oα  , (3.1.9) £ αPJ Âα  ¤ αPJ Aα ^ . (3.1.10) Proof. We prove (3.1.9). The proof of (3.1.10) is analogous. Let a P A. For each α0 P J we Chapter 3. Fuzzy Formal Concept Analysis 84 have μpYαPJOαqpaq  infoPO rμYαPJOαpoq ñ μIf po, aqs  inf oPO  sup αPJ μOαpoq

ñ μIf po, aq  ¤  sup αPJ μOαpoq

ñ μIf po, aq  ¤  μOα0 poq ñ μIf po, aq  , as ñ is decreasing in the first component. Applying the infimum over o P O and then the infimum over α P J on the right-hand side yields μpYαPJOαqpaq ¤ μXαPJOαpaq . Since a P A is arbitrary, ¤ αPJ Oα   £ αPJ Oα . Conversely, we want to prove £ αPJ Oα  ¤ αPJ Oα  . (3.1.11) Notice that if for all o P O and for all a P A we have μXαPJOαpaq ¤   μ αPJ Oα poq  ñ μIf po, aq  , (3.1.12) then taking the infimum over o P O on the right-hand side of (3.1.12) yields (3.1.11). Hence it is sufficient to prove (3.1.12). Suppose for the sake of contradiction that (3.1.12) does not hold, that is, for some o P O and a P A, κ : inf αPJ  inf õPO pμOαpõq ñ μIf põ, aqq  ¡  sup αPJ μOαpoq

ñ μIf po, aq  . Define x0  sup αPJ μOαpoq. Let pαnq be a sequence on J such that   μOαn poq  converges to x0 and, for each n ¡ 0, define xn  μOαn poq. By hypothesis, for each n ¡ 0, px0 ñ μIf po, aqq infαPJ  inf õPO pμOαpõq ñ μIf põ, aqq  p κq ¤ inf õPO   μOαn põq ñ μIf põ, aq  ¤   μOαn poq ñ μIf po, aq   pxn ñ μIf po, aqq . Thus, px0 ñ μIf po, aqq κ ¤ lim sup nÑ8 pxn ñ μIf po, aqq , Chapter 3. Fuzzy Formal Concept Analysis 85 and so x ÞÑ pxñ μIf po, aqq is not upper semicontinuous, contradicting Proposition 2.4.17. Hence, (3.1.12) holds. Theorem 3.1.13. Let Cf  xO,A, Ify be a fuzzy formal context. Using a lower semicontinuous t-norm and its residuum to define the maps  and ^, define the order ¤ on the set BfpCfq of all the fuzzy formal concepts of Cf by xO1, A1y ¤ xO2, A2y iff O1  O2 piff A2  A1q . (3.1.14) Then LCf : xBfpCfq,¤y is a complete lattice, called the fuzzy concept lattice of Cf . If K is an index set and Cκ  xOκ, Aκy P BpCq for each κ P K then inf κPK Cκ  C£ κPK Oκ, ¤ κPK Aκ ^G , (3.1.15) sup κPK Cκ  C¤ κPK Oκ ^ , £ κPK Aκ G . (3.1.16) Proof. Similar to that of the Basic Theorem (Theorem 1.1.18). 3.2 Formal concept analysis on complete residuated lattices As we have already seen it is possible to extend classical FCA by replacing classical sets with fuzzy sets. Moreover, in Sect. 2.5 we have presented a generalized version of fuzzy sets, called L-fuzzy sets, where the unit interval is replaced by a complete lattice L. The present section follows BĚLOHLÁVEK; VYCHODIL's "What is a fuzzy concept lattice?" in which some distinct approaches to L-fuzzy concept analysis are presented (2005). 3.2.1 The approach of Burusco and Fuentes-Gonzáles The field of Fuzzy FCA was launched by (FUENTES-GONZÁLEZ; BURUSCO, 1994). Their approach was based on an algebraic system xL,¤,1 ,O, 0, 1y in which 1. xL,¤, 0, 1y is a complete lattice; 2. O is a t-conorm6 on L; and 6 According to Def. 2.5.2. Chapter 3. Fuzzy Formal Concept Analysis 86 3. 1 is a complementation7 on L. Then, given classical sets X, Y and a L-fuzzy relation IL  LXY 8, they define maps  : LX Ñ LY and ^ : LY Ñ LX by Apyq  © xPX pA1pxqOIpx, yqq , B^pxq  © yPY pB1pyqOIpx, yqq . This approach leads to a complete L-fuzzy concept lattice. Nonetheless, it does not present some useful properties, namely, ^ and ^ are not closure operators9. 3.2.2 The approach of Pollandt and Bělohlávek In the late 1990's (POLLANDT, 1997) and (BĚLOHLÁVEK, 1999) independently developed an approach which could10 be stated as a straightforward generalization of the approach presented in Sec. 3.1 to complete residuated lattices. In fact, let xL,_,^, ,ñ, 0, 1y be a complete residuated lattice11. An L-context is a triple xX, Y, Iy in which I P LXY is an L-fuzzy binary relation. The maps  : LX Ñ LY , ^ : LY ñ LX are defined for each A P LX , B P LY by Apyq  © xPX pApxq ñ Ipx, yqq , B^pxq  © yPY pBpyq ñ Ipx, yqq . In working with this approach, p,^ q constitutes a Galois connection12, so that ^, ^ are closure operators. 7 FUENTES-GONZÁLEZ; BURUSCO's complementation, or negation, is introduced by (LÓPEZ; FUENTEZ-GONZÁLES, 1991) and is similar the notion of negation introduced in Def. 2.3.10. They define a negation as a map 1 : LÑ L that satisfies, for all x, y P L: a) px1q1  x; b) px_ yq1  x1 ^ y1; (_ and ^ are the meet and join of L) c) 01  1. 8 So that, according to the approach presented thus far, we could say that @ X,Y, LXY D is a L-fuzzy formal context. 9 Notice that, according to FUENTES-GONZÁLEZ; BURUSCO's approach, implications are generalized based on the form A_ B. As we have seen in Example 3.1.5, one such implication may not yield monotonic maps ^, ^, hence it they are not closure operators. 10 By reversing the thermodynamic arrow of time. 11 Recall that the notion of residuated lattices, presented in Sec. 2.5 (Def. 2.5.5), generalizes the notion o lower semicontinuous t-norms and their residua. 12 In an extended notion of Galois connections, according to (BĚLOHLÁVEK, 1999). Chapter 3. Fuzzy Formal Concept Analysis 87 With this approach the Basic Theorem on Concept Lattices (Theorem 1.1.18) is given two different versions: one in which L-fuzzy concepts are ordered by L-fuzzy set inclusion13 of their extents (BĚLOHLÁVEK, 1999) ; and another in which a so-called L-order14 on L-fuzzy concepts is used (BĚLOHLÁVEK, 2004) . Example 3.2.1. Recall that, according to Example 2.5.7, the following lattice diagram, t-norm and residuum constitute a residuated lattice. 0 a b 1 ^ 0 a b 1 0 0 0 0 0 a 0 a 0 a b 0 0 b b 1 0 a b 1 ñ 0 a b 1 0 1 1 1 1 a b 1 b 1 b a a 1 1 1 0 a b 1 Consider the following L-fuzzy context. O A a1 a2 o1 0 a o2 1 b o3 a 1 By evaluating every L-fuzzy formal concept and ordering the results, we found the following diagram. u a2 1 o1 v a2 u a1 v a1 1 a2 v o1 1 o3 u o1 1 o2 v o2   u o3 13 In the sense that A  B in LY iff Apyq ¤ Bpyq for all y P Y . 14 On a complete residuated lattice xL,_,^, ,ñ, 0, 1y, the notions of equality, order and subsetness are generalized to "L-notions", and after several other definitions and results, BĚLOHLÁVEK concludes that "xxBpCq,y, y is [a] completely lattice L-ordered set," with formulae for infimum, supremum, and conditions under which a "completely lattice L-ordered set" is isomorphic to this structure (2004, p. 11). Chapter 3. Fuzzy Formal Concept Analysis 88 It is a bothersome, and even impracticable task to compute every single concept of a (L) fuzzy context. Clearly, computer algorithms must be used in this task, and that is precisely the purpose of the next and final chapter of this text. 89 4 Computing the Fuzzy Concept Lattice 4.1 Computational algorithms As mentioned in Sect. 1.2, (LINDIG, 2000) introduces an algorithm for computing the concept lattice of classical FCA, which is based on the idea of finding upper neighbours of each given node of the lattice. In fact, the algorithm is not restricted to concept lattices, but it works for any finite lattice of closed sets with respect to a given closure operator1. Although we have not presented the classical algorithm in Sect. 1.2, we now present an extended version of that algorithm which, when restricted to truth-values 0 and 1, corresponds to the classical case. This extended algorithm was introduced by (BĚLOHLÁVEK et al., 2010). For practical purposes, we consider a finite linearly ordered set L  ta1, ..., anu, where a1 ... an. If i  1, ..., n  1 we write a i for ai 1. Furthermore, we consider a nonempty finite set Y  ty1, ..., ymu (which we may interpret as a set of attributes) and a closure operator C : LY Ñ LY , i. e., if A,B are L-fuzzy sets (members of LY ) then 1. A  CpAq , 2. CpAq  CpCpAqq , 3. A  B implies CpAq  CpBq . Here, as before, A  B abbreviates @y P Y,Apyq ¤ Bpyq, where ¤ is the order on L. Furthermore, because we shall work with exactly one closure operator C, we shall write A for CpAq. We denote by C the collection of closed L-fuzzy sets of Y , i. e., C  tA P LY : A  Au . We know from Theorem 0.2.20 that xfixpY q,y is a (complete) lattice. If A,B P C and B is an upper neighbour2 of A, we write A B . If Apyq 1 denote by rysA the L-fuzzy set rysA   AY  Apyq  y

 , 1 Recall basic properties of closure operator on Def. 0.2.17 and Theorem 0.2.20. 2 That is, A  B and there is no B1 such that A  B1  B. For basic notions concerning upper neighbours and covering relations, see Def. 0.2.5 and Lem. 0.2.7. Chapter 4. Computing the Fuzzy Concept Lattice 90 that is, we take the L-fuzzy set A, increase the truth-value of y to the next possible value, and then take the closure of this set. It is clear that A  rysA (by monotonicity of the closure operator), but it may or may not happen that A rysA. Example 4.1.1. Consider the following fuzzy context: a1 a2 a3 o1 0.5 0 0 o2 0.5 1 0 o3 0.5 0 0.5 We now define Y  A  ta1, a2, a3u, L  t0, 0.5, 1u, define ^, by means of 3-valued Łukasiewicz's implication and let the closure operator C be the composition ^. Let A  0.5{a1   0{a2   0.5{a3. Then ra1sA   1 a1   0 a2   0.5 a

 1 a1   0.5 a2   1 a3 , Notice that A  ra1sA  ra2sA  ra3sA, whence ra3sA is not an upper neighbour of A. Let A  B and define yi  maxtyj P Y : Apyjq Bpyjqu , (4.1.2) and let gApBq : yi. Belohlávek et. al. prove that if A B then B  gApBq, and gApBq is called an upper neighbour generator, so that B is generated by yi. Thus far we know that an upper neighbour B of A can always be described in the form rysA, but a closed L-fuzzy set rysA may not be an upper neighbour of A. Belohlávek et. al. then find the following criteria for verifying whether y is an upper neighbour generator. Theorem 4.1.3. Let MA  ty P Y : A rysAu  ty P Y : A B and y  gApBqu Chapter 4. Computing the Fuzzy Concept Lattice 91 be the set of upper neighbour generators of A. Let yi P Y be such that yi  gApryisAq . (4.1.4) Then yi P MA iff for each yk P MA such that k i we have pryisAq pykq  Apykq . (4.1.5) Example 4.1.6. Let us consider Ex. 4.1.1 again. Recall that A  0.5{a1 0{a2 0.5{a3. There, ra1sA  ra2sA, and so we have redundant information. Such a redundancy can be avoided when we consider (4.1.4). Since ra1sA  1 a1   0.5 a2   0.5 a3 , we have gApra1sAq  a2 . This, means that a1 R MA, and we need not concern ourselves with a1 when finding upper neighbours of A, because ra2sA gives the same result. What about a3? We see that ra3sA  1 a1   0.5 a2   1 a3 , As we have already seen, A  ra2sA  ra3sA, so that A ¢ ra3sA, and so a3 R MA, even though a3  gApra3sAq, thus satisfying (4.1.4). According to (4.1.5) there must exist k 3 such that yk P MA and pra3sAq pykq ¡ Apykq . The only values k 3 are k  1 and k  2, but we have already seen that a1 R MA. Therefore a2 P MA, and we conclude that MA  ta2u , meaning that A has a unique upper neighbour, namely ra2sA. Theorem 4.1.3 gives us all the information necessary for finding a minimal set MA of upper neighbour generators of a given closed L-fuzzy set A and, from it, we can find the set UA of upper neighbours of A as shown in Alg. 3. Notice that if the conditional in row 7 of Alg. 3 returns False, then there exists yk in the intersection and either: 1. k i, so that (4.1.5) does not hold3; or 3 If k  1, condition (4.1.5) is vacuously satisfied. For k  2, ..., i 1, yk remains in Min iff yk P MA. Chapter 4. Computing the Fuzzy Concept Lattice 92 Algorithm 3 – UpperNeighbours input : Set A output :The set UA of upper neighbours of A 1 UA ÐH; 2 Min Ð ty P Y : Apyq 1u ; /* Candidates to MA */ 3 for i  1 : m do 4 if Apyiq 1 then 5 B Ð ryisA; 6 Increased Ð tyk P Y : k  i and Apykq Bpykqu; 7 if MinX Increased  H then 8 append B to UA; 9 else 10 remove y from Min; 11 end 12 end 13 end 2. k ¡ i, so that (4.1.4) does not hold. Now, we should be able to determine a simple recursive procedure to build up the lattice as follows: 1. Compute H - the smallest closed set - and store it. 2. For a given closed set A  Y , find its set UA of upper neighbours and, for each B P UA: Store B in the set C of closed sets (if it has not been previously computed); Store A in the set LB of lower neighbours of B. This can be accomplished by combining Algorithm 3 with Algorithms 4 and 5. Variable F is shared by Algorithms 4 and 5, as well as UA,LA for each A P F4. Furthermore, sets L, Y and the closure operator C : LY Ñ LY are assumed to be known by all three algorithms5. Example 4.1.7. Let us consider once again Example 4.1.1. Recall that the fuzzy formal context was given by 4 For implementation purposes, think of a k  3 matrix M , in which k increases as we compute new elements of F . Then, in each row of M we have xA,UA,LA, y, where A P F . 5 The universal set Y must be known explicitly, whereas it is sufficient that L and C be programmed in a closure operator procedure. Chapter 4. Computing the Fuzzy Concept Lattice 93 Algorithm 4 – GenerateFrom input : Set A action :Evaluates sublattice of elements greater than A 1 if A  Y then 2 UA Ð UpperNeighbourspAq; 3 NÐ UAzF ; 4 foreach B P UA do 5 append A to LB; 6 if B P N then 7 append B to F ; 8 end 9 foreach B P N do 10 GenerateFrom (B); 11 end 12 end 13 end Algorithm 5 – Lattice input : output : txA,UA,LAy : A P Fu 1 F ÐH; 2 AÐH; 3 append A to F ; 4 GenerateFrompAq; a1 a2 a3 o1 0.5 0 0 o2 0.5 1 0 o3 0.5 0 0.5 Table 3 – Revisiting example 4.1.1. We consider Y  ta1, a2, a3u and L  t0, 0.5, 1u with 3-valued Łukasiewicz implication in order to define the closure ^. Then, application of Alg. 5 to Table 3 returns the triples6, and corresponding lattice diagram, presented in Fig. 11. 6 Here the subscripts are indices representing the orders in which the nodes have been found. A  pα1, α2, α3q expresses A  α1 a1   α2 a2   α3 a3 . Chapter 4. Computing the Fuzzy Concept Lattice 94 A UA LA A0  p0.5, 0.0, 0.0q tA1, A2u H A1  p0.5, 0.5, 0.0q tA3, A4u tA0u A2  p0.5, 0.0, 0.5q tA4u tA0u A3  p1.0, 0.0, 0.0q tA5u tA1u A4  p1.0, 0.5, 0.5q tA5, A7u tA1, A2u A5  p1.0, 1.0, 0.5q tA6u tA3, A4u A6  p1.0, 1.0, 1.0q H tA5, A7u A7  p1.0, 0.5, 1.0q tA6u tA4u 0 0.5 a1 1 0.5 a2 2 0.5 a3 3 1.0 a2 4 1.0 a1 5 6 7 1.0 d3 Figure 11 – Output (left) and Lattice Diagram (right) of Alg. 5 applied to Table 3. Now, we explore an application of the theory we have presented thus far. 4.2 An application of fuzzy FCA Let us now explore an application of Fuzzy FCA. In this section we shall consider an example of medical diagnosis of pneumonia in children, based on a work by PEREIRA et al. (2004), which comprises a study of the diagnosis of pneumonia in children by means of fuzzy relations. The approach used in deciding how to compute and interpret the fuzzy concept lattice is entirely ours. 4.2.1 Posing the problem Pereira's study comprises 153 children who were randomly divided in two groups: an analysis sample (115 cases) and a validation sample (38 cases). These patients were independently diagnosed by two paediatricians according to the following possible diagnoses: pneumonia, non-pneumonia diseases, and healthy. Only cases of agreement were included in the group. The clinical signs observed were 1. Radiological signs, measured by alveolar and interstitial infiltrates, atelectasis, pleural effusion, pneumatoceles, airtrapping, pneumothorax; 2. Dyspnea, taking into account mild discomfort, lower rib in-drawing with tachypnea, intercostal in-drawing with severe tachypnea and/or presence of nasal flaring, full retraction of ribs plus cyanosis and/or poor peripheral blood perfusion; 3. Auscultation signs, increasing with rales, crackles, bronchial breathing; 4. Heart rate, ranging from normal to highly tachycardic; Chapter 4. Computing the Fuzzy Concept Lattice 95 Clinical sign Parameters s1 X-ray 0 – 7 s2 Dyspnea 0 – 4 s3 Auscultation 0 – 3 s4 Heart rate 0 – 4 s5 Temperature 0 – 3 s6 Toxemia 0 – 4 s7 Respiratory rate 0 – 4 Table 4 – Clinical signs observed for diagnosis of pneumonia 5. Body temperature, considered to be either normal (T ¤ 37oC), mild fever (37oC T 38.5oC), fever (38.5oC ¤ T 40oC) or severe fever (40oC T ); 6. Toxemia, according to the presence of pallor, pallor and listlessness, irritability, drowsiness; and 7. Respiratory rate, according to age group, ranging from normal to highly tachypneic. These signs, together with the range of parameters (measure scales) are presented in Table 4. The scalings have been normalized so that if the original values of a clinical sign were natural numbers 0–ns, after normalization we have Vs  tk{ns : k  0, ..., nsu . Now, let X, Y, Z be respectively the sets of clinical signs, patients and diagnoses. The data collected from the patients consisted in a fuzzy binary relation S on X  Y and a crisp binary relation T on Y  Z. Then, a fuzzy relation R on X  Z was obtained by means of the fuzzy relation composition7 Rpx, zq  a yPY rSpx, yq ^ T py, zqs , which is expressed in Table 5. 7 This composition of fuzzy binary relations extends the composition of classical binary relations pT  Sq  tpx, zq P X  Z : Dy P Y such that px, yq P S and py, zq P T u . When functions are defined as binary relations such that px, y1q and px, y2q imply y1  y2, the composition of functions is their composition as binary relations, and px, yq P f is expressed as fpxq  y. Chapter 4. Computing the Fuzzy Concept Lattice 96 Clinical sign Diagnosis Pneumonia Non-pneumonia disease Healthy s1: X-ray 3{7 0 0 s2: Dyspnea 1 0.25 0 s3: Auscultation 2{3 0 0 s4: Heart rate 1 1 0.5 s5: Temperature 2{3 1 0 s6: Toxemia 0.75 0.5 0 s7: Respiratory rate 1 0.75 0 Table 5 – Fuzzy relation of clinical signs and diagnoses Table 5 can be considered as a fuzzy formal context, in which O is the set of clinical signs, A is the set of diagnoses and If  R. Our goal is to build the fuzzy concept lattice by means of Alg. 5 - hence, of course, using Algorithms 3 and 4. Recall that Alg. 3 depends on L-fuzzy sets and we use the closure operator ^. Thus, we need to choose a set L (which by hipothesis of the algorithms used is finite and linearly ordered) and the residuum of a lower semicontinuous t-norm. We have chosen to work with two different residua: Gödel's implication and Łukasiwcicz's implication. 4.2.2 Computing the fuzzy concept lattice First we consider Gödel's implication. Recall that pxñg yq  $& %1, if x ¤ yy, if x ¡ y . Because the only possible values this implication assumes are 1 and the value of the consequent (y), we may assume that L corresponds to the set of values on Table 5, as it already includes 1. This choice allows us to make the diagram on Fig. 12. Now, let us consider Łukasiewicz implication pxñł yq  p1 x  yq ^ 1 . It is based on a sum and a difference, and so the safest manner to be assured that repeated applications of ñł will always result in a value that is a member of L is to choose a set L closed under additions and subtractions. Because of the parameters on Table 4, which are later normalized, our first attempt was to make L  r0, 1s closed under additions and subtractions of 1{3, 1{4 and 1{7. The least common multiple of 3, 4 and 7 is 84, so we chose L  tk{84 : k  0, 2, ..., 84u . Chapter 4. Computing the Fuzzy Concept Lattice 97 0 1.00 s1 0.43 p 1 0.67 p 1.00 s32 0.75 p 3 0.25 np 4 1.00 p 0.43 s1   0.67 s3 5 6 1.00 s2 7 0.75 s6 8 1.00 s7 9 0.25 s2   0.67 s5   0.50 s6   0.75 s7 10 0.50 h 1.00 s411 1.00 h 0.50 s4 12 1.00 s6 13 0.50 np 14 0.75 np 15 1.00 np 1.00 s5 Figure 12 – Medical Diagnosis: Gödel's implication This has been a misleading attempt, as it turns out the resulting fuzzy concept lattice has 55 160 nodes, which would be impracticable to read - not to mention the difficulty of even drawing such a diagram. Thus, some slightly different approach must be used. A better attempt has been to choose a small natural number n0 ¡ 0, define Ln0 1  tk{n0 : k  0, ..., n0u , (4.2.1) and then to redefine the formal context by taking each value v of it and mapping it to the smallest l P Ln 1 such that v l. For example, if we consider L5 (so that n0  4), we convert Table 5 into Table 6. Clinical sign Diagnosis Pneumonia Non-pneumonia disease Healthy s1: X-ray 0.5 0 0 s2: Dyspnea 1 0.25 0 s3: Auscultation 0.75 0 0 s4: Heart rate 1 1 0.5 s5: Temperature 0.75 1 0 s6: Toxemia 0.75 0.5 0 s7: Respiratory rate 1 0.75 0 Table 6 – Relation of clinical signs and diagnoses, with values in L5 Chapter 4. Computing the Fuzzy Concept Lattice 98 0 25 50 75 100 125 150 175 Number of Values 0 10 20 30 40 50 Nu m be r o f C on ce pt s ( ×1 04 ) Figure 13 – Number of concepts with multivalued Łukasiewicz implications When Ln is defined8 as in (4.2.1) andñł is defined on Ln, we say to be working with the n-valued Łukasiewicz implication. We have numerically evaluated the number of concepts corresponding to Table 5 by using n-valued Łukasiewicz implications, n  3k for k  1, ..., 58. The data is shown in Fig. 13. It turns out that the data can be interpolated9 by a third degree polynomial10 ypnq  an3   bn2   cn , where y corresponds to the number of concepts and we have a  8.62  102 , b  5.56  101 , c  2.22  101 . Figure 14, presents the lattices generated by 3, 4-valued Łukasiewicz implications, whereas Fig. 15 corresponds to 5-valued implication. 8 With n  n0   1. 9 With relative numerical errors of the order of 1016, which can be regarded as roundoff errors. 10 For finite O,A and considering Ln, the number of concepts can never exceed nk, where k  mint|O|, |A|u (= 3 for Table 6), |S| being the number of elements (or cardinality) of S. Initial numerical experiments suggest that the number y of concepts can be expressed as ypnq  ķ i1 ain i . Chapter 4. Computing the Fuzzy Concept Lattice 99 0.5 p 1.0 s1 1.0 p 1.0 s3 0.50 np 1.0 s2   1.0 s6 1.0 np 1.0 s5   1.0 s7 0.5 h 0.5 s1   0.5 s3 0.5 s2   1.0 s4   0.5 s5   0.5 s6   0.5 s7 1.0 h 0.5 s4 (a) 3-valued implication 1.00 s1   1.00 s3 0.67 p 0.33 np 1.00 p 1.00 s2 0.67 np 1.00 s6 0.33 h 0.67 s1   0.67 s2 1.00 s7 0.67 s2 0.67 s5   0.67 s6   0.67 s7 0.33 s2   1.00 s4   0.33 s5   0.33 s6   0.33 s7 1.00 h 0.67 s4 0.67 h 0.33 s1   0.33 s3 1.00 np 1.00 s5 (b) 4-valued implication Figure 14 – Medical Diagnosis: multivalued Łukasiewicz implications 4.2.3 Interpreting the diagrams We shall focus our attention on Fig. 15. For this reason we have labelled the nodes of the diagram of 5-valued Łukasiewicz implication with a top-bottom, left-right order (which is different from the order provided by the Upper Neighbours algorithm). Recall that in Theorem 3.1.13 we have found equations (3.1.15, 3.1.16), which give us the infimum and supremum of any collection xCκyκPK of formal concepts as follows. inf κPK Cκ  C£ κPK Oκ, ¤ κPK Aκ ^G , sup κPK Cκ  C¤ κPK Oκ ^ , £ κPK Aκ G . This means that the attributes of sup κPK Cκ correspond to the intersection of the attributes of the Cκ, and dually the objects in inf κPK Cκ correspond to the intersection of the objects of the Cκ. This has a very important consequence on how one should interpret the attributes and objects of a given node. Let us consider, for instance, node C14. We see that this node is the infimum of nodes C4 and C9, so that one question we should ask ourselves Chapter 4. Computing the Fuzzy Concept Lattice 100 C1 0.50 p 1.00 s1 C2 0.75 p 1.00 s3 C3 0.25 np C5 1.00 p 1.00 s2 C6 0.50 np 1.00 s6 C4 0.25 h 0.75 s1 C8C7 0.75 s3 C11 1.00 s7 C10 0.75 s2 C14 0.75 s6 C16 0.75 s5   0.75 s7 C15 0.50 s2 C18 1.00 s4   0.50 s5   0.50 s6   0.50 s7 C19 0.25 s2   0.75 s4   0.25 s5   0.25 s6   0.25 s7C20 1.00 h 0.50 s4 C17 0.75 h 0.25 s1   0.25 s3 C13 0.50 h 0.50 s1   0.50 s3 C9 0.75 np C12 1.00 np 1.00 s5 Figure 15 – Medical Diagnosis: 5-valued Łukasiewicz implication in considering node C14 is: "what symptoms (objects) are common to [possibly different] persons presenting the attribute of being healthy with degree 0.25 [node C4] and some non-pneumonia disease with degree 0.75 [node C9]?" The answer is that every symptom, up to their respective degrees observed node C14 - i.e., in every node below C14 including itself - is common to both of the conditions of being "0.25 healthy" and of presenting some "0.75 non-pneumonia disease." It might be possible however that a person with either condition would present feebler degrees of some signs. The same reasoning could be applied to answer the dual question: "if one observes clinical sings as they are presented in node C14, what does this indicate in terms of the attributes 'healthy', 'non-pneumonia disease' and 'pneumonia'?" Well, it indicates that each condition may be present up to the degree one reads in nodes above C14 (again, including itself). Now, let us read some actual information from the diagram. According to C1, only patients with (perhaps not fully developed) pneumonia present all seven clinical signs under consideration to their most severe degree. Nevertheless, but it is not usual, even among patients suffering from pneumonia, to present all these signs to such a high degree. In fact, node C5 informs us that, although all seven signs are usually present in pneumonia Chapter 4. Computing the Fuzzy Concept Lattice 101 patients, only s2 and s4 (dyspnea and tachycardia) are in their respective worst degrees. But then, that a clinical sign is not present in its most severe degree is not sufficient reason to disregard it. In fact, PEREIRA et al. conclude that [although] heart rate, body temperature, toxemia, and respiratory rate [signs s4 through s6] are important clinical signs to separate 'non-pneumonia disease' from healthy subjects, fuzzy relations complementarily inform that these clinical signs do also have a strong relationship with the diagnosis of pneumonia11. This is precisely what we see in C15. It informs us that, although a somewhat high respiratory rate (s7  0.75) together with severe fever (s5  1) may be more prone to indicate some non-pneumonia disease (np  1), pneumonia cannot be ruled out based on this information alone (p  0.75). The attentive reader may have noticed a disturbing information on the diagram which may seem a defect of the theory. According to C20, even healthy subjects do present some degree of tachycardia. Fortunately for the theory, and unfortunately for some of the children from whom data has been collected for Table 5 to be composed, Pereira informs us that [...] analysis of the patients' records showed that, actually, six children in this group had mild tachycardia (three at a level of 0.25 and three at a level of 0.50), with the weight of four of them being below the 25th percentile. Thus, these children probably were not perfectly healthy but were malnourished and perhaps had the not rarely accompanying condition of anemia, which could account for the tachycardia12. 4.2.4 Final considerations concerning applications Why did we choose to interpret the information in Fig. 15, rather than Fig. 12 or one of the diagrams in Fig. 14? The choice of 5-valued Łukasiewicz implication of 3, 4-valued implications is for a simple reason. By allowing Łukasiewicz implication to work with more possible truth-values we get more detailed information. However, we do not allow ourselves to increase the number of truth-values too much as this would give us excessive information, which would then be useless. Now, comparing Gödel's implication and Łukasiewicz implication, both have their own advantages and disadvantages. First of all, the lattice constructed from either 11 See (PEREIRA et al., 2004, p. 708) 12 See (PEREIRA et al., 2004, p. 705) . Chapter 4. Computing the Fuzzy Concept Lattice 102 lattice allows us to read back the information of the corresponding fuzzy context13. As an example, concept C5 in Fig. 15 is C5  B 0.50 s1   1.00 s2   0.75 s3   1.00 s4   0.75 s5   0.75 s6   1.00 s7 , 1.00 p   0.25 np   0.00 h F . Because it is the greatest concept (with the greatest fuzzy extent) which has p as a full member of its intent (if we move up from C5, the membership of p decreases), we conclude that the first column of Table 6 is the sequence x0.50, 1.00, 0.75, 1.00, 0.75, 0.75, 1.00y , corresponding to the extent of C5. Now, because when working with Gödel's implication (Fig. 12) we work with the original information (Table 5), Gödel's implication has a clear advantage over 5-valued Łukasiewicz implication as application of the later implies lost of (parts of) the original information. Only rough approximations remain. An attempt to remedy this problem would incur in the aforementioned problem of excessive information. On the other hand, Łukasiewicz implication breaks down the information at hand, giving us an understanding of truly fuzzy information, such as the degrees of clinical signs associated with some non-pneumonia disease. Thus, a good practice would be the following: if everything you need is to consider attributes (or objects) with membership degree 1 (e. g., if we are considering a fuzzy context of relations between models of cars and ages of consumers, typically it is the full model that shall be considered), it may be more interesting to construct the lattice by using Gödel's implication as the resulting lattice will preserve the original information - if there is excess of information, the fuzzy context may be adapted. If on the other hand both objects and attributes are fuzzy, the approach that uses Łukasiewicz implication may prove itself more useful. 13 That is, from Fig. 12 we can read Table 5, and from Fig. 15 we can read Table 6. 103 Conclusion Formal Concept Analysis is an elegant mathematical theory closely related to the theory of complete lattices. As shown in this work the theory can be generalized, allowing us to work with formal contexts in which objects and attributes are only partially related to each other and, in fact, the "degree" of such a relationship may be drawn from arbitrary complete residuated lattices. In order to guide the reader in a process of understanding how such a generalization can be achieved, we have attempted to build the main ideas up from little assumptions on the reader's mathematical knowledge (although the ability to follow mathematical reasoning is required). Thus, the reader is introduced to the ideas of orders, (complete) lattices and closure operators on such lattices. That - together with notions of naïve set theory - is what is necessary for an understanding of classical FCA. Afterwards, we introduced the theory of fuzzy sets which, together with fuzzy connectives (particularly t-norms and their residua), provided us with the tools necessary to extend the classical theory. It is our belief that the examples provided throughout the text have been illuminating as a guide for intuition. Although we have developed the results in Section 3.1 independently, we are aware that they fall within the scope of a more general approach based on complete residuated lattices, which was developed (independently) by (POLLANDT, 1997) and (BĚLOHLÁVEK, 1999). In fact, we have shown that the unit interval together with a lower semicontinuous t-norm and its residuum constitute a complete residuated lattice. Nonetheless, this more general approach tends to remain on purely theoretical grounds. We have presented one example based on a non-linear (complete) residuated lattice merely for illustrative purposes. We are aware that potential connections between the theory here presented and other fields such as Bayesian networks and ontologies have not even been drafted in this text. We notice nonetheless that POELMANS et al. have previously performed a survey on fuzzy formal concept analysis (2014) in which they have presented references for connections with ontology engineering. To the best of our knowledge, there are no efficient algorithms for computing the L-fuzzy concept lattice when the residuated lattice L is not linearly ordered. It is our intention to explore more on non-linearly ordered lattices L in the future and, perhaps, to develop an algorithm for (directly and efficiently) solving the problem of computing the Chapter 4. Computing the Fuzzy Concept Lattice 104 concept lattice in such cases. 105 Bibliography ANGIONI, L. Introdução à teoria da predicação em Aristóteles. Campinas: Editora da UNICAMP, 2006. Citado na página 65. BARROS, L. C. de; BASSANEZI, R. C.; LODWICK, W. A. A first course in fuzzy logic, fuzzy dynamical systems, and biomathematics: theory and applications. Springer Berlin Heidelberg, 2017. (Studies in Fuzziness and Soft Computing). 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