A model-theoretic analysis of Fidel-structures for mbC Marcelo E. Coniglio Institute of Philosophy and the Humanities (IFCH), and Centre for Logic, Epistemology and the History of Science (CLE) University of Campinas (UNICAMP), Brazil Email: coniglio@cle.unicamp.br Aldo Figallo-Orellano Centre for Logic, Epistemology and the History of Science (CLE) University of Campinas (UNICAMP), Brazil and Departament of Matematics, National University of the South (UNS) Bahıa Blanca, Argentina Email: aldofigallo@gmail.com Abstract In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to analyze the class of mbC-structures. Thus, substructures, union of chains, direct products, direct limits, congruences and quotient structures can be analyzed under this perspective. In particular, a Birkhoff-like representation theorem for mbC-structures as subdirect poducts in terms of subdirectly irreducible mbC-structures is obtained by adapting a general result for first-order structures due to Caicedo. Moreover, a characterization of all the subdirectly irreducible mbC-structures is also given. An alternative decomposition theorem is obtained by using the notions of weak substructure and weak isomorphism considered by Fidel for Cn-structures. 1 Paraconsistency and non-deterministic semantics Paraconsistency is the study of logic systems having a negation ¬ which is not explosive, that is, there exist formulas α and β in the language of the logic such that β is not derivable from the contradictory set {α,¬α}. In other words, the logic has contradictory but non-trivial theories. There are several approaches to paraconsistency in the literature since the introduction in 1948 of Jaskowski's system of Discussive logic (see [20]), such as Relevant logics, Adaptive logics, 1 Many-valued logics, and many others.1 The well-known 3-valued logic LP (Logic of Paradox) was introduced by G. Priest in [24] with the aim of formalizing the philosophical perspective underlying G. Priest and R. Sylvan's Dialetheism (see, for instance, [28] and [25]). As it is well-known, the main thesis behind Dialetheism is that there are true contradictions, that is, that some sentences can be both true and false at the same time and in the same way. Since then, the logic LP was intensively studied and developed by several authors poposing, for instance, extensions to first-order languages and applications to Set Theory (see, among others, [30, 31, 23]). The publication in 1963 of N. da Costa's Habilitation thesis Sistemas Formais Inconsistentes (Inconsistent Formal Systems, in Portuguese, see [15]) constitutes a landmark in the history of paraconsistency. In that thesis, da Costa introduces the hierarchy Cn (for n ≥ 1) of Csystems. This approach to paraconsistency differs from others, as it is based on the idea of locally recovering the classical reasoning (in particular, the explosion law for negation) by means of a derived unary connective of well-behavior, (*)◦. Being so, a contradiction is not explosive in general in such systems (namely, α,¬α 0 β for some α and β). But assuming additionally that α is well-behaved, then such a contradiction must be trivializing; namely, α,¬α, α◦ ` β for every β. The idea of C-systems was afterwards generalized by W. Carnielli and J. Marcos in [12] through the class of Logics of Formal Inconsistency, in short LFIs. In such logics, da Costa's well-behavior derived connective (*)◦ is replaced by a (possibly primitive) consistency unary conective ◦. The basic idea of LFIs, as in da Costa's C-systems, is that α,¬α 0 β in general, but α,¬α, ◦α ` β always. Of course the C-systems are particular cases of LFIs. Giving a semantical interpretation for C-systems, and for LFIs in general, is not a simple task: most of the LFIs introduced in the literature (see [12, 10, 9]) are not algebraizable by means of the standard techniques of algebraic logic (including Blok and Pigozzi's method, see [5]). Being so, the use of semantics of a non-deterministic character have shown to be a useful alternative way for dealing with LFIs. Several non-deterministic semantical tools were introduced in the literature in order to analyze such systems, allowing so decision procedures for them: non-truthfunctional bivaluations (proposed by N. da Costa and E. Alves in [16, 17]), possible-translations semantics (proposed by W. Carnielli in [7]), and non-deterministic matrices (or Nmatrices), proposed by A. Avron and I. Lev in [1]. In particular, the Nmatrix semantics can be analyzed from a general (non-deterministic) algebraic perspective through the notion of swap structures (see [9, Chapter 6] and [14]). However, as shown in [2], not every LFI can be characterized by a single finite Nmatrix. In particular, da Costa's logic C1 cannot be characterized by a single finite Nmatrix. Another interesting approach to non-deterministic semantics for non-classical logics was proposed for G. Priest in [26], through the notion of plurivalent semantics. Let M = 〈A, D〉 be a matrix semantics for a propositional signature Ξ (that is, A is an algebra over Ξ and D is a non-empty subset of the domain A of A, called the set of designated values). Then, a plurivalent semantics over M is a pair M. = 〈M, .〉 such that . ⊆ V × A is a relation from the set V of propositional variables to A. It is assumed that, for every p ∈ V, there is a ∈ A such that 1For a good introductory article on Paraconsistency see [29] and the references therein. 2 p . a.2 Given M., a non-deterministic (or plurivalent) interpretation [*].M for the algebra of formulas over Ξ generated by V is defined recursively: [p].M = {a ∈ A : p . a}, if p ∈ V; and [c(φ1, . . . , φn)] . M = ⋃ {cA(a1, . . . , an) : ai ∈ [φi].M for 1 ≤ i ≤ n}, for every n-ary connective c in Ξ. Given a set of formulas Γ ∪ {φ}, Γ infers φ in M., denoted by Γ |=.M φ, if [φ].M ∩D 6= ∅ whenever [β].M ∩ D 6= ∅ for every β ∈ Γ. The plurivalent consequence relation |=Mp generated from M is defined as expected: Γ |=Mp φ iff Γ |=.M φ for every M.. Priest have shown in [26] how to produce a family of plurivalent logics related to the first-degree entailment FDE. Additionally, in [27] he applies the notion of plurivalent semantics in order to analyze Indian Buddhist logic from the pespective of formal logic. Before all these efforts, M. Fidel have proved in 1969 (despite it was only published in 1977, see [19]), for the first time in the literature, the decidability of da Costa's calculi Cn by means of a novel algebraic-relational class of structures called Cn-structures. Afterwards, this kind of structure was called Fidel-structures or F-structures (see [22]). A Cn-structure is a triple 〈A, {Na}a∈A, {N (n)a }a∈A〉 such that A is a Boolean algebra with domain A and each Na and N (n) a is a non-empty subset of A. The intuitive meaning of b ∈ Na and c ∈ N (n)a is that b and c are possible values for the paraconsistent negation ¬a of a and for the well-behavior a◦ of a in Cn, respectively. The use of relations instead of functions for interpreting the paraconsistent negation ¬ and the well-behavior connective (*)◦ of the Cn calculi is justified by the fact that these connectives are not truth-functional. That is, they cannot be characterized by means of truth-functions, and so the use of relations seems to be a good choice, constituting the first non-deterministic semantics for da Costa's calculi Cn proposed in the literature. Given that every Cn can be characterized by F-structures over the 2-element Boolean algebra, as Fidel has shown in [19], this result evidences the greater expressive power of F-structures with respect to Nmatrices. A discussion about this topic can be found in [9, Sections 6.6 and 6.7]. Fidel-structures can be defined for a wide class of logics, not only paraconsistent ones. Concerning LFIs, Fidel structures were defined in [8] and [9, Chapter 6] for several LFIs which are weaker than da Costa's C1, starting from a basic but very interesting LFI called mbC. This paper proposes the study of Fidel-structures for mbC from the point of view of Model Theory. Under this perspective, substructures, union of chains, direct products, direct limits, congruences and quotient structures can be defined. In particular, a generalization to first-order structures of Birkhoff's representation theorem for algebras, due to Caicedo, can be obtained for mbC-structures (see Theorem 56). Moreover, a characterization of the subdirectly irreducible mbC-structures is given in Theorem 55. This representation theorem is compared to a similar one, obtained by Fidel for the calculi Cn. 2 The logic mbC The logic mbC is the most basic LFI analyzed by Carnielli, Coniglio and Marcos in [10], which later on it was studied by several authors. This logic is based on propositional positive classical 2The general case in which this conditions is dropped is briefly analyzed by Priest in [26]. 3 logic and, despite its apparent simplicity, it enjoys extremely interesting features. This section is devoted to briefly describe the logic mbC for the reader's convenience. Consider the propositional signature Σ = {∧,∨,→,¬, ◦} for LFIs, and let V = {pi : i ∈ N} be a denumerable set of propositional variables. The algebra of propositional formulas over Σ generated by V will be denoted by For(Σ). Definition 1 (Logic mbC, [10]) The calculus mbC over the language For(Σ) is defined by means of the following Hilbert calculus : (A1) α→ (β → α) (A2) (α→ β)→ ((α→ (β → γ))→ (α→ γ)) (A3) α→ (β → (α ∧ β)) (A4) (α ∧ β)→ α (A5) (α ∧ β)→ β (A6) α→ (α ∨ β) (A7) β → (α ∨ β) (A8) (α→ γ)→ ((β → γ)→ ((α ∨ β)→ γ)) (A9) α ∨ (α→ β) (A10) α ∨ ¬α (A11) ◦α→ (α→ (¬α→ β)) (MP) From α and α→ β infer β It is worth noting that mbC is obtained from a calculus for the positive cassical logic CPL+ by adding the axiom schemas (A10) and (A11), concerning the paraconsistent negation ¬ and the consistency operator ◦. As observed in [12], each calculus Cn is a particular case of LFI in which the consistency connective ◦ is defined in term of the others: for instance, ◦α def= α◦ = ¬(α∧¬α) in C1. The logic C1 can be seen as an axiomatic extension of mbC up to language (see [10, 9]). Being weaker than C1, the logic mbC cannot be characterized by any standard algebraic semantics, even in the wide sense of Blok-Pigozzi [5]. Moreover, it cannot be characterized by a single finite logical matrix. From this, it is clear that alternative semantics for mbC are necessary. In [8] and [9], Fidel-structures for mbC and for several axiomatic extensions of it were presented. However, the formal study of the properties of the class of F-structures for such LFIs was never developed, in contrast with the study of the F-structures for Cn carry out by Fidel in [19]. 4 In the following sections the class of F-structures for mbC will be studied as a basic but relevant example through an original approach based on elementary concepts of Model Theory and Category Theory.3 3 Fidel-structures for mbC In this section, the class of F-structures for mbC introduced in [8] and [9] will be recast in the language of first-order structures. This constitutes a novel approach to Fidel-structures in general. Consider classical first-order theories defined over first-order signatures based on the following logical symbols: the connectives ∧, ∨, → and ∼ (for conjunction, disjunction, implication and negation, respectively), the quantifiers ∀ and ∃, and the symbol ≈ for the equality predicate which is always interpreted as the identity relation. Assume also a denumerable set Vind = {vi : i ∈ N} of variables. The letters u, w, z will be used to refer to arbitrary variables. If Ξ is a first-order signature then Ξ-str denotes the category of Ξ-structures, that is, the category of first-order structures over the signature Ξ. Definition 2 The signature for F-structures for LFIs is the first-order signature Θ composed by the following symbols: (i) Two binary predicate symbols N and O, for the paraconsistent negation and the consistency operator, respectively; (ii) Two binary function symbols u and t, and an unary function symbol −, for Boolean meet, Boolean join and Boolean complement, respectively; (iii) Two constant symbols 0 and 1 for the bottom and the top element, respectively. Observe that the subsignature ΘBA of Θ obtained by dropping the predicate symbols N , O is the usual signature for Boolean algebras. Definition 3 An F-structure for mbC (in short, an mbC-structure) is a Θ-first order structure E = 〈A,uE ,tE ,−E ,0E ,1E , NE , OE〉 such that: (a) the ΘBA-reduct A = 〈A,uE ,tE ,−E ,0E ,1E〉 of E is a Boolean algebra; that is, A satisfies the usual equations axiomatizing Boolean algebras in the signature ΘBA, see for instance [13, Example 1.4.3];4 3For general notions on Model Theory and Category Theory the reader can consult [13] and [21], respectively. 4In difference with several authors, we admit the trivial one-element Boolean algebra, see Remark 39. 5 (b) E satisfies the following Θ-sentences: (i) ∀u∃wN(u,w), (ii) ∀u∃wO(u,w), (iii) ∀u∀w(N(u,w)→ (u t w ≈ 1)), (iv) ∀u∀w(N(u,w)→ ∃z(O(u, z) ∧ ((u u w u z) ≈ 0)). The class of mbC-structures will be denoted by FmbC. Notation 4 From now on, an mbC-structure E will be denoted by E = 〈A, NE , OE〉 such that A = 〈A,uE ,tE ,−E ,0E ,1E〉 is a Boolean algebra. We will frequently write #A instead of #E for # ∈ {u,t,−,0,1}. Given an mbC-structure E and a, b ∈ A, alternatively we can write b ∈ NEa instead of NE(a, b). Similar notation will be adopted for the predicate symbol O, alternatively writing b ∈ OEa instead of OE(a, b). This is in line with the traditional presentation of Fstructures where the non-truth-functional unary connectives are interpreted by families of nonempty subsets of the domain of the structure (recall the definition of Cn-structure outlined in Section 1). The intuitive reading for b ∈ NEa and c ∈ OEa is that b is a possible negation ¬a of a, and that c is a possible consistency ◦a of a coherent with b. This is justified by Definition 6 below. Remark 5 Observe that NE(0E , b) iff b = 1E , for every F-structure E for mbC. On the other hand, if (1E ,1E) ∈ NE then (1E ,0E) must belong to OE . Indeed, it must exists z ∈ A such that (1E , z) ∈ OE and 1E u 1E u z = 0E . Therefore z = 0E and so (1E ,0E) ∈ OE . Recall now the semantics for mbC defined from F-structures (see [8, 9]). Given an mbCstructure E , consider the Boolean implication defined as usual by a ⇒E b def= −Ea tE b for every a, b ∈ A. For # ∈ {∧,∨,→} let #E be the corresponding operation in E , that is: ∧E = uE ; ∨E = tE ; and →E =⇒E . As stated above, For(Σ) denotes the algebra of formulas for the logic mbC. Definition 6 A valuation over an mbC-structure E is a map v : For(Σ) → A satisfying the following properties, for every formulas α and β: (1) v(α#β) = v(α)#Ev(β), for # ∈ {∧,∨,→}; (2) v(¬α) ∈ NEv(α) (that is, N E(v(α), v(¬α))); (3) v(◦α) ∈ OEv(α) (that is, v(◦α) is such that O E(v(α), v(◦α))) and v(α) uE v(¬α) uE v(◦α) = 0E . 6 Observe that item (3) of the previous definition is well-defined by item (b)(iv) of Definition 3 and item (2) of the last definition. The semantical consequence relation associated to F-structures for mbC is naturally defined: Definition 7 Let Γ ∪ {α} ⊆ For(Σ) be a finite set of formulas. (i) Given a Fidel-structure E for mbC, we say that α is a semantical consequence of Γ (w.r.t. E), denoted by Γ mbCE α, if, for every valuation v over E: v(α) = 1 whenever v(γ) = 1 for every γ ∈ Γ. (ii) We say that α is a semantical consequence of Γ (w.r.t. Fidel-structures for mbC), denoted by Γ mbCF α, if Γ mbC E α for every F-structure E for mbC. Theorem 8 (Soundness and completeness of mbC w.r.t. F-structures, [8, 9]) Let Γ ∪ {α} be a finite set of formulas in For(Σ). Then: Γ `mbC α iff Γ mbCF α. Moreover, let A2 be the two-element Boolean algebra with domain {0, 1}, and let mbCF2 be the semantical consequence relation with respect to the class of mbC-structures defined over A2. By adapting the proof of [9, Theorem 6.2.16] it is easy to obtain the following result: Theorem 9 (Soundness and completeness of mbC w.r.t. F-structures over A2) Let Γ ∪ {α} be a finite set of formulas in For(Σ). Then: Γ `mbC α iff Γ mbCF2 α. The latter result give us a decision procedure for checking validity in mbC. 4 On the axiomatization of mbC-structures In this section we briefly discuss the class FmbC of F-structures from the point of view of the syntactic form of its axioms (recall Definition 3). From this, some well-known results from Model Theory can be applied to FmbC in a direct way, as we shall see along this paper. Recall from [13, Page 407] that a basic Horn formula over a first-order signature Ξ is a formula of the form σ1 ∨ . . .∨ σn (for n ≥ 1) such that at most one formula is atomic and the rest is the negation of an atomic formula over Ξ. In particular, formulas of the form σ1 ∧ . . .∧ σn → σn+1, where each σi is atomic, are (logically equivalent to) basic Horn formulas. A Horn formula over Ξ is any formula over Ξ built up from basic Horn formulas by using exclusively the connective ∧ and the quantifiers ∀ and ∃. A Horn sentence is a Horn formula with no free variables. Remark 10 (FmbC as a Horn theory) From Definition 3, it is easy to see that FmbC can be axiomatized by means of Horn sentences. Indeed, the axioms of Boolean algebras are sentences of the form ∀x1 * * * ∀xnσ, where σ is an atomic formula. On the other hand, axioms (b)(i) and (b)(ii) are of the form ∀u∃wσ, where σ is an atomic formula. Axiom (b)(iii) is of the form ∀u∀w(σ1 → σ2), where σ1 and σ2 are atomic formula. Finally, axiom (b)(iv) is logically equivalent to a sentence of the form ∀u∀w∃z((σ1 → σ2) ∧ (σ1 → σ3)) where each σi is atomic. This means that FmbC can be axiomatized by Horn sentences. 7 Now, recall from [13, Pages 142-143] that an universal-existencial sentence, or a ∀∃-sentence, or a Π02-sentence over a first-order signature Ξ, is a sentence of the form ∀x1 * * * ∀xn∃y1 * * * ∃ykσ, where σ is a formula over Ξ without quantifiers. Remark 11 (FmbC as a Π02-theory) It is immediate to see that the class of mbC-structures can be axiomatized by means of ∀∃-sentences (that is, Π02-sentences). Indeed, the axioms of Boolean algebras are of the form ∀x1 * * * ∀xnσ, where σ is an atomic formula. On the other hand, axioms (b)(i) and (b)(ii) are of the form ∀u∃wσ, where σ is an atomic formula. Axiom (b)(iii) is of the form ∀u∀wσ, where σ is without quantifiers. Finally, axiom (b)(iv) is logically equivalent to a sentence of the form ∀u∀w∃zσ where σ has not quantifiers. This means that FmbC can be axiomatized by Π02-sentences over signature Θ. Finally, it is worth noting that FmbC can also be axiomatized by sentences over signature Θ of the form ∀x1 * * * ∀xn(σ1 → ∃y1 * * * ∃ykσ2)), where σ1 and σ2 are positive formulas (that is, built up from conjunctions and disjunctions only) without quantifiers. The fact that FmbC can be axiomatized by sentences of a special form will be used along this paper, as it will be pointed out. 5 Homomorphisms and substructures From the definitions of the previous section, the category of mbC-structures can be defined in a natural way. Definition 12 Let E = 〈A, NE , OE〉 and E ′ = 〈A′, NE ′ , OE ′〉 be two mbC-structures. An mbChomomorphism h from E to E ′ is a homomorphism h : E → E ′ in the category of Θ-structures. Remark 13 By definition, an mbC-homomorphism h : E → E ′ is a function h : A → A′ satisfying the following conditions, for every a, b ∈ A: (i) h : A → A′ is a homomorphism between Boolean algebras, (ii) if NE(a, b) then NE ′ (h(a), h(b)); (iii) if OE(a, b) then OE ′ (h(a), h(b)). From the notions above, it is defined a category FmbC of mbC-structures having the class FmbC of F-structures as objects and with mbC-homomorphisms as its morphisms. Clearly, it is a full subcategory of the category Θ-str of Θ-structures. For every mbC-structure E = 〈A, NE , OE〉, the identity homomorphism given by the identity mapping over A will be denoted by idE : E → E . If h : E → E ′ and h′ : E ′ → E ′′ are two homomorphisms then the composite homomorphism from E to E ′′ will be denoted by h′ ◦h. As a consequence of the definitions, wellknown basic notions and results from Model Theory can be applied to FmbC. The only detail to be taken into account in the constructions is that mbC-structures are first-order structures over Θ satisfying certain Θ-sentences, as it was discussed in Section 4. 8 Definition 14 Let E = 〈A, NE , OE〉 and E ′ = 〈A′, NE ′ , OE ′〉 be two mbC-structures. The structure E is said to be a substructure of E ′, denoted by E ⊆ E ′, if the following conditions hold: (i) A is a Boolean subalgebra of A′ (which will be denoted as A ⊆ A′),5 (ii) NE = NE ′ ∩ (A′)2 and OE = OE ′ ∩ (A′)2. Remark 15 The notion of substructure considered in Model Theory (and, in particular, in Definition 14) differ slightly from that used in some areas of Mathematics. For instance, in Graph Theory a graph H = 〈V,E〉 (where V and E denote, respectively, the set of vertices and edges) is said to be a subgraph of another grap G = 〈V ′, E′〉 provided that V ⊆ V ′ and E ⊆ E′. Then, it can be possible to have a subgraph H of a graph G such that H is not a substructure of G seen as first-order structures (over the signature of graphs having, besides the equality predicate symbol ≈, a binary predicate symbol for the edge relation). Indeed, H is a substructure of G if and only if H is what is called in Graph Theory a induced subgraph of G. In Category Theory and in Universal Algebra the subobjects (the subalgebras, respectively) are characterized by means of the the notion of monomorphism. Recall from Category Theory (see, for instance, [21]) that a monomorphism in a category C is a homomorphism h : A → B in C such that, for every pair of parallel homomorphisms h′, h′′ : C → A for which h ◦ h′ = h ◦ h′′ in C, it is the case that h′ = h′′. On the other hand, recall the following notion from Model Theory (see, for instance, [13]): Definition 16 Let A and A′ be two first-order structures over a signature Ξ. A homomorphism h : A → A′ in Ξ-str is an embedding if it is injective and, for every n-ary predicate symbol P and every (a1, . . . , an) ∈ |A|n, (a1, . . . , an) ∈ PA if and only if (h(a1), . . . , h(an)) ∈ PA ′ .6 It is well-known that a homomorphism h : A → A′ in Ξ-str is an embedding if and only if it is a monomorphism in the subcategory Ξ-emb of Ξ-str formed by Ξ-structures as objects and embeddings as morphisms. Clearly, if Ξ has only function symbols besides the identity predicate ≈ (this is the case of Universal Algebra) then Ξ-emb is the subcategory of Ξ-str in which every morphism is a monomorphism. Moreover, the induced substructures correspond to the subobjects in Ξ-str . Remark 17 As a direct consequence of the definitions, a homomorphism h : E → E ′ in FmbC is an embedding if and only if it is an injective homomorphism where conditions (ii) and (iii) of Remark 13 are replaced by (ii)' NE(a, b) if and only if NE ′ (h(a), h(b)); 5This means that A ⊆ A′, 0A = 0A ′ , 1A = 1A ′ and, for every a, b ∈ A: a#Ab = a#A ′ b and −Aa = −A ′ a for # ∈ {u,t}. Note that we write sA instead of sE when s correspond to a symbol of the subsignature ΘBA of Θ. 6Since in Model Theory the equality predicate ≈ is considered as a predicate symbol which is always interpreted as the standard equality, the injectivity of an embedding is a consequence of the definition. 9 (iii)' OE(a, b) if and only if OE ′ (h(a), h(b)). Clearly, E ⊆ E ′ if and only if A ⊆ A′ and the inclusion map i : A → A′ induces an embedding i : E → E ′ of Θ-structures. Remark 18 From the definitions above, it is clear that in the category FmbC the monomorphism (that is, the substructures in Θ–str) are not strong enough: in order to obtain a substructure in FmbC the monomorphism must be, in addition, an embedding. Consider, for instance, the mbC-structures E = 〈A2, NE , OE〉 and E ′ = 〈A2, NE ′ , OE ′〉 defined over the twoelement Boolean algebra A2 such that NE = {(0, 1), (1, 0)}, OE = {(0, 1), (1, 1)} and NE ′ = OE ′ = {(0, 1), (1, 0), (1, 1)}. Clearly, the identity h : {0, 1} → {0, 1} induces a monomorphism h : E → E ′ in FmbC (since it is an injective homomorphism in FmbC). But E is not a substructure of E' since, for instance, (1, 0) ∈ OE ′ ∩ |E|2 but (1, 0) /∈ OE . A weaker notion of substructure was considered in the literature of Model Theory under the name of weak substructures. Thus, A is said to be a weak substructure of A′ in Ξ-str provided that |A| ⊆ |A′| and PA ⊆ PA′ , for every predicate symbol P . This is equivalent to say that the inclusion mapping i : |A| → |A′| induces a homomorphism i : A→ A′ in Ξ-str. For instance, in Graph Theory a graph H is a subgraph of a graph G provided that H is a weak substructure of G, as observed in Remark 15. In FmbC it can be considered a intermediate notion between weak substructures and substructures (as defined in Model Theory): Definition 19 Let E = 〈A, NE , OE〉 and E ′ = 〈A′, NE ′ , OE ′〉 be two mbC-structures. We say that E is a weak substructure of E ′ in FmbC, denoted by E ⊆W E ′, if A is a Boolean subalgebra of A′, NE ⊆ NE ′, and OE ⊆ OE ′. This is equivalent to say that the inclusion map i : A → A′ induces an injective homomorphism i : E → E ′ in FmbC. Remark 20 Observe that if a Ξ-structure A is a substructure of A′ then it is a weak substructure of A′. On the other hand, if E is a substructure of an mbC-structure E' (hence, a weaksubstructure) in the category Θ-str then E is not necessarily an mbC-structure and so it is not necessarily a weak substructure of E ′ in FmbC. For instance, consider the four-element Boolean algebra A4 with domain FOUR = {0, a, b, 1} (observe that A4 is isomorphic, as a Boolean algebra, to the powerset of {0, 1}) and let E ′ = 〈A4, NE ′ , OE ′〉 be the mbC-structure over A4 such that NE ′ = {(0, 1), (a, b), (b, a), (1, a)} and OE ′ = {(0, a), (a, 0), (b, 0), (1, 0)}. Let E = 〈A2, NE , OE〉 be the Θ-structure with domain {0, 1} such that NE = {(0, 1)} = NE ′∩ ( {0, 1} )2 and OE = {(1, 0)} = OE ′ ∩ ( {0, 1} )2 . Then E is a substructure of E' which is not an mbCstructure, hence it is not a weak substructure of E' in FmbC. Finally, it is worth noting that if E and E' are two mbC-structures such that E is a weak substructure of E' then every valuation over E (recall Definition 6) is a valuation over E'. It is interesting to notice that any mbC-structure E over A2 can be seen as a substructure (in FmbC) of an mbC-structure E(A) over A, for any Boolean algebra A with more than two elements: 10 Proposition 21 Let A be a Boolean algebra with more that two elements. Given an mbCstructure E = 〈A2, NE , OE〉 defined over A2 let E(A) def = 〈A, NEA, OEA〉 be the Θ-structure defined as follows: NEA = N E ∪ {(a,∼a) : a ∈ |A| \ {0, 1}}; OEA = O E ∪ {(a, 1) : a ∈ |A| \ {0, 1}}. Then, E(A) is an mbC-structure over A such that E ⊆ E(A) in FmbC. Proof. It is straightforward from the definitions. 2 As a consequence of the latter result, combined with Theorem 9, it can be seen that the logic of the mbC-structures over A is exactly mbC. Theorem 22 (Soundness and completeness of mbC w.r.t. F-structures over a nontrivial Bolean algebra A) Let A be a non-trivial Boolean algebra (that is, 0A 6= 1A). Let

mbCFA be the semantical consequence relation with respect to the class FA of mbC-structures defined over A, and let Γ ∪ {α} be a finite set of formulas in For(Σ). Then: Γ `mbC α iff Γ mbCFA α. Proof. If A = A2 then the result follows by Theorem 9. Assume now that A has more than two elements. The 'only if' part is a consequence of Theorem 8. Now, suppose that Γ 0mbC α. By Theorem 9, there exists an mbC-structure E over A2, and a valuation v over E such that v[Γ] ⊆ {1} but v(α) = 0. By Proposition 21, there exists an mbC-structure E(A) over A such that E ⊆ E(A). In particular, E is a weak substructure of E(A). By the last observation in Remark 20 it follows that v is also a valuation over E(A). This shows that Γ 6 mbCFA α. 2 It is worth noting that Fidel considered in [19] the notion of F-substructures stated in Definition 19 in order to obtain a decomposition result for F-structures for da Costa's calculi Cn in terms of irreducible structures. The adaptation of this result to mbC-structures will be briefly analyzed in Subsection 8.4. Recall that an epimorphism in a category C is a homomorphism h : A → B such that, for every pair of parallel homomorphisms h′, h′′ : B → C such that h′ ◦ h = h′′ ◦ h, it is the case that h′ = h′′, see for instance [21]. Then: Proposition 23 A homomorphism h : E → E ′ is an epimorphism in FmbC if and only if h is onto as a mapping. Recall now that an isomorphism in a category C is a homomorphism h : A→ B such that there exists a homomorphism h′ : B → A where h′ ◦ h = idA and h ◦ h′ = idB, see for instance [21]. Then: Proposition 24 A homomorphism h : E → E ′ is an isomorphism in FmbC if and only if h is an embedding which is onto, that is, h is a bijective embedding. 11 This means that, in the category FmbC of mbC-structures, h is an isomorphism if and only if it is both a monomorphism and an epimorphism. 6 Union of chains of mbC-structures As a consequence of the proposed approach to F-structures as being a class of Θ-structures axiomatized by a set of sentences of a certain form (recall Section 4), some basic results from Model Theory can be applied to its study. In this section, the union of chains of mbC-structures will be analyzed. Definition 25 A chain of F-structures for mbC is a family (Eλ)λ<μ for an ordinal μ such that Eξ ⊆ Eλ whenever ξ < λ < μ. A chain can be displayed as E0 ⊆ E1 ⊆ . . . ⊆ Eλ ⊆ . . . for λ < μ. From Model Theory, it is known that, for every chain (Eλ)λ<μ of F-structures for mbC (seen as first-order structures), there exists a Θ-structure E which is its union, such that Eλ ⊆ E for every λ < μ. The structure E is defined as follows: (i) A = ⋃ λ<μAλ; (ii) NE = ⋃ λ<μN Eλ and OE = ⋃ λ<μO Eλ ; (iii) for a, b ∈ A: a uE b = a uEλ b; a tE b = a tEλ b; and −Ea = −Eλa, if a, b ∈ Aλ; (iv) 0E = 0E0 and 1E = 1E0 . Observe that items (ii) and (iii) are well-defined given that, for ξ < λ < μ, Eξ ⊆ Eλ and so Aξ ⊆ Aλ. Clearly 0E = 0Eλ and 1E = 1Eλ for every λ < μ. Given that, as observed in Remark 11, the class FmbC of mbC-structures can be axiomatized by means of ∀∃-sentences (that is, Π02-sentences), that class is closed under union of chains. This is a consequence of [13, Theorem 3.2.3]. In other words, the following result holds: Proposition 26 Let (Eλ)λ<μ be a chain of mbC-structures, and let E be its union. Then E is the least mbC-structure having every Eλ as a substructure. The last result can be interpreted in terms of the propositional logics generated by single mbCstructures (see Proposition 28 below). Indeed: recall from Definition 7 the propositional consequence relation mbCE generated by an mbC-structure E .7 Then, the following useful result can be stated: 7The propositional consequence relation mbCK generated by a class K of mbC-structures should not be confused with the first-order consequence relation |=K defined over first-order Θ-sentences as follows: Υ |=K σ if, for every E ∈ K, it holds: E |= σ whenever E |= % for every % ∈ Υ. 12 Lemma 27 Let E and E ′ be two mbC-structures such that E ⊆ E ′. Then mbCE ′ ⊆ mbCE . Proof. Suppose that E ⊆ E ′ and let Γ ∪ {α} be a finite set of formulas in For(Σ) such that Γ mbCE ′ α. Consider a valuation v over E such that v[Γ] ⊆ {1E}. Then, the mapping v : For(Σ) → A′ such that v(γ) = v(γ) for every γ ∈ For(Σ) is a valuation over E ' such that v[Γ] ⊆ {1E ′}. By hypothesis, v(α) = 1E ′ whence v(α) = 1E . This means that Γ mbCE α. 2 From this, Proposition 26 can be interpreted in terms of the propositional logics associated to mbC-structures: Proposition 28 Let (Eλ)λ<μ be a chain of mbC-structures, and let E be its union. Then

mbCE = ⋂ λ<μ mbC Eλ . Proof. It is clear from Lemma 27 that mbCE ⊆ mbCEλ for every λ < μ, whence mbC E ⊆⋂ λ<μ mbC Eμ . Now, let Γ∪{α} be a finite set of formulas in For(Σ) such that Γ 1 mbC E α. Then, there is a valuation v over E such that v[Γ] ⊆ {1E} and v(α) 6= 1E . Since Γ∪ {α} is a finite set, then v[Γ∪ {α}] ⊆ An for some natural number n < μ. Define a mapping v : For(Σ)→ An such that v(γ) = v(γ) whenever γ ∈ Γ ∪ {α} and, for every formula γ 6∈ Γ ∪ {α}: (0) if γ ∈ V then v(γ) is arbitrary; (1) if γ = γ1#γ2 then v(γ) = v(α)# En v(β), for # ∈ {∧,∨,→}; (2) if γ = ¬γ1 then v(γ) is such that NEn(v(γ1), v(γ)); (3) if γ = ◦γ1 then v(γ) is such that OEn(v(γ1), v(γ)) and v(γ1) uEn v(¬γ1) uEn v(γ) = 0En .8 It is clear by the very definition that v is a valuation over En such that v[Γ] ⊆ {1En} and v(α) 6= 1En . This means that Γ 1mbCEn α. Therefore ⋂ λ<μ mbC Eλ ⊆ mbC E . 2 7 Congruences and quotient structures In this section, the well-known lattice isomorphism between the lattice of filters over a Boolean algebra A and the lattice of Boolean congruences over A will be extended to mbC-structures. Additionally, the quotient mbC-structures by mbC-congruences will be defined. Definition 29 Let θ be a relation on an mbC-structure E = 〈A, NE , OE〉. Then θ is said to be an mbC-congruence over E if the following conditions hold: (i) θ is a Boolean congruence over A;9 8In order to define v by induction on the complexity of γ, the complexity measure on For(Σ) must be defined in a way such that ◦β has a complexity degree strictly greater than that of ¬β, for every β ∈ For(Σ). 9That is, θ is an equivalence relation which is preserved by the operations of the Boolean algebra A. 13 (ii) if (x, x′), (y, y′) ∈ θ and NE(x, y) then NE(x′, y′); (iii) if (x, x′), (y, y′) ∈ θ and OE(x, y) then OE(x′, y′). Given a Boolean algebra A, let ConB(A) be the set of Boolean congruences defined on A. It is well-known that the poset (ConB(A),⊆) partially ordered by the inclusion relation is a distributive lattice. Definition 30 Let A be a Boolean algebra, and let F ⊆ A. Then F is a filter over A if the following holds: (i) 1A ∈ F ; (ii) if x, y ∈ F then x u y ∈ F ; and (iii) if x ∈ F and x ≤ y then y ∈ F . We denote by F (A) the set of filters over A. The following is a well-known result: Theorem 31 Given a Boolean algebra A, there exists a lattice isomorphism between F (A) and ConB(A) given by F 7→ R(F ), where R(F ) = {(x, y) ∈ A2 : x u z = y u z for some z ∈ F}. The inverse mapping is given by θ 7→ [1A]θ, where [a]θ denotes the θ-equivalence class of a ∈ A. Definition 32 Given an mbC-structure E = 〈A, NE , OE〉, a set F ⊆ A, is said to be an mbCfilter if the following conditions hold: (i) F is a filter over the Boolean algebra A; (ii) R(F ) verifies conditions (ii) and (iii) of Definition 29, where R(F ) is defined as in Theorem 31. Let us denote by FmbC(E) and by ConmbC(E) the set of mbC-filters and the set of mbCcongruences over a given mbC-structure E , respectively. Thus, the following result holds: Theorem 33 Let E = 〈A, NE , OE〉 be an mbC-structure. Then, there exists a lattice isomorphism between FmbC(E) and ConmbC(E). Proof. It is proved by an easy adaptation of proof of the corresponding result for Boolean algebras. 2 Now, we are going to define quotient mbC-structures. Let E be an mbC-structure, and let θ be an mbC-congruence on it. Then A/θ is a Boolean algebra with the operations induced from A; this Boolean algebra will be denoted by A/θ. Consider the following relations over A/θ induced from E : NE/θ def = {([x]θ, [y]θ) ∈ A/θ ×A/θ : (x, y) ∈ NE} and OE/θ def = {([x]θ, [y]θ) ∈ A/θ ×A/θ : (x, y) ∈ OE}. 14 From Definition 29, it follows that (x, y) ∈ NE if and only if ([x]θ, [y]θ) ∈ NE/θ; the same holds for the predicate O. From this, it is easy to check that E/θ = 〈A/θ,NE/θ, OE/θ〉 is an mbCstructure. Now, consider the canonical projection q : A → A/θ given by q(x) = [x]θ. It is clear that q is a homomorphism of Boolean algebras. Moreover, it is an mbC-homomorphism q : E → E/θ which is onto, that is, an epimorphism in FmbC (by Proposition 23). It is well-known that given a Boolean homomorphism h : A → A′, the relation Ker(h) = {(x, y) ∈ A × A : h(x) = h(y)} is a Boolean congruence. This allows us to prove the socalled First Isomorphism theorem for Boolean algebras. In order to generalize this result to mbC-structures, the homomorphisms must satisfy additional coherence properties: Definition 34 Let h : E → E ′ be an mbC-homomorphism.10 Then h is said to be congruential if it satisfies the following: If NE(x, y) and (x, x′), (y, y′) ∈ Ker(h) then NE(x′, y′), and If OE(x, y) and (x, x′), (y, y′) ∈ Ker(h) then OE(x′, y′). Proposition 35 Let h : E → E ′ be an mbC-homomorphism. Then, h is congruential if and only if the relation Ker(h) = {(x, y) ∈ A×A : h(x) = h(y)} is an mbCcongruence. Proof. It follows from the very definitions. 2 Theorem 36 (First Isomorphism theorem) Let h : E → E ′ be an mbC-homomorphism which is congruential. Then, there is a unique mbC-monomorphism h : E/Ker(h) → E ′ such that h ◦ q = h. In particular, if h is surjective then h is an isomorphism between E/Ker(h) and E ′. Proof. As mentioned above, it is well-known that the Boolean congruence θ = Ker(h) is such that there is a unique Boolean monomomorphism h : A/θ → A′ with h ◦ q = h. Thus, in view of Proposition 35, it suffices to prove that h is an mbC-monomomorphism. The details are left to the reader. 2 8 Birkhoff's decomposition theorems for mbC-structures In this section, Caicedo's generalization to first-order structures (see [6]) of Birkhoff's decomposition theorem for algebras (see [4]) will be applied to the specific case of mbC-structures. 10Observe that, in particular, h : A → A′ is a Boolean homomorphism. 15 In order to adapt Caicedo's result to the present framework, some important constructions over mbC-structures will be analyzed: direct and subdirect poducts, as well as the associated notion of subdirectly irreducible mbC-structures. Finally, in Subsection 8.4, a related decomposition result will be obtained by adapting the notions introduced by Fidel in [19] in terms of weak substructures, recall Definition 19. Once again, the fact that FmbC can be axiomatized by means of sentences of a special form (recall Section 4) will we used, by adapting well-known results from Model Theory. 8.1 Direct products of mbC-structures As observed in Remark 10 the class FmbC of mbC-structures can be axiomatized by means of Horn sentences. Then, as a consequence of [13, Proposition 6.2.2], the class FmbC is closed under (arbitrary) direct products. That is: Theorem 37 The category FmbC has arbitrary products.11 For the reader's convenience the standard construction of products in FmbC will be given in Definition 38 below. Given a family {Ai}i∈I of Boolean algebras, consider the standard construction A = ∏ i∈I Ai of its product such that its support is ∏ i∈I Ai = {x ∈ ( ⋃ i∈I Ai) I : x(i) ∈ Ai for every i ∈ I},12 and the operations are defined pointwise; in particular, 0A(i) = 0Ai and 1A(i) = 1Ai for every i ∈ I. If I = ∅ then ∏ i∈I Ai is the trivial one-point Boolean algebra A⊥ (see Remark 39). The canonical i-projection πi : ∏ i∈I Ai → Ai is given by πi(x) = x(i) for every x ∈ ∏ i∈I Ai and every i ∈ I. Definition 38 Let Ei = 〈Ai, NEi , OEi〉 (for i ∈ I) be an mbC-structure. The direct product of the family {Ei}i∈I is the structure ∏ i∈I Ei = 〈 ∏ i∈I Ai, NΠi∈IEi , OΠi∈IEi〉 defined as follows: (i) ∏ i∈I Ai is the standard product of the family {Ai}i∈I of Boolean algebras; (ii) (x, y) ∈ NΠi∈IEi if and only if (x(i), y(i)) ∈ NEi for every i ∈ I; (iii) (x, y) ∈ OΠi∈IEi if and only if (x(i), y(i)) ∈ OEi for every i ∈ I. It is an easy exercise to check that the construction described in Definition 38 is in fact the product of the given family of mbC-structures (that is, 〈 ∏ i∈I Ei, {πi : i ∈ I}〉 satisfies the universal property of the product in FmbC). 11In addition, also as a consequence of [13, Proposition 6.2.2], it follows that the class FmbC is closed under reduced products; in particular, it is closed under ultraproducts. 12As usual, if I and Z are two sets, then ZI denotes the set of mappings from I to Z. 16 Remark 39 Observe that the product of the empty family of mbC-structures is the terminal object 1⊥ = 〈A⊥, N1⊥ , O1⊥〉 given by the one-element Boolean algebra A⊥ with domain A⊥ = {∗}, and where N1⊥ = O1⊥ = {(∗, ∗)}. Note that 0A⊥ = 1A⊥ = ∗. 8.2 Subdirect products and subdirectly irreducible mbC-structures Recall from Remark 17 the characterization of embeddings in the category of mbC-structures. As discussed in Section 5, the substructures in FmbC are defined by means of embeddings. It is worth noting that in [6] the embeddings are called substructure monomorphisms. Definition 40 (Subdirect product, [6]) For i ∈ I let Ei = 〈Ai, NEi , OEi〉 be an mbCstructure. A subdirect product of the family {Ei}i∈I is an embedding h : E → ∏ i∈I Ei (for some mbC-structure E) such that πi ◦ h is onto for every i ∈ I. It is also called a subdirect decomposition of E. Definition 41 (Subdirectly irreducible structures, [6]) An mbC-structure E is said to be subdirectly irreducible (s.i.) in FmbC if for every subdirect descomposition h : E → ∏ i∈I Ei in FmbC with I 6= ∅, there is an i ∈ I such that πi ◦ h is an isomorphism in FmbC. Theorem 42 ([6, Lemma 1]) Let E = 〈A, NE , OE〉 be an mbC-structure such that E 6= 1⊥. Then, E is subdirectly irreducible in FmbC if and only if there exists a predicate P ∈ {N,O,≈} and (x, y) ∈ A2 such that (x, y) 6∈ P E , and for every onto homomorphism h : E → E ′ in FmbC which is not an isomorphism, (h(x), h(y)) ∈ P E ′. Remark 43 It is interesting to characterize, in terms of Theorem 42, the mbC-structures which are not subdirectly irreducible. Thus, it follows from the previous result that an mbC-structure E = 〈A, NE , OE〉 6= 1⊥ is not subdirectly irreducible in FmbC if only if the following conditions hold: 1. For every (x, y) ∈ A2, (x, y) /∈ NE implies that there exists an onto homomorphism h : E → E ′ in FmbC which is not an isomorphism, such that (h(x), h(y)) /∈ NE ′ (hence E ′ 6= 1⊥); 2. For every (x, y) ∈ A2, (x, y) /∈ OE implies that there exists an onto homomorphism h : E → E ′ in FmbC which is not an isomorphism, such that (h(x), h(y)) /∈ OE ′ (hence E ′ 6= 1⊥); 3. For every (x, y) ∈ A2, x 6= y implies that there exists an onto homomorphism h : E → E ′ in FmbC which is not an isomorphism, such that h(x) 6= h(y) (hence E ′ 6= 1⊥). In the rest of this section, the problem of characterizing the subdirectly irreducible mbCstructures will be considered. 17 Proposition 44 The terminal mbC-structure 1⊥ = 〈A⊥, N1⊥ , O1⊥〉 is subdirectly irreducible in FmbC. Proof. It follows from the definition of 1⊥ and from Definition 41: the only subdirect decomposition of 1⊥ is id1⊥ , where the codomain of id1⊥ is the product 1⊥ of the empty family of mbC-structures. 2 Now, let us consider the general case for mbC-structures defined over an arbitrary Boolean algebra A 6= A⊥. Given a Boolean algebra A, let EmaxA def = 〈A, NmaxA , OmaxA 〉 be the greatest mbC-structure defined over A, where NmaxA = {(x, y) ∈ A2 : x t y = 1} and OmaxA = A2. By simplicity, EmaxA2 will be denoted by E max 2 = 〈A2, Nmax2 , Omax2 〉, recalling that A2 denotes the two-element Boolean algebra. Observe that EmaxA⊥ is 1⊥. Lemma 45 Let h : Emax2 → E be an mbC-homomorphism such that E = 〈A2, NE , OE〉 is defined over A2. Then h is the identity map, E = Emax2 and h is an isomorphism in FmbC. Proof. Since h is, in particular, a Boolean homomorphism, h(0) = 0 and h(1) = 1, hence h is the identity map. Since h preserves the predicates and h is the identity map, Pmax2 = h[P max 2 ] ⊆ P E ⊆ Pmax2 , and so P E = Pmax2 for P ∈ {N,O}. 2 Proposition 46 Emax2 is subdirectly irreducible in FmbC. Proof. Consider in Theorem 42 the predicate symbol N and the point (0, 0). By item (b)(iii) of Definition 3, (0, 0) /∈ Nmax2 . Let h : Emax2 → E be an onto homomorphism which is not an isomorphism in FmbC. In particular, h : A2 → A is an onto homomorphism of Boolean algebras which is not an isomorphism in FmbC, hence A is the one-point Boolean algebra A⊥ or A is A2. By Lemma 45 A must be A⊥ (otherwise h will be an isomorphism in FmbC). From this E is the terminal mbC-structure 1⊥ and so (h(0), h(0)) ∈ NE = {(∗, ∗)}. By Theorem 42, it follows that E is subdirectly irreducible in FmbC. 2 The following is a well-known result concerning Boolean algebras which will be useful for our purposes: Proposition 47 Let A be a Boolean algebra such that A 6= A⊥. (1) Let x, y ∈ A such that x 6= y. Then, there exists a homomorphism h : A → A2 of Boolean algebras such that h(x) 6= h(y). (2) Let x ∈ A such that x 6= 1. Then, there exists a homomorphism h : A → A2 of Boolean algebras such that h(x) = 0. (3) If h : A → A′ is a non-injective homomorphism of Boolean algebras, there exists z ∈ A such that z 6= 0 but h(z) = 0. (4) If h : A → A′ is a non-injective homomorphism of Boolean algebras, there exists z ∈ A such that z 6= 1 but h(z) = 1. 18 From now on, the cardinal of a set X will be denoted by card(X). Proposition 48 Let EmaxA an mbC-structure over a Boolean algebra A such that card(A) > 2. Then, EmaxA is not subdirectly irreducible in FmbC. Proof. Let (x, y) /∈ NmaxA . Then, x t y 6= 1 and so, by Proposition 47(2), there exists a homomorphism h : A → A2 of Boolean algebras such that h(x t y) = 0, whence h(x) = h(y) = 0. Clearly h induces an onto homomorphism h : EmaxA → Emax2 in FmbC which is not an isomorphism (since card(A) > 2), such that (h(x), h(y)) = (0, 0) 6∈ Nmax2 . Now, let (x, y) ∈ A2 such that x 6= y. By Proposition 47(1), there exists a homomorphism h : A → A2 of Boolean algebras such that h(x) 6= h(y). It is immediate to see that h induces an onto homomorphism h : EmaxA → Emax2 in FmbC which is not an isomorphism, such that h(x) 6= h(y). Finally, there is no (x, y) ∈ A2 such that (x, y) /∈ OmaxA . Therefore, it follows that EmaxA is not s.i. in FmbC, by Remark 43. 2 Proposition 49 (1) Let (x, y) ∈ NmaxA \ {(0, 1)}. Then EmaxA,N(x,y) def = 〈A, NmaxA \ {(x, y)}, OmaxA 〉 is an mbCstructure which is subdirectly irreducible in FmbC. (2) Let (x, y) ∈ A2\{(1, 0)}. Then EmaxA,O(x,y) def = 〈A, NmaxA , OmaxA \{(x, y)}〉 is an mbC-structure which is subdirectly irreducible in FmbC. Proof. (1) Let E def= EmaxA,N(x,y). Clearly, E is an mbC-structure. In order to apply Theorem 42, consider the predicate symbol N and the point (x, y). By definition, (x, y) /∈ NE . Now, suppose that we have an onto mbC-homomorphism h : E → E ′ which is not an isomorphism in FmbC. Then, h : A → A′ is an onto homomorphism of Boolean algebras. There are two cases: Case 1: h is an isomorphism of Boolean algebras. Then, the only reason for h (an isomorphism of Boolean algebras) not being an isomorphism in FmbC is that there exist (c, d) ∈ A2 and a predicate symbol P ∈ {N,O,≈} such that (c, d) /∈ P E but (h(c), h(d)) ∈ P E ′ . Given that OE = A2 it follows that P 6= O. On the other hand, since h is injective then P 6= ≈. Therefore, P = N and (c, d) = (x, y), by definition of E . This implies that (h(x), h(y)) ∈ NE ′ (and E ′ = EmaxA′ ). Case 2: h is not an isomorphism of Boolean algebras. Then, h is not injective. By Proposition 47(3), there exists z ∈ A such that z 6= 0 but h(z) = 0. There are two subcases to analyze: Case 2.1: Either z 6≤ x or z 6≤ y. Case 2.1.1: z 6≤ x. Let x′ = xtz. Then, x′ 6= x (otherwise, if x = x′ then z = zu(xtz) = zux and so z ≤ x, a contradiction). Clearly h(x′) = h(x). Moreover, x′ t y = (x t z) t y = 1 (since xty = 1). Hence (x′, y) 6= (x, y) such that (x′, y) ∈ NE and so (h(x), h(y)) = (h(x′), h(y)) ∈ NE ′ since h is homomorphism in FmbC. Case 2.1.2: z 6≤ y. As in the proof of Case 2.1.1 it can be shown that (h(x), h(y)) ∈ NE ′ (now by taking the pair (x, y t z)). 19 Case 2.2: Both z ≤ x and z ≤ y. Let x′ = x u ∼z. Then x 6= x′ (otherwise, if x = x′ then z = z u x = z u (x u ∼z) = 0, a contradiction). On the other hand x′ t y = (x u ∼z) t y = (x t y) u (∼z t y) = 1 u (∼z t y) = ∼z t y = ∼z t (y t z) = 1. Hence (x′, y) 6= (x, y) such that (x′, y) ∈ NE and so (h(x), h(y)) = (h(x′), h(y)) ∈ NE ′ since h is homomorphism in FmbC. By Theorem 42, E is s.i. in FmbC. (2) The proof is analogous to that of (1), but now considering the predicate symbol O and the point (x, y). 2 Remark 50 In Proposition 49 item (1) the point (x, y) = (0, 1) was not considered because of the first observation in Remark 5 (namely, (0, 1) ∈ NE for every E). Similarly, in Proposition 49(2) it is required that (x, y) 6= (1, 0) because of the second observation in Remark 5. Indeed, since (1, 1) ∈ NmaxA then (1, 0) must belong to OmaxA \ {(x, y)} in order to get an mbCstructure. Proposition 51 Let Emax∗A def = 〈A, NmaxA \ {(1, 1)}, OmaxA \ {(1, 0)}〉. Then, Emax∗A is an mbCstructure which is subdirectly irreducible in FmbC. Proof. It is easy to see that Emax∗A is indeed an mbC-structure. We will use Theorem 42 in order to show that E = Emax∗A is subdirectly irreducible in FmbC. Thus, consider the predicate symbol O and the point (1, 0) ∈ A2. Clearly, (1, 0) /∈ OE . It will be shown that, for every onto mbC-homomorphism h : E → E ′ which is not an isomorphism, (1, 0) = (h(1), h(0)) ∈ OE ′ . Thus, let h : E → E ′ be an onto mbC-homomorphism which is not an isomorphism. There are two cases: Case 1: h is an isomorphism of Boolean algebras. By adapting the argument used in the Case 1 of the proof of Proposition 49, and taking into account that E is obtained from EmaxA by removing just one point from NmaxA and just one point from O max A , there are two subcases to analyze: Case 1.1: E ′ is isomorphic to EmaxA in FmbC via h. That is, E ′ = EmaxA′ . Case 1.2: E ′ is isomorphic to EmaxA,N(1,1) in FmbC via h (recall Proposition 49). That is, E ′ = EmaxA′,N(h(1),h(1)) = E max A′,N(1,1). In both cases, (h(1), h(0)) ∈ OE ′ . Observe that the case that E ′ is isomorphic to EmaxA,O(1,0) is not allowed, given that (1, 1) ∈ NE ′ and so (1, 0) must be in OE ′ (see Remark 5). Case 2: h is not an isomorphism of Boolean algebras. Then, h is not injective. By Proposition 47(4), there exists z ∈ A such that z 6= 1 but h(z) = 1. From this, (z, 0) ∈ OE (since (z, 0) 6= (1, 0)) and so (h(z), h(0)) = (1, 0) ∈ OE ′ , since h is an homomorphism in FmbC. That is, (h(1), h(0)) ∈ OE ′ . By Theorem 42, E is s.i. in FmbC. 2 Proposition 52 Let E = 〈A, NE , OE〉 be an mbC-structure over A such that card ( NmaxA \ NE ) ≥ 1 and card ( OmaxA \ OE ) ≥ 1, and E 6= Emax∗A (see Proposition 51). Then E is not subdirectly irreducible in FmbC. 20 Proof. Once again, Remark 43 will be used in order to prove that E is not subdirectly irreducible. Let (x, y) ∈ NmaxA \ NE and (z, t) ∈ OmaxA \ OE . Let (c, d) 6∈ NE , and let E ′ = 〈A, NE , OmaxA 〉. It is easy to see that E ′ is an mbC-structure. Then, the identity map h : A→ A induces an onto homomorphism h : E → E ′ which is not an isomorphism, since (h(z), h(t)) = (z, t) ∈ OE ′ but (z, t) /∈ OE . Observe that (h(c), h(d)) = (c, d) /∈ NE ′ , since NE ′ = NE . Now, let (c, d) 6∈ OE . We have two subcases to analyze: Case 1: NE = NmaxA \ {(1, 1)}. Then OE 6= OmaxA \ {(1, 0)} since, by hypothesis, E 6= Emax∗A . Case 1.1: (c, d) = (1, 0). Then (1, 0) /∈ OE and so there exists (z, t) 6= (1, 0) such that (z, t) /∈ OE (given that E 6= Emax∗A ). Thus, consider E ′ = 〈A, NE , OE ∪ {(z, t)}〉. It is clear that E ′ is an mbC-structure such that the identity map h : A → A is an onto homomorphism h : E → E ′ which is not an isomorphism, since (h(z), h(t)) = (z, t) ∈ OE ′ but (z, t) /∈ OE . Observe that (h(c), h(d)) = (c, d) = (1, 0) /∈ OE ′ . Case 1.2: (c, d) 6= (1, 0). Observe that NmaxA = NE∪{(1, 1)}. Let E ′ = 〈A, NmaxA , OE∪{(1, 0)}〉. It is easy to prove that E ′ is an mbC-structure. In addition, the identity map h : A→ A is an onto homomorphism h : E → E ′ which is not an isomorphism, since (h(1), h(1)) = (1, 1) ∈ NE ′ but (1, 1) /∈ NE . Observe that (h(c), h(d)) = (c, d) /∈ OE ′ given that (c, d) 6= (1, 0). Case 2: NE 6= NmaxA \{(1, 1)}. Let (x, y) ∈ NmaxA \NE such that (x, y) 6= (1, 1) (such point must exists, by hypothesis). Then xu y 6= 1. Let E ′ = 〈A, NE ′ , OE ′〉 such that NE ′ def= NE ∪ {(x, y)} and OE ′ is defined according to the following subcases: Case 2.1: If d = 0 then OE ′ def = OE ∪ {(x,∼(xu y))}. Observe that (x,∼(xu y)) 6= (c, d) since x u y 6= 1 (hence, ∼(x u y) 6= 0 = d). Case 2.2: If d 6= 0 then OE ′ def= OE ∪ {(x, 0)}. Clearly (x, 0) 6= (c, d). Observe that E ′ is an mbC-structure: indeed, if (x,w) is the new point added to OE in OE ′ then (x u y) u w = 0. Thus, the property required in Definition 3(b)(iv) is satisfied for the new point (x, y) added to NE in NE ′ . Moreover, the identity map h : A → A is an onto homomorphism h : E → E ′ which is not an isomorphism, since (h(x), h(y)) = (x, y) ∈ NE ′ but (x, y) /∈ NE . Note that (h(c), h(d)) = (c, d) /∈ OE ′ given that the point added to OE in each case is different to (c, d). Finally, if (c, d) ∈ A2 such that c 6= d then the identity map h : A→ A is an onto homomorphism h : E → EmaxA which is not an isomorphism, such that h(c) 6= h(d). By Remark 43, E is not subdirectly irreducible. 2 Proposition 53 Let E = 〈A, NE , OE〉 be an mbC-structure over A such that NE = NmaxA and card ( OmaxA \OE ) ≥ 2. Then, E is not subdirectly irreducible in FmbC. Proof. By hypothesis, there exist (x, y) 6= (z, t) such that (x, y), (z, t) ∈ OmaxA \ OE . Let (c, d) /∈ NE . Then, the identity map h : A → A is an onto homomorphism h : E → EmaxA which is not an isomorphism, since (h(z), h(t)) = (z, t) ∈ OmaxA but (z, t) /∈ OE . Observe that (h(c), h(d)) = (c, d) /∈ NE . Now, let (c, d) /∈ OE . By hypothesis, there exists (c′, d′) 6= (c, d) such that (c′, d′) /∈ OE . Let E ′ = 〈A, NE , OE ∪ {(c′, d′)}〉. Clearly E ′ is an mbC-structure such that the identity map h : 21 A→ A is an onto homomorphism h : E → E ′ which is not an isomorphism, since (h(c′), h(d′)) = (c′, d′) ∈ OE ′ but (c′, d′) /∈ OE . Observe that (h(c), h(d)) = (c, d) /∈ OE ′ . The case for the identity predicate ≈ is treated as in the proof of Proposition 52. 2 Proposition 54 Let E = 〈A, NE , OE〉 be an mbC-structure over A 6= A2 such that OE = OmaxA and card ( NmaxA \NE ) ≥ 2. Then E is an mbC-structure which is not subdirectly irreducible in FmbC. Proof. The proof is analogous to that of Proposition 53. 2 Observe that, if (x, y), (z, t) ∈ Nmax2 such that (x, y) 6= (z, t) then 〈A2, Nmax2 \{(x, y), (z, t)}, Omax2 〉 is not an mbC-structure. This is why is required that A 6= A2 in the last proposition. By combining the previous results, it is possible to determine, among all the mbC-structures, which of them are s.i. and which of them are not: Theorem 55 Let E be an mbC-structure defined over a Boolean algebra A. Then E is subdirectly irreducible in FmbC if and only if exactly one of the following conditions holds: (1) E = 1⊥; or (2) E = Emax2 ; or (3) E = EmaxA,N(x,y) def = 〈A, NmaxA \ {(x, y)}, OmaxA 〉 for some (x, y) ∈ NmaxA \ {(0, 1)}; or (4) E = EmaxA,O(x,y) def = 〈A, NmaxA , OmaxA \ {(x, y)}〉 for some (x, y) ∈ A2 \ {(1, 0)}; or (5) E = Emax∗A def = 〈A, NmaxA \ {(1, 1)}, OmaxA \ {(1, 0)}〉. Proof. It is a direct consequence of propositions 44, 46, 48, 49, 51, 52, 53 and 54. 2 8.3 Subdirect decomposition theorem for mbC-structures Finally, we are ready to obtain a decomposition theorem for mbC-structures in terms of subdirectly irreducible structures. It is an instance of a general theorem obtained by Caicedo in [6]. Indeed, as it was observed at the end of Section 4, the class FmbC of mbC-structures can be axiomatized by sentences over signature Θ of the form ∀x1 * * * ∀xn(σ1 → ∃y1 * * * ∃ykσ2)), where σ1 and σ2 are quantifier-free positive formulas over Θ. Thus, by combining Corollary 5 and Theorem 4 in [6], the following result is obtained: Theorem 56 (Birkhoff-Caicedo's decomposition theorem for mbC-structures) Any non-trivial mbC-structure E = 〈A, NE , OE〉 is a subdirect product of at most א0 + card(A) non-trivial subdirectly irreducible structures. The relationship between Birkhoff-Caicedo's decomposition theorem for mbC-structures and the usual Birkhoff's decomposition theorem for varieties of algebras is not immediate. In the case of mbC-structures, it seems that there are more than necessary s.i. structures. Indeed, the single mbC-structure Emax2 is enough in order to characterize the logic mbC. 22 Theorem 57 (Soundness and completeness of mbC w.r.t. Emax2 ) Let Γ ∪ {α} be a finite set of formulas in For(Σ). Then: Γ `mbC α iff Γ mbCEmax2 α. Proof. The 'only if' part is a consequence of Theorem 8. Conversely, suppose that Γ 0mbC α. By adapting the proof of [9, Theorem 6.2.16], there exists an mbC-structure E over A2 and a valuation v over E such that v[Γ] ⊆ {1} but v(α) = 0. Given that E is a weak substructure of Emax2 , it follows that v is also a valuation over Emax2 , by the last observation in Remark 20. This shows that Γ 6 mbCEmax2 α. 2 By a similar argument, from Theorem 22 it can be proven the adequacy of mbC w.r.t. the mbC-structure EmaxA , for any Boolean algebra A with more than two elements. Theorem 58 (Soundness and completeness of mbC w.r.t. EmaxA ) Let A be a Boolean algebra with more than two elements, and let Γ ∪ {α} be a finite set of formulas in For(Σ). Then: Γ `mbC α iff Γ mbCEmaxA α. Proof. The 'only if' part is a consequence of Theorem 8. Now, assume that Γ 0mbC α. By Theorem 22, there exists an mbC-structure E over A and a valuation v over E such that v[Γ] ⊆ {1} but v(α) = 0. Since E is a weak substructure of EmaxA , it follows that v is also a valuation over EmaxA , by the last observation in Remark 20. Therefore Γ 6 mbCEmaxA α. 2 In the next subsection an alternative decomposition theorem for mbC-structures will be presented, by adapting a decomposition theorem due to Fidel. As it will be argued below, this alternative decomposition theorem is closer to the traditional Birkhoff's decomposition theorem of Universal Algebra. 8.4 Fidel's decomposition theorem for mbC-structures In 1977, Fidel (see [19]) obtained for the first time the decidability of the hierarchy of paraconsistent calculi Cn of da Costa (see [15]) in terms of certain F-structures called Cn-structures. In that paper, Fidel showed that every Cn-structure is weakly isomorphic to a weak substructure of a product of a special Cn-structure defined over the two-element Boolean algebra A2 called C, see [19, Theorem 8]. By a weak isomorphism we mean a homomorphism which is bijective as a mapping.13 By adopting the notion of weak substructure in FmbC (recall Definition 19), it is possible to obtain another decomposition theorem for mbC-structures, alternative to the one given in Theorem 56. Thus, it will be shown in Theorem 60 that each mbC-structure is weakly isomorphic to a weak substructure in FmbC of a product of mbC-structures over A2. The same result can be rephrased without using the notion of weak isomorphism (see Theorem 65 below). 13Such homomorphisms are called isomorphisms in [19, Definition 6]. 23 Definition 59 Let h : E → E ′ be an mbC-homomorphism. Then h is said to be a weak isomorphism if h is a bijective mapping. Theorem 60 (Weak subdirect decomposition theorem for mbC-structures) Let E be an mbC-structure. Then, there exists a set I such that E is weakly isomorphic to a weak substructure of ∏ i∈I Ei, where each Ei is defined over A2 for every i ∈ I. Proof. Let E = 〈A, NE , OE〉 be an mbC-structure. By Birkhoff's representation theorem for Boolean algebras [3], there exists a set I and a monomorphism of Boolean algebras h : A →∏ i∈I Ai such that Ai = A2 for every i ∈ I. If I = ∅ then A ' A⊥ and so E is the terminal mbC-structure 1⊥, hence the result holds with I = ∅. Now, suppose that I 6= ∅. For each i ∈ I consider the structure Ei = 〈A2, NEi , OEi〉 such thatNEi = {((πi◦h)(a), (πi◦h)(b)) : (a, b) ∈ NE} and OEi = {((πi ◦ h)(a), (πi ◦ h)(b)) : (a, b) ∈ OE}. Since (0, 1) ∈ NE and (1, b) ∈ NE for some b ∈ A then (0, 1) ∈ NEi and (1, (πi ◦ h)(b)) ∈ NEi . Analogously, it is proved that (0, x) ∈ OEi and (1, y) ∈ OEi for some x, y ∈ TWO. Suppose now that (x, y) ∈ NEi . Then, there exists (a, b) ∈ NE such that x = (πi ◦ h)(a) and y = (πi ◦ h)(b). Since E is an mbC-structure, there exists c ∈ A such that (a, c) ∈ OE and a uE b uE c = 0. From this, (x, (πi ◦ h)(c)) ∈ OEi such that 0 = (πi ◦ h)(0) = x uA2 y uA2 (πi ◦ h)(c)). This shows that Ei = 〈A2, NEi , OEi〉 is an mbC-structure, for every i ∈ I. Now, let E ′ = ∏ i∈I Ei be the direct product in FmbC of the family {Ei}i∈I , see Definition 38. Clearly, h is an mbC-homomorphism from E to E ′: if (a, b) ∈ NE then ((πi ◦ h)(a), (πi ◦ h)(b)) ∈ NEi for every i ∈ I. Hence, (h(a), h(b)) ∈ NE ′ by Definition 38. Analogously, if (a, b) ∈ OE then (h(a), h(b)) ∈ OE ′ . Consider now the structure E ′′ = 〈h(A), NE ′′ , OE ′′〉 defined as follows: NE ′′ = {(h(a), h(b)) : (a, b) ∈ NE} and OE ′′ = {(h(a), h(b)) : (a, b) ∈ OE}. It is easy to see that E ′′ is an mbC-structure: in order to prove that property (b)(iv) of Definition 3 holds, suppose that NE ′′ (x, y). Then, there exists a unique (a, b) ∈ A2 such that x = h(a), y = h(b) and NE(a, b). From this, there exists c ∈ A such that OE(a, c) and a uE b uE c = 0. This means that OE ′′(x, h(c)) such that x uh(A) y uh(A) h(c) = h(0) = 0. The properties (b)(i)-(iii) are proved analogously. Since h is injective, it follows that E is weakly isomorphic to E ′′ such that E ′′ ⊆W ∏ i∈I Ei, recalling Definition 19 of weak substructure in FmbC. 2 In order to better understand the real significance of the latter decomoposition result, it is convenient to introduce some definitins in the general framework of Model Theory. Firstly, observe that the notion of weak isomorphism of Definition 59 makes sense in any class of firstorder structures: Definition 61 Let A and A′ be two first-order structures over a signature Ξ. A homomorphism h : A→ A′ in Ξ-str is said to be a weak isomorphism of Ξ-structures if it a bijective mapping. The notion of direct image of a structure by a homomorphism can be weakened in a suitable way: 24 Definition 62 Let A and A′ be two first-order structures over a signature Ξ, and let h : A→ A′ be a homomorphism in Ξ-str. The weak image of A by h is the Ξ-structure h(A)w with domain h[|A|]; for each function symbol f of arity n, f is interpreted in h(A)w by restricting the domain and image of fA ′ to h[|A|]; the constants are interpreted as in A′;14; and for every n-ary predicate symbol P and every (b1, . . . , bn) ∈ ( h[|A| )n , (b1, . . . , bn) ∈ P h(A)w if and only if there exists (a1, . . . , an) ∈ |A|n such that (h(a1), . . . , h(an)) = (b1, . . . , bn) and (a1, . . . , an) ∈ PA. Recall the notion of weak substructure in Ξ-str considered right before Definition 19. The proof of the following results is straightforward. Proposition 63 Let h : A→ A′ be a homomorphism in Ξ-str. Then h(A)w is a weak substructure of the direct image h(A) of A by h, hence it is a weak substructure of A′. Proposition 64 Let h : A → A′ be a weak isomorphism in Ξ-str. Then A is isomorphic (via h) to h(A)w, a weak substructure of A ′. The last result relates weak isomorphisms with embeddings (recall Definition 16) in a clear way. Using the previous notions and results, Theorem 60 can be recast as follows (here, A ' A′ denotes that the Ξ-structures A and A′ are isomorphic in Ξ-str): Theorem 65 (Weak subdirect decomposition theorem for mbC-structures, version 2) Let E be an mbC-structure. Then, there exists a set I and an mbC-structure E' such that E ' E ′ ⊆W ∏ i∈I Ei, where Ei ⊆W Emax2 for every i ∈ I. From Theorem 9 we known that the mbC-structures defined over the two-element Boolean algebra are enough to semantically characterize the logic mbC. Then, Theorem 65 reflects in a precise way the relationship between the semantical structures and the logic in a similar way to the traditional algebraic approach to logic. In that sense, Theorem 65 is more informative than Theorem 56. 9 Concluding remarks In this paper, the class of F-structures for mbC was analyzed under the perspective of Model Theory. This approach is based on the observation that F-structures are nothing more than first-order structures satisfying specific Horn sentences of its underlying language, as it was seen in Section 4. Under this broad perspective, the present study could be adapted to the study of 14Observe that the interpretation of function symbols and constants is well-defined since f is a homomorphism of first-order structures, hence it is an algebraic homomorphism. 25 the class of F-structures for another LFIs as the ones proposed in [8] and [9, Chapter 6]. The fact that all these LFIs are axiomatic extensions of mbC imposes additional restrictions to the corresponding class of F-structures (which are still axiomatized by Horn sentences of the same kind), and so it would be expected that the class of irreducible structures should be reduced. Being so, Caicedo's version of Birkhoff's decomposition theorem (recall Theorem 56) would keep closer to Fidel's one (recall theorems 60 and 65). As observed at the end of Section 8, it can be argued that Fidel's result reflects the meaning of Birkhoff's decomposition theorem for algebras in a more faithful way than the theorem obtained by using the notions from Model Theory, which reveals some limitations of the model-theoretic approach to F-structures. The fact that the notion of weak substructures play a fundamental role in Fidel's decomposition theorem suggests that new concepts and tools should be developed for standard Model Theory in order to analyze the (meta)theory of propositional non-algebraizable logics under the perspective of F-structures. Related to this, there is another interesting topic of future research, now concerning the development of a new approach to Model Theory for first-order LFIs (and non-algebraizable logics in general) by means of Fidel-structures. To fix ideas, consider the first-order version of mbC, namely the logic QmbC (see [11]). This logic can be semantically characterized by the socalled paraconsistent Tarskian structures, which are (standard) first-order structures together with a paraconsistent two-valued mbC-valuation extended naturally to first-order languages. Such valuations could be replaced by valuations over a given F-structure for mbC (in particular, an mbC-structure over the two-element Boolean algebra). This perspective, which generalizes the standard approach to Model Theory over ordered algebras, open interesting lines of research. Thus, the notions and results obtained here for F-structures for mbC could be adapted for the development of the new Model Theory based on F-structures for QmbC, as well as for other quantified non-algebraizable logics. In particular, interesting results on Model Theory for quantified LFIs such as the Keisler-Shelah Theorem for QmbC obtained by T. Ferguson in [18] can be adapted to the proposed framework. Finally, the connections between Priest's plurivalent semantics (recall Section 1) and other non-deterministic semantics, specifically Nmatrices and Fidel-structures, deserves future research. Moreover, a formal study of plurivalent semantics from the pespective of Model theory and Universal Algebra is an interesting task to be done. Acknowledgments We are grateful to X. Caicedo for his remarks, suggestions and criticisms, which helped to improve this paper. The first author was financially supported by an individual research grant from CNPq, Brazil (308524/2014-4). The second author acknowledges support from a postdoctoral grant from FAPESP, Brazil (2016/21928-0). 26 References [1] A. Avron and I. Lev. Canonical propositional Gentzen-type systems. In R. Gore, A. Leitsch, and T. Nipkow, editors, Proceedings of the 1st International Joint Conference on Automated Reasoning (IJCAR 2001), volume 2083 of LNAI, pages 529–544. Springer Verlag, 2001. [2] A. 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