Results for 'co-Heyting algebras'

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  1.  6
    Model completion of scaled lattices and co‐Heyting algebras of p‐adic semi‐algebraic sets.Luck Darnière - 2019 - Mathematical Logic Quarterly 65 (3):305-331.
    Let p be prime number, K be a p‐adically closed field, a semi‐algebraic set defined over K and the lattice of semi‐algebraic subsets of X which are closed in X. We prove that the complete theory of eliminates quantifiers in a certain language, the ‐structure on being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field for. We classify these ‐structures up to elementary equivalence, and get in particular (...)
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  2. Heyting Mereology as a Framework for Spatial Reasoning.Thomas Mormann - 2013 - Axiomathes 23 (1):137- 164.
    In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspects of spatial reasoning and therefore has to be enhanced by further non-mereological concepts to overcome its congenital limitations. The allegedly unmereological (...)
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  3.  49
    Negotiating the World: Some philosophical considerations on dealing with differential academic language proficiency in schools.Roel Van Goor & Frieda Heyting - 2008 - Educational Philosophy and Theory 40 (5):652-665.
    Differential academic language proficiency is an issue of major educational concern, bearing on problems varying from pupil performance, to social prospects, and citizenship. In this paper we develop a conception of the language‐acquiring subject, and we discuss the consequences for understanding differential language proficiency in schools. Starting from Wittgenstein's meaning‐as‐use theory we show that learning a language requires an activity that relates the subject both to the community of language users, and to the things language is about. In opposition to (...)
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  4.  19
    Negotiating the world: Some philosophical considerations on dealing with differential academic language proficiency in schools.Frieda Heyting Roel van Goor - 2008 - Educational Philosophy and Theory 40 (5):652-665.
    Differential academic language proficiency is an issue of major educational concern, bearing on problems varying from pupil performance, to social prospects, and citizenship. In this paper we develop a conception of the language-acquiring subject, and we discuss the consequences for understanding differential language proficiency in schools. Starting from Wittgenstein's meaning-as-use theory we show that learning a language requires an activity that relates the subject both to the community of language users, and to the things language is about. In opposition to (...)
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  5.  16
    Heyting Algebras: Duality Theory.Leo Esakia - 2019 - Cham, Switzerland: Springer Verlag.
    This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key (...)
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  6.  67
    Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.
    Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
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  7.  20
    Linear Heyting algebras with a quantifier.Laura Rueda - 2001 - Annals of Pure and Applied Logic 108 (1-3):327-343.
    A Q -Heyting algebra is an algebra of type such that is a Heyting algebra and the unary operation ∇ satisfies the conditions ∇0=0, a ∧∇ a = a , ∇=∇ a ∧∇ b and ∇=∇ a ∨∇ b , for any a , b ∈ H . This paper is devoted to the study of the subvariety QH L of linear Q -Heyting algebras. Using Priestley duality we investigate the subdirectly irreducible linear Q -Heyting (...)
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  8.  7
    On Heyting Algebras with Negative Tense Operators.Federico G. Almiñana, Gustavo Pelaitay & William Zuluaga - 2023 - Studia Logica 111 (6):1015-1036.
    In this paper, we will study Heyting algebras endowed with tense negative operators, which we call tense H-algebras and we proof that these algebras are the algebraic semantics of the Intuitionistic Propositional Logic with Galois Negations. Finally, we will develop a Priestley-style duality for tense H-algebras.
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  9. Bi-Heyting algebras, toposes and modalities.Gonzalo E. Reyes & Houman Zolfaghari - 1996 - Journal of Philosophical Logic 25 (1):25 - 43.
    The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.
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  10.  46
    Inquisitive Heyting Algebras.Vít Punčochář - 2021 - Studia Logica 109 (5):995-1017.
    In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures and provide several alternative characterizations of these algebras. For instance, it is shown that a (...)
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  11.  48
    Symmetrical Heyting algebras with a finite order type of operators.Luisa Iturrioz - 1995 - Studia Logica 55 (1):89 - 98.
    The main purpose of this paper is to introduce a class of algebraic structures related to many-valued ukasiewicz algebras. They are symmetrical Heyting algebras with a set of modal operators indexed by a finite completely symmetric poset. A representation theorem is given for these (not functionally complete) algebras.
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  12.  30
    Heyting Algebras with a Dual Lattice Endomorphism.Hanamantagouda P. Sankappanavar - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (6):565-573.
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  13.  20
    Heyting Algebras with a Dual Lattice Endomorphism.Hanamantagouda P. Sankappanavar - 1987 - Mathematical Logic Quarterly 33 (6):565-573.
  14.  23
    Heyting Algebras with Operators.Yasusi Hasimoto - 2001 - Mathematical Logic Quarterly 47 (2):187-196.
    In this paper, we will give a general description of subdirectly irreducible Heyting algebras with operators under some weak conditions, which includes the finite case, the normal case and the case for Boolean algebras with diamond operator. This can be done by normalizing these operators. This answers the question posed in Wolter [4].
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  15.  14
    Symmetrical Heyting algebras with operators.Luisa Iturrioz - 1983 - Mathematical Logic Quarterly 29 (2):33-70.
  16.  39
    Finitely generated free Heyting algebras: the well-founded initial segment.R. Elageili & J. K. Truss - 2012 - Journal of Symbolic Logic 77 (4):1291-1307.
    In this paper we describe the well-founded initial segment of the free Heyting algebra ������α on finitely many, α, generators. We give a complete classification of initial sublattices of ������₂ isomorphic to ������₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ������α.
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  17.  66
    Expansions of Semi-Heyting Algebras I: Discriminator Varieties.H. P. Sankappanavar - 2011 - Studia Logica 98 (1-2):27-81.
    This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give (...)
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  18.  39
    Locally Finite Reducts of Heyting Algebras and Canonical Formulas.Guram Bezhanishvili & Nick Bezhanishvili - 2017 - Notre Dame Journal of Formal Logic 58 (1):21-45.
    The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting (...)
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  19.  17
    Semi-Heyting Algebras and Identities of Associative Type.Juan M. Cornejo & Hanamantagouda P. Sankappanavar - 2019 - Bulletin of the Section of Logic 48 (2).
    An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ ≈ x ∧ y, x ∧ ≈ x ∧ [ → ], and x → x ≈ 1.
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  20.  22
    Computable Heyting Algebras with Distinguished Atoms and Coatoms.Nikolay Bazhenov - 2023 - Journal of Logic, Language and Information 32 (1):3-18.
    The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there (...)
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  21.  98
    Varieties of monadic Heyting algebras. Part I.Guram Bezhanishvili - 1998 - Studia Logica 61 (3):367-402.
    This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].
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  22.  22
    Contrapositionally complemented Heyting algebras and intuitionistic logic with minimal negation.Anuj Kumar More & Mohua Banerjee - 2023 - Logic Journal of the IGPL 31 (3):441-474.
    Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension (...)
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  23.  3
    Degree of Satisfiability in Heyting Algebras.Benjamin Merlin Bumpus & Zoltan A. Kocsis - forthcoming - Journal of Symbolic Logic:1-19.
    We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top $ is no larger than $\frac {2}{3}$. Finally, (...)
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  24.  38
    Free‐decomposability in varieties of semi‐Heyting algebras.Manuel Abad, Juan Manuel Cornejo & Patricio Díaz Varela - 2012 - Mathematical Logic Quarterly 58 (3):168-176.
    In this paper we prove that the free algebras in a subvariety equation image of the variety equation image of semi-Heyting algebras are directly decomposable if and only if equation image satisfies the Stone identity.
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  25.  19
    The Automorphism Group of the Fraïssé Limit of Finite Heyting Algebras.Kentarô Yamamoto - 2023 - Journal of Symbolic Logic 88 (3):1310-1320.
    Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.
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  26.  79
    Finitely generated free Heyting algebras.Fabio Bellissima - 1986 - Journal of Symbolic Logic 51 (1):152-165.
    The aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of "special elements" of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.
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  27.  62
    Varieties of monadic Heyting algebras part II: Duality theory.Guram Bezhanishvili - 1999 - Studia Logica 62 (1):21-48.
    In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting algebras (...)
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  28.  34
    n‐linear weakly Heyting algebras.Sergio A. Celani - 2006 - Mathematical Logic Quarterly 52 (4):404-416.
    The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras (...)
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  29.  17
    On self‐distributive weak Heyting algebras.Mohsen Nourany, Shokoofeh Ghorbani & Arsham Borumand Saeid - 2023 - Mathematical Logic Quarterly 69 (2):192-206.
    We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the (...)
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  30.  13
    Hyper-MacNeille Completions of Heyting Algebras.J. Harding & F. M. Lauridsen - 2021 - Studia Logica 109 (5):1119-1157.
    A Heyting algebra is supplemented if each element a has a dual pseudo-complement \, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the (...)
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  31.  18
    Lukasiewicz and Symmetrical Heyting Algebras.Luisa Iturrioz - 1976 - Mathematical Logic Quarterly 23 (7‐12):131-136.
  32.  32
    Characteristic Formulas of Partial Heyting Algebras.Alex Citkin - 2013 - Logica Universalis 7 (2):167-193.
    The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with (...)
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  33.  26
    Lukasiewicz and Symmetrical Heyting Algebras.Luisa Iturrioz - 1977 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 23 (7-12):131-136.
  34.  24
    Quasivarieties of Heyting algebras.Andrzej Wronski - 1981 - Bulletin of the Section of Logic 10 (3):128-131.
  35.  39
    On some Classes of Heyting Algebras with Successor that have the Amalgamation Property.José L. Castiglioni & Hernán J. San Martín - 2012 - Studia Logica 100 (6):1255-1269.
    In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IPC S (n), associated with these varieties.We use that every algebra in any of the varieties of S-algebras studied (...)
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  36.  44
    Varieties of monadic Heyting algebras. Part III.Guram Bezhanishvili - 2000 - Studia Logica 64 (2):215-256.
    This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice (MHA) of all varieties of monadic Heyting algebras. For every n , we introduce and investigate varieties of depth n and cluster n, and present two partitions of (MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of (MHA) and investigate finite and critical varieties of monadic (...) algebras in detail. In particular, we prove that there exist exactly thirteen critical varieties in (MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of (MHA) is also given. All these provide us with a satisfactory insight into (MHA). Since (MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC. (shrink)
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  37.  15
    Frontal Operators in Weak Heyting Algebras.Sergio A. Celani & Hern?N. J. San Mart?N. - 2012 - Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator r preserving finite meets which also satisfies the equation?? b V, for all a,b? A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will (...)
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  38.  20
    Remarks on Heyting algebras with tense operators.A. V. Figallo & G. Pelaitay - 2012 - Bulletin of the Section of Logic 41 (1/2):71-74.
  39.  8
    On decidable varieties of Heyting algebras.Devdatt P. Dubhashi - 1992 - Journal of Symbolic Logic 57 (3):988-991.
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  40.  8
    Expansions of Dually Pseudocomplemented Heyting Algebras.Christopher J. Taylor - 2017 - Studia Logica 105 (4):817-841.
    We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally (...)
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  41.  21
    Expansions of Dually Pseudocomplemented Heyting Algebras.Christopher J. Taylor - 2017 - Studia Logica 105 (4):817-841.
    We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally (...)
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  42. Decidability problem for finite Heyting algebras.Katarzyna Idziak & Pawel M. Idziak - 1988 - Journal of Symbolic Logic 53 (3):729-735.
    The aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.
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  43. Decision methods for linearly ordered Heyting algebras.Sara Negri & Roy Dyckhoff - 2006 - Archive for Mathematical Logic 45 (4):411-422.
    The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic.
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  44. On a Generalization of Heyting Algebras I.Amirhossein Akbar Tabatabai, Majid Alizadeh & Masoud Memarzadeh - forthcoming - Studia Logica:1-45.
    \(\nabla \) -algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems. In a series of two papers, we will systematically study the algebro-topological properties of different varieties of \(\nabla \) -algebras. In the present paper, we start with investigating the structure of these varieties by characterizing their subdirectly irreducible and simple elements. Then, we prove (...)
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  45.  47
    On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond.Luck Darnière & Markus Junker - 2010 - Archive for Mathematical Logic 49 (7-8):743-771.
    We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in (...)
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  46.  95
    Varieties of three-valued Heyting algebras with a quantifier.M. Abad, J. P. Díaz Varela, L. A. Rueda & A. M. Suardíaz - 2000 - Studia Logica 65 (2):181-198.
    This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in (...)
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  47.  29
    Varieties of Three-Values Heyting Algebras with a Quantifier.Manuel Abad, J. P. Diaz Varela & L. A. Rueda - 2000 - Studia Logica 65 (2):181-198.
    This paper is devoted to the study of some subvarieties of the variety Q of Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q subscript 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Q is far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible (...)
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  48.  38
    On Some Compatible Operations on Heyting Algebras.Rodolfo Cristian Ertola Biraben & Hernán Javier San Martín - 2011 - Studia Logica 98 (3):331-345.
    We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one (...)
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  49.  57
    Subalgebras of Heyting and De Morgan Heyting Algebras.Valeria Castaño & Marcela Muñoz Santis - 2011 - Studia Logica 98 (1-2):123-139.
    In this paper we obtain characterizations of subalgebras of Heyting algebras and De Morgan Heyting algebras. In both cases we obtain these characterizations by defining certain equivalence relations on the Priestley-type topological representations of the corresponding algebras. As a particular case we derive the characterization of maximal subalgebras of Heyting algebras given by M. Adams for the finite case.
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  50.  60
    Frontal Operators in Weak Heyting Algebras.Sergio A. Celani & Hernán J. San Martín - 2012 - Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point (...)
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