Completeness for monadic fuzzy logics via functional algebras

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Completeness for monadic fuzzy logics via functional algebras

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Diego Castaño a,b,∗ , Cecilia Cimadamore a,b , José Patricio Díaz Varela a,b , Laura Rueda a,b

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a Departamento de Matemática, Universidad Nacional del Sur, 8000 Bahía Blanca, Argentina b INMABB, UNS-CONICET, 8000 Bahía Blanca, Argentina

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Received 16 October 2019; received in revised form 30 January 2020; accepted 8 February 2020

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Abstract

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We study S5-modal (monadic) expansions of extensions of Hájek’s basic logic BL. Hájek proposed Hilbert-style systems axiomatizing these logics and we prove that completeness theorems for these logics follow from algebraic representation results, namely, functional representations of finitely subdirectly irreducible algebras. We prove a general theorem linking these concepts and give two major applications, namely, for the S5-modal expansions of Łukasiewicz and Gödel logics. © 2020 Published by Elsevier B.V.

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Keywords: Monadic logic; Basic logic; Completeness theorem; Functional algebras; Łukasiewicz logic; Gödel logic

1. Introduction and preliminaries

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We denote by BL the basic logic defined by Hájek in [18]. BL is an example of an implicative logic, so it is an algebraizable logic whose equivalent algebraic semantics Alg∗ BL is the variety of BL-algebras that we usually denote by BL. As a result of the general theory of algebraizability, every axiomatic extension of BL is also algebraizable and its equivalent algebraic semantics is a subvariety of BL. More precisely, if C is an axiomatic extension of BL, the equivalent algebraic semantics Alg∗ C is the subvariety of BL determined by the equations of the form ϕ ≈ 1 where ϕ ranges over the axiom schemata of C. In [18] Hájek also introduced the logic S5(C) as an S5-modal expansion of an axiomatic extension C of BL. This logic is defined semantically over the language of basic logic augmented with the unary connectives  and ♦. The structures on which formulas are interpreted are defined as triples K = (W, e, L) where L is a BL-chain (i.e., a totally ordered BL-algebra) in Alg∗ C, W is a non-empty set and e : W × P rop → L is an evaluation function defined on pairs (w, p) consisting of an element w ∈ W and a propositional variable p ∈ P rop. Hájek defined the truth degree ϕK,w of a formula ϕ in K at the world w. This is done recursively on the structure of ϕ. For propositional variables p ∈ P rop we have that pK,w = e(w, p). The definition of the truth value is then extended for the logical connectives of the basic logic in the usual way, and for the modal connectives by

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* Corresponding author at: Departamento de Matemática, Universidad Nacional del Sur, 8000 Bahía Blanca, Argentina.

E-mail address: [email protected] (D. Castaño). https://doi.org/10.1016/j.fss.2020.02.002 0165-0114/© 2020 Published by Elsevier B.V.

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ψK,w := inf ψK,w , 

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w ∈W

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and

♦ψK,w := sup ψK,w . w  ∈W

Note that the infima and suprema above may not exist in general; hence, we restrict our attention only to safe structures, that is, structures K for which ϕK,w is defined for every formula ϕ at every world w. Originally Hájek was only interested in the tautologies of this logic, but in [19] he considered, albeit implicitly, a global consequence relation S5(C ) . Given a set of formulas  we say that a safe structure K = (W, e, L) is a model of  if for every ϕ ∈  and every w ∈ W we have that ϕK,w = 1. Thus, given a set of formulas  and a formula ϕ, we write  S5(C ) ϕ if and only if for every model K = (W, e, L) of  we have that ϕK,w = 1 for every w ∈ W . Already in [18] Hájek noted that S5(C) is actually equivalent to the monadic fragment in one variable (without constants) of the first-order logic C∀ because there is a natural correspondence between formulas of both logics, between corresponding models and between the corresponding truth degrees. Thus, since the latter is finitary (being a fragment of a finitary logic), the consequence relation S5(C ) is also finitary. Hájek also introduced a Hilbert-style syntactic calculus in the language of S5(C) whose consequence relation we denote by S5(C ) . The axioms of this calculus are the instantiations of the axiom schemata of C for formulas in the language of S5(C), plus the following axioms:

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ϕ → ϕ ϕ → ♦ϕ (ν → ϕ) → (ν → ϕ) (ϕ → ν) → (♦ϕ → ν) (ϕ ∨ ν) → (ϕ ∨ ν) ♦(ϕ ∗ ϕ) ≡ (♦ϕ) ∗ (♦ϕ)

where ϕ is any formula, ν is any propositional combination of formulas beginning with  or ♦, and α ≡ β abbreviates (α → β) ∧ (β → α). The inference rules for this calculus are modus ponens (ϕ, ϕ → ψ / ψ ) and necessitation (ϕ / ϕ). In [19] Hájek claimed to have proved that S5(C ) = S5(C ) for every axiomatic extension C of BL, thus proving the completeness of the calculus for S5(C). However, the proof of Lemma 2 in [19] is not totally convincing.1 We aim to prove this using algebraic models, and this article proves some partial results towards this goal. It is easy to show that the logic S5(C ) is an implicative logic and, thus, the class of S5(C)-algebras Alg∗ S5(C) is its equivalent algebraic semantics (see [17]). In [10] we introduced the variety MBL of monadic BL-algebras as BL-algebras endowed with two unary operations ∀ and ∃ that satisfy the following identities (M1) (M2) (M3) (M4) (M5)

∀x → x ≈ 1, ∀(x → ∀y) ≈ ∃x → ∀y, ∀(∀x → y) ≈ ∀x → ∀y, ∀(∃x ∨ y) ≈ ∃x ∨ ∀y, ∃(x ∗ x) ≈ ∃x ∗ ∃x.

The main goal of that article was to prove that MBL is the equivalent algebraic semantics of S5(BL) (the semantic version of S5(BL)). However, that result relied on [19]. Nonetheless, what is easily provable using the properties in [10, Lemma 2.2] is that MBL is the equivalent algebraic semantics Alg∗ S5(BL) of the implicative logic S5(BL) (the syntactic version of S5(BL)). Note that the unary operations ∀ and ∃ in monadic BL-algebras correspond to the logical connectives  and ♦, respectively. For the sake of preserving the notation of previous articles we identify ∀ and ∃ with  and ♦, respectively, using the former for the algebras and the latter for the logic. As a consequence of the general theory of algebraizability all axiomatic extensions of S5(BL) are algebraizable logics and their equivalent algebraic semantics are precisely the subvarieties of MBL. In particular, given any axiomatic extension C of BL, the logic S5(C ) is an axiomatic extension of S5(BL) with an equivalent algebraic semantics Alg∗ S5(C), denoted also by MBLC , which is a subvariety of MBL. In Section 2 we prove that a com-

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1 Hájek’s proof in [19] does not use the axiom ♦(ϕ ∗ ϕ) ≡ (♦ϕ) ∗ (♦ϕ), but we showed in [10] that this axiom is independent of the other ones.

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pleteness theorem holds, that is, S5(C ) = S5(C ) , if and only if the variety MBLC is generated, as a quasivariety, by a special kind of algebras (C-functional algebras). Two important applications of this theorem are given in Sections 3 and 4 to two of the most significant extensions of basic logic: Łukasiewicz and Gödel logics. We collect here some basic results on monadic BL-algebras that are used throughout the article. We refer the reader to [10] for a more thorough treatment of these algebras. For brevity, if A is a BL-algebra and we enrich it with a monadic structure, we denote the resulting algebra by (A, ∃, ∀). The next lemma collects some of the basic properties that hold true in any monadic BL-algebra. Lemma 1.1 ([10, Lemma 2.3]). Let (A, ∃, ∀) ∈ MBL and a, b ∈ A. Then

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(M6) (M7) (M8) (M9) (M10) (M11) (M12)

∀∃a = ∃a. a → ∃a = 1. ∀1 = 1. ∃∀a = ∀a. ∀0 = 0. ∃1 = 1. ∃0 = 0.

(M13) (M14) (M15) (M16) (M17) (M18)

∃∃a = ∃a and ∀∀a = ∀a. If a ≤ b, then ∀a ≤ ∀b and ∃a ≤ ∃b. ∃(a ∨ b) = ∃a ∨ ∃b. ∃(∃a ∧ ∃b) = ∃a ∧ ∃b. ∃(a ∧ ∃b) = ∃a ∧ ∃b. ∀(a ∧ b) = ∀a ∧ ∀b.

Given a monadic BL-algebra (A, ∃, ∀), we write ∃A = {∃a : a ∈ A} and ∀A = {∀a : a ∈ A}. Lemma 1.2 ([10, Lemma 2.6]). If (A, ∃, ∀) ∈ MBL, then ∃A = ∀A and this set is the universe of a subalgebra of A denoted by ∃A. Lemma 1.3 ([10, Lemma 2.9]). Let (A, ∃, ∀) ∈ MBL. Then the congruence lattice of (A, ∃, ∀) is isomorphic to the congruence lattice of the BL-algebra ∃A. The last lemma implies that a monadic BL-algebra (A, ∃, ∀) is simple, subdirectly irreducible or finitely subdirectly irreducible if and only if the BL-algebra ∃A is simple, subdirectly irreducible or finitely subdirectly irreducible, respectively. In particular, (A, ∃, ∀) is finitely subdirectly irreducible if and only if ∃A is totally ordered. Theorem 1.4 ([10, Theorem 2.13]). Given a finitely subdirectly irreducible monadic BL-algebra (A, ∃, ∀), there exists a subdirect representation of the underlying BL-algebra A ≤ i∈I Ai , where each Ai is a totally ordered BL-algebra and ∃A is embedded in Ai via the corresponding projection map. 2. Completeness via functional algebras

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In this section we define C-functional algebras and prove that, given an axiomatic extension C of BL, the completeness theorem holds, that is, S5(C ) = S5(C ) , if and only if the variety MBLC is generated as a quasivariety by its C-functional algebras. To define C-functional algebras, we need to recall how to define monadic BL-algebras based on m-relatively complete subalgebras of a BL-algebra. In [10] we give a characterization of those subalgebras of a given BL-algebra that may be the range of the quantifiers ∀ and ∃. Given a BL-algebra A, we say that a subalgebra C ≤ A is m-relatively complete if the following conditions hold:

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(s1) For every a ∈ A, the subset {c ∈ C : c ≤ a} has a greatest element and {c ∈ C : c ≥ a} has a least element. (s2) For every a ∈ A and c, d ∈ C such that c ≤ d ∨ a, there exists c ∈ C such that c ≤ d ∨ c and c ≤ a. (s3) For every a ∈ A and c ∈ C such that a ∗ a ≤ c, there exists d ∈ C such that a ≤ d and d ∗ d ≤ c. Furthermore if C is totally ordered, condition (s2) may be replaced by the following simpler equivalent form:

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(s2 ) If 1 = c ∨ a for some c ∈ C, a ∈ A, then c = 1 or a = 1.

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Given a BL-algebra A and an m-relatively complete subalgebra C ≤ A, if we define on A the operations ∃a := min{c ∈ C : c ≥ a},

∀a := max{c ∈ C : c ≤ a},

then (A, ∃, ∀) is a monadic BL-algebra such that ∀A = ∃A = C. Let A be a BL-algebra and let X be a nonempty set. Let S be the set of functions f ∈ AX such that inf{f (x) : x ∈ X} and sup{f (x) : x ∈ X} both exist in A. For every f ∈ S, we define: (∀∧ f )(x) = inf{f (y) : y ∈ X} and (∃∨ f )(x) = sup{f (y) : y ∈ X}, x ∈ X. Note that ∀∧ f and ∃∨ f are constant maps. We are interested in subalgebras B ≤ AX such that B ⊆ S and (B, ∃∨ , ∀∧ ) is a monadic BL-algebra. Note that we do not require A to be complete, since we would like to build algebraic models that correspond to all safe structures used to interpret the formulas of the logics S5(C). The most important example of this kind of algebras arises when A is a BL-chain. In that case, as is shown in [10], for any subalgebra B of AX such that B ⊆ S and B is closed under ∃∨ and ∀∧ , the structure (B, ∃∨ , ∀∧ ) is a monadic BL-algebra (in fact, a finitely subdirectly irreducible one). These algebras are important because they encode the information of the models used to interpret first-order formulas in one-variable, as defined by Hájek in [18] (note that the truth values in Hájek’s predicate logic belong to a BL-chain). We call functional monadic BL-algebra to the algebras B that arise in this way. Furthermore, given an axiomatic extension C of BL, we say that a monadic BL-algebra is C-functional if it is built in the previously defined way from a BL-chain A that belongs to Alg∗ C (the equivalent algebraic semantics of C, a subvariety of BL). Theorem 2.1. Let C be an axiomatic extension of BL. Then the following are equivalent:

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(1) for any formula ϕ and any set of formulas , we have  S5(C ) ϕ if and only if  S5(C ) ϕ; (2) the variety MBLC is generated, as a quasivariety, by its C-functional algebras.

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Proof. We prove first that (2) implies (1). The soundness implication is straightforward and does not depend on the hypothesis. We prove the completeness implication. Assume  S5(C ) ϕ. Since S5(C ) is finitary, we may also assume that  is finite. By way of contradiction, suppose  S5(C ) ϕ. Then  MBLC ϕ (this is the consequence relation associated to the variety MBLC ), which is equivalent to saying that MBLC does not satisfy the quasi-equation γ1 ≈ 1 & . . . & γn ≈ 1 ⇒ ϕ ≈ 1, where {γ1 , . . . , γn } = . By hypothesis, there is a functional algebra (B, ∃∨ , ∀∧ ) ∈ MBLC with B ≤ AX and A ∈ Alg∗ C, and a valuation h into (B, ∃∧ , ∀∧ ) such that h() ⊆ {1} but h(ϕ) = 1. Consider the structure K = (X, e, A) with e(x, p) = h(p)(x) for x ∈ X and p ∈ P rop. For every γ ∈  and x ∈ X we have that γ K,x = h(γ )(x) = 1, but ϕK,x = h(ϕ)(x) = 1 for some x ∈ X, which is a contradiction. Conversely, we now prove that (1) implies (2). Assume MBLC is not generated as a quasivariety by its C-functional members. Thus, there is a quasi-identity γ1 ≈ 1 & . . . & γn ≈ 1 ⇒ ϕ ≈ 1 that is true on every C-functional algebra, but is not true in MBLC . From the properties of algebraizable logics, we get that {γ1 , . . . , γn } S5(C ) ϕ. However, we claim that {γ1 , . . . , γn } S5(C ) ϕ. Indeed, assume K = (W, e, L) is a model of {γ1 , . . . , γn }, where L is a BL-chain in Alg∗ C. Then γi K,w = 1 for every w ∈ W and 1 ≤ i ≤ n. Let F m be the set of propositional formulas in the language of S5(C). For each ψ ∈ F m, define fψ : W → L such that fψ (w) = ψK,w for every w ∈ W . Consider A := {fψ : ψ ∈ F m}. Then A ⊆ LW and, in addition, A is a subuniverse of the BL-algebra LW . Moreover, it is straightforward to check that ∃∨ fψ = f♦ψ and ∀∧ fψ = fψ . Then (A, ∃∨ , ∀∧ ) is a C-functional algebra. Consider now the interpretation e : F m → A given by e (ψ) = fψ for every ψ ∈ F m. Then e (γi ) = 1 for 1 ≤ i ≤ n. By hypothesis, e (ϕ) = 1, that is, fϕ = 1, so ϕK,w = 1 for every w ∈ W . This completes the proof that {γ1 , . . . , γn } S5(C ) ϕ.  3. Monadic Łukasiewicz logic

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One of the main axiomatic extensions of basic logic is Łukasiewicz logic L, which may be obtained by adding the axiom schema ¬¬p → p to BL. Its corresponding equivalent algebraic semantics is the variety MV of MV-algebras, which is precisely the subvariety of BL determined by the identity ¬¬x ≈ x. We refer the reader to [11] for all basic properties of MV-algebras.

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We recall here some useful notions of BL-algebras and MV-algebras that will be needed. Given a BL-algebra A, the congruences on A are in one-to-one correspondence with implicative filters of A. An implicative filter (or filter, for short) of A is a subset F ⊆ A such that 1 ∈ F and whenever a, a → b ∈ F , we have b ∈ F . Associated to a filter F we have the quotient algebra A/F defined in the usual way via the congruence relation x ≡F y if and only if x → y, y → x ∈ F . A filter F is prime if F = A and a → b ∈ F or b → a ∈ F for every a, b ∈ A. For any BLalgebra A, F is prime if and only if A/F is totally ordered and nontrivial. Given X ⊆ A, we denote by Fg(X) the least implicative filter containing X, which is the intersection of all implicative filters containing X. It is well-known that Fg(X) = {a ∈ A : a ≥ x1 x2 · · · xn for some x1 , . . . , xn ∈ X}. In case X = {x}, we write Fg(x) instead of Fg(X). In this case we have Fg(x) = {a ∈ A : a ≥ x n for some n}. Another property that will be useful is the following: Fg(X ∪ {a}) ∩ Fg(X ∪ {b}) = Fg(X ∪ {a ∨ b}) for any a, b ∈ A, X ⊆ A. We consider now the monadic expansion S5(L) of L and prove the completeness of the proposed calculus. As we mentioned earlier, the syntactic calculus S5(L) is algebraizable and the class of monadic MV-algebras is its equivalent algebraic semantics. A monadic MV-algebra is a monadic BL-algebra that satisfies the identity ¬¬x ≈ x. In other words a monadic MV-algebra is a monadic BL-algebra whose BL-reduct is an MV-algebra. These algebras were first considered by Rutledge in his Ph.D. thesis [24]. He proved the functionality of subdirectly irreducible algebras and used it to show the weak standard completeness of the calculus. We give here a much simpler proof of the representation theorem, and use it to derive strong completeness of the calculus with respect to the chain-based models. We will need two special properties of monadic MV-algebras, which we include in the following lemma.

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Lemma 3.1 (see [10, Lemma 5.4]). Let (A, ∃, ∀) be a monadic MV-algebra and a, b ∈ A. (a) ∀a = ¬∃¬a. (b) ∃(∃a → b) = ∃a → ∃b.

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In this section, we prove that the amalgamation property for a first-order class of structures implies an infinite version of the amalgamation property, which suits better our goals. This result is a straightforward application of model theoretic techniques, but we could not find it in the literature, so we include a proof for the sake of completeness. Let A be a structure in the first-order language L. Let {ca : a ∈ A} be a set of new constant symbols and put LA = L ∪ {ca : a ∈ A}. Given an L-structure B and a family {ba : a ∈ A} of elements of B, we denote by (B, {ba }a∈A ) the expansion of B to the language LA that results by defining ba as the interpretation of the constant ca for every a ∈ A. The (Robinson) diagram of A, written diag(A), is the set of all atomic sentences and negations of atomic sentences in the language LA which are true in (A, {a}a∈A ). Lemma 3.2 (Diagram lemma). For every pair of L-structures A, B, and any function f : A → B, the following conditions are equivalent:

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1. f is an embedding of A into B, 2. (B, {f (a)}a∈A ) is a model of diag(A).

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Let K be a class of structures in a first-order language. A V-formation in K is a 5-tuple (A, A1 , A2 , α1 , α2 ) consisting of three structures A, A1 , A2 in K and two embeddings α1 : A → A1 and α2 : A → A2 . An amalgam in K of the V-formation is a triple (B, β1 , β2 ) consisting of a structure B in K and embeddings β1 : A1 → B and β2 : A2 → B satisfying β1 ◦ α1 = β2 ◦ α2 .

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The class K has the amalgamation property provided every V-formation in K has an amalgam in K. For our purposes we need a generalization of the notion of V-formation and amalgam. Given a set I , an I -formation in K is a triple (A, {Ai }i∈I , {αi }i∈I ) where A and all Ai are a structures in K and each αi : A → Ai is an embedding. An amalgam in K of the I -formation is a pair (B, {βi }i∈I ) where B is a structure in K and βi : Ai → B are embeddings such that βi ◦ αi = βj ◦ αj for every i, j ∈ I . We say that K has the amalgamation property over I if every I -formation in K has an amalgam in K.

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Theorem 3.3. Let K be a class of structures in a first-order language with the amalgamation property. Then K has the amalgamation property over any finite set I . If, in addition, K is an elementary class, then K has the amalgamation property over any set I .

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Proof. The fact that K has the amalgamation property over any finite set I follows straightforwardly by induction on the size of I . Now assume the additional hypothesis that K is an elementary class. We will prove that K has the amalgamation property over I using a compactness argument. Let (A, {Ai : i ∈ I }, {αi : i ∈ I }) be an I -formation in K. For simplicity, we may assume that αi is the inclusion map for every i ∈ I . Thus A ≤ Ai  for every i ∈ I . We may further assume that Ai ∩ Aj = A for i = j . Let ca be a new constant symbol for every a ∈ i∈I Ai and consider the language  L = LK ∪ {ca : a ∈ Ai }, i∈I

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where TK is a set of axioms for K. We claim that T is finitely satisfiable. Indeed, if I0 is a finite subset of I , we know that the I0 -formation (A, {Ai : i ∈ I0 }, {αi : i ∈ I0 }) has an amalgam (C, {γi : i ∈ I0 }) in K. Expand C to an L-structure C defining caC := γi (a) for  every a ∈ Ai and i ∈ I0 . For i ∈ / I0 and a ∈ Ai \ A, define caC arbitrarily. Then C is a model of TK ∪ i∈I0 diag(Ai ). By the compactness theorem, T is satisfiable. Let B be a model of T and B its LK -reduct. Note that B is in K. In addition, since B is a model of diag(Ai ) the map βi : Ai → B defined by βi (x) = cxB , x ∈ Ai , is an embedding. Moreover, for every a ∈ A and any i ∈ I we have

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which is independent of i. This shows that (B, {βi : i ∈ I }) is an amalgam in K of (A, {Ai : i ∈ I }, {αi : i ∈ I }).

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For our purposes, we would like to apply the last theorem to the class of totally ordered MV-algebras. In [23] Pierce proved that the class of totally ordered Abelian -groups has the amalgamation property. The following result is then a straightforward consequence of Chang and Mundici’s well known correspondence between totally ordered Abelian -groups and totally ordered MV-algebras (see [11]). A proof of this result using geometrical tools can be found in [6, Theorem 5.2].

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Theorem 3.4. The class of totally ordered MV-algebras has the amalgamation property.

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Using now Theorem 3.3 we get the following stronger version.

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Corollary 3.5. The class of totally ordered MV-algebras has the amalgamation property over any set.

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3.2. Functionality of finitely subdirectly irreducibles

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where LK is the language of the structures in the class K. Now consider the L-theory  T = TK ∪ diag(Ai )

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With the aid of the general amalgamation property seen in the previous subsection we prove here that finitely subdirectly irreducible monadic MV-algebras are isomorphic to L-functional algebras. As a consequence we derive the strong completeness of monadic Łukasiewicz logic.

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The key result is the following.

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Lemma 3.6. Let (A, ∃, ∀) be a finitely subdirectly irreducible monadic MV-algebra. For each a ∈ A, there is a prime filter P on the MV-algebra A such that a/P = ∃a/P and P ∩ ∃A = {1}.

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Proof. Let F be the family of all implicative filters F of A such that F ∩ ∃A = {1} and ∃a → a ∈ F . We want to prove that there is a prime filter P in F . n We note first that F is nonempty for Fg(∃a → a) ∈ F . Indeed, if c ∈ Fg(∃a → a) ∩ ∃A, then c ≥ (∃a → a)2 for some positive integer n, so 2n

c = ∃c ≥ ∃(∃a → a)

2n

= (∃(∃a → a))

= (∃a → ∃a)

2n

=1

2n

= 1,

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

39 40 41 42 43 44 45 46 47 48 49 50 51 52

8 9

12

where we used (M5) repeatedly and Lemma 3.1. It is clear that the union of a chain of implicative filters in F is again an implicative filter in F . Thus, by Zorn’s Lemma, there is a maximal P in F . We claim that P is a prime filter. Assume P is not prime. Thus, there are x, y ∈ A such that x → y, y → x ∈ / P . Thus P is properly contained in Fg(P ∪ {x → y}) and also in Fg(P ∪ {y → x}). By the maximality of P in F , Fg(P ∪ {x → y}) ∩ ∃A = {1} and Fg(P ∪ {y → x}) ∩ ∃A = {1}. Let 1 = c ∈ Fg(P ∪ {x → y}) ∩ ∃A and 1 = d ∈ Fg(P ∪ {y → x}) ∩ ∃A. Then

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c ∨ d ∈ Fg(P ∪ {x → y}) ∩ Fg(P ∪ {y → x}) = Fg(P ∪ {(x → y) ∨ (y → x)}) = P .

20

Thus c ∨ d ∈ P ∩ ∃A = {1}, that is, c ∨ d = 1. Finally, since ∃A is totally ordered, c = 1 or d = 1, which is a contradiction. This shows that P is a prime filter. 

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Using the previous lemma we get a stronger version of Theorem 1.4 for monadic MV-algebras.

25

Corollary 3.7. Given a finitely subdirectly irreducible monadic MV-algebra (A, ∃, ∀), there exists a subdirect embedding α : A → i∈I Ai , where each Ai is a totally ordered MV-algebra and πi ◦ α|∃A : ∃A → Ai is an embedding, where πi is the i-th projection map. Moreover, for every a ∈ A, there are i, j ∈ I such that (πi ◦ α)(∃a) = (πi ◦ α)(a) and (πj ◦ α)(∀a) = (πj ◦ α)(a). 

Proof. It suffices to consider the subdirect representation of A given by A → P ∈PA/P , where P is the family of all prime filters of A such that P ∩ ∃A = {1}. By Theorem 1.4, we already know that P = {1}, so the representation is subdirect. The condition P ∩ ∃A = {1} guarantees that the projections on A/P are one-one on ∃A. Now, given a ∈ A, from the previous lemma there is P ∈ P such that ∃a/P = a/P . In addition, there is Q ∈ P such that ∃¬a/Q = ¬a/Q, so ∀a/Q = ¬∃¬a/Q = ¬¬a/Q = a/Q, where we used Lemma 3.1. 

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Theorem 3.8. Every finitely subdirectly irreducible monadic MV-algebra (A, ∃, ∀) is isomorphic to an L-functional monadic MV-algebra, that is, there exists an MV-chain B, an index set I and an embedding α : A → BI such that α(∃a)(i) = sup{α(a)(j ) : j ∈ I },

and

41 42

for every a ∈ A and i ∈ I .

43



Proof. Corollary 3.7 produces a family {Ai : i ∈ I } of MV-chains and an embedding α : A → i∈I Ai such that πi ◦ α|∃A : ∃A → Ai is an embedding for every i ∈ I . Consider the I -formation (∃A, {Ai : i ∈ I }, {πi ◦ α|∃A : i ∈ I }) in the elementary  class K of MV-chains. By Corollary 3.5, the I -formation has an amalgam (B, {βi : i ∈ I }) in K. Let β := βi : i∈I Ai → BI and note that β ◦ α : A → BI is an embedding. We claim that

for every a ∈ A and i ∈ I .

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α(∀a)(i) = inf{α(a)(j ) : j ∈ I }

(β ◦ α)(∃a)(i) = sup{(β ◦ α)(a)(j ) : j ∈ I }, and (β ◦ α)(∀a)(i) = inf{(β ◦ α)(a)(j ) : j ∈ I }

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Note first that, because of the amalgamation of the I -formation, (β ◦ α)(∃a)(i) = β(α(∃a))(i)

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= βi (α(∃a)(i))

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= βi (πi (α(∃a)))

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= βj (πj (α(∃a)))

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= βj (α(∃a)(j ))

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= β(α(∃a))(j )

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= (β ◦ α)(∃a)(j ) for every i, j ∈ I . We also know that there is i0 ∈ I such that (πi0 ◦ α)(∃a) = (πi0 ◦ α)(a). Then (β ◦ α)(a)(i0 ) = (β ◦ α)(∃a)(i0 ) = (β ◦ α)(∃a)(j ) ≥ (β ◦ α)(a)(j ) for every j ∈ I . This shows that (β ◦ α)(∃a)(i0 ) = sup{(β ◦ α)(a)(j ) : j ∈ I }. The condition for ∀a follows identically.  Corollary 3.9. The variety of monadic MV-algebras is generated, as a quasivariety, by its L-functional members. Using this corollary and Theorem 2.1 we get the following completeness result.

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Theorem 3.10 (Strong completeness for monadic Łukasiewicz logic). For any formula ϕ and any set of formulas , we have  S5(L) ϕ if and only if  S5(L) ϕ. Remark 3.11. Observe that the last result is a strong completeness theorem with respect to chain-based models. On the other hand, Rutledge’s result in [24] is different from ours in two ways. It is stronger in that the models considered are standard models, i.e. based on the standard MV-algebra [0, 1], but it is also weaker, in that it assumes  = ∅ (weak completeness). 4. Monadic Gödel logic

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Another important axiomatic extension of BL is the one obtained by adding the axiom p → p 2 . This is usually called Gödel logic and denoted by G. The corresponding equivalent algebraic semantics is the variety of Gödel algebras, that is, the subvariety of BL determined by the identity x 2 ≈ x (or, equivalently, by the identity x ∗ y ≈ x ∧ y). We denote this variety by G. In addition, Gödel algebras may also be defined as prelinear Heyting algebras, that is, G is the variety generated by totally ordered Heyting algebras. Equivalently, G is the subvariety of the variety of Heyting algebras determined by the prelinearity equation (x → y) ∨ (y → x) ≈ 1. This variety is generated by the BL-algebra based on the Gödel t-norm [0, 1]G (see, e.g., [18]). In this section we consider the axiomatic extension S5(G) of S5(BL) by the axiom p → p2 . Its corresponding equivalent algebraic semantics is the subvariety of MBL given by the identity x 2 ≈ x. We denote this subvariety by MG and call its members monadic Gödel algebras. We prove that the variety MG is generated as a quasivariety by its G-functional algebras, thus proving the completeness theorem stating that S5(G ) = S5(G ) (applying Theorem 2.1). We derive the generation by G-functional algebras from two facts: the variety MG has the finite embeddability property and finite subdirectly irreducible algebras in MG are G-functional.

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Remark 4.1. Monadic Heyting algebras were introduced by Monteiro and Varsavsky in [22] and later studied in depth by Bezhanishvili in [2]. Monadic Gödel algebras coincide with monadic prelinear Heyting algebras that satisfy the identity (M4) (see, e.g., [9,10]).

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4.1. Finite embeddability property

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In this section we prove that the variety MG has the finite embeddability property (FEP for short). As an immediate consequence, we obtain that the variety is generated, as a quasivariety, by its finite members. Note that the finite model property for S5(G) has been already established in [7,8]. Let us observe first that MG is not a locally finite variety. To see that, consider the monadic Gödel algebra A := ([0, 1]N G , ∃∨ , ∀∧ ) where [0, 1]G is the standard linearly ordered Gödel algebra of the real interval. Let S be the subalgebra of A generated by the element f0 (n) = 1 − n1 , n ∈ N, and let us define recursively a sequence f1 , f2 , . . .  1 if n ≤ i . of elements in S. For each i ∈ N, let fi = fi−1 ∨ (fi−1 → ∀∧ fi−1 ). It is easy to see that fi (n) = 1 1 − n if n > i Then S is infinite. Lemma 4.2. The class of finitely subdirectly irreducible members of MG has the FEP.

17 18 19 20

Proof. Let (A, ∃, ∀) be a finitely subdirectly irreducible monadic Gödel algebra, and let B := {a1 , . . . , ar } be a finite subset of A. Let us denote by C the subalgebra generated by B in A. Since the variety of Gödel algebras is locally finite, we know that C is finite. We want to define two unary operations ∃C and ∀C on C such that (C, ∃C , ∀C ) is a subdirectly irreducible monadic Gödel algebra; moreover, we want these operations to agree with ∃ and ∀ when possible, that is,

23

• if x ∈ C and ∃x ∈ C, then ∃C x = ∃x; • if x ∈ C and ∀x ∈ C, then ∀C x = ∀x. Let D = ∃A ∩ C, which is a subuniverse of C. Let us see that D is an m-relatively complete subalgebra of C.

28 29 30 31

(s1) This is immediate since D is finite. (s2) Observe that D is a chain, since ∃A is a chain. Consequently, we only need to prove that (s2 ) is satisfied. To do that, assume 1 = d ∨ x where d ∈ D and x ∈ C, and note that d ∈ ∃A. Since ∃A is an m-relatively complete subalgebra of A, d = 1 or x = 1. (s3) This is immediate since x ∗ y = x ∧ y. As a consequence, if we define on C the operations

36 37 38 39 40 41 42

∃C a := min{d ∈ D : d ≥ a},

∀C a := max{d ∈ D : d ≤ a},

then (C, ∃C , ∀C ) is a monadic Gödel algebra such that ∀C C = ∃C C = D = ∀C = ∃C. It remains to show that the new operations on C agree with the original operations whenever possible. Let x ∈ C such that ∃x ∈ C. Then ∃x ∈ D. Since x ≤ ∃x, ∃C x ≤ ∃x by definition of ∃C . On the other hand, we know that x ≤ ∃C x = ∃y for some y ∈ A. So, ∃x ≤ ∃∃y = ∃y = ∃C x. Thus ∃x = ∃C x. Suppose now that x ∈ C and ∀x ∈ C. Since ∀x ∈ D, by definition of ∀C y, we have that ∀x ≤ ∀C x. In addition, ∀C x = ∀y for some y ∈ A. Then, ∀y = ∀C x ≤ x, so ∀C x = ∀y = ∀∀y ≤ ∀x. Hence ∀C x = ∀x.  Corollary 4.3. The variety MG has the FEP. Corollary 4.4. The variety MG is generated, as a quasivariety, by its finite members. 4.2. Functionality of finite subdirectly irreducibles

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In this section we show that all finite subdirectly irreducible monadic Gödel algebras are isomorphic to G-functional algebras. Note that we are dealing here with finite algebras which, in addition, are (finitely) subdirectly irreducible. We do this in three steps.

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Lemma 4.5. Every finite monadic Gödel chain is isomorphic to a G-functional algebra.

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2 3 4 5 6 7 8 9 10 11 12 13

Proof. Let A be a finite monadic Gödel chain and ∃A = {c0 , c1 , . . . , ck } where 0 = c0 < c1 < . . . < ck = 1. Put Ai = {x ∈ A : ci−1 < x < ci } for 1 ≤ i ≤ k. Observe that, if Ai is not empty, then ∃Ai = {ci } and ∀Ai = {ci−1 }. N We will define an embedding ϕ from A into [0, 1]N G , ∃∨ , ∀∧ ). First consider the constant sequences gi ∈ [0, 1] i given by gi (n) = k for every n ∈ N and 0 ≤ i ≤ k. Put ϕ(ci ) = gi for 0 ≤ i ≤ k. Now, for 1 ≤ i ≤ k and j ≥ 1,

16 17 18 19

22

25

Observe that

13

(1)

gi−1 < fi

(2)

< fi

(j )

< . . . < fi

(j )

(j +1)

< fi

9 10 11 12

14

< . . . < gi .

(j )

15

(1)

(2)

(r )

Note also that ∃∨ fi = gi and ∀∧ fi = gi−1 for 1 ≤ i ≤ k, j ≥ 1. Let Ai = {ai , ai , . . . , ai i } for 1 ≤ i ≤ k where (j ) (j ) (1) (2) (r ) ai < ai < . . . < ai i . Then define ϕ(ai ) = fi for 1 ≤ i ≤ k, 1 ≤ j ≤ ri . It is straightforward to show that ϕ is the desired embedding. 

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16 17 18 19 20

Remark 4.6. Note that we have proved a stronger result. Every finite monadic Gödel chain is isomorphic to a subalgebra of [0, 1]N G , ∃∨ , ∀∧ ).

21 22 23

Given a monadic Gödel chain (A, ∃, ∀) and a natural number n, we define the algebra (A, ∃, ∀)n

= (An , ∃, ∀) where

24 25

the quantifiers are defined by

26

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6

8

23 24

5

7

20 21

4

consider the sequences ∈ [0, 1]N given by ⎧  j ⎪ ⎨i − 1 1− 1 if n is even, k k n+1 (j ) fi (n) =  j ⎪ ⎩i − 1 1 if n is odd. k k n+1

(j ) fi

14 15

3

∃(a1 , . . . , an )(i) = ∃(a1 ∨ . . . ∨ an ),

∀(a1 , . . . , an )(i) = ∀(a1 ∧ . . . ∧ an ),

1 ≤ i ≤ n.

We have that C = ∃(An ) = ∀(An ) = {(c, . . . , c) : c ∈ ∃A}. Moreover, it is easy to check that C is the universe of an m-relatively complete subalgebra of An and that ∃(a1 , . . . , an ) = min{(c, . . . , c) ∈ C : (a1 , . . . , an ) ≤ (c, . . . , c)}, ∀(a1 , . . . , an ) = max{(c, . . . , c) ∈ C : (a1 , . . . , an ) ≥ (c, . . . , c)}, for every (a1 , . . . , an

) ∈ An .

Thus

(A, ∃, ∀)n

is a monadic Gödel algebra.

27 28 29 30 31 32 33 34 35

Lemma 4.7. For every finite monadic Gödel chain (A, ∃, ∀) and every natural number n, the monadic Gödel algebra (A, ∃, ∀)n is isomorphic to a G-functional algebra.

36

Proof. By the previous lemma, (A, ∃, ∀) is isomorphic to a G-functional algebra. Let B be a Gödel chain and ϕ : A → BX an embedding such that ϕ(∃a) = ∃∨ ϕ(a) and ϕ(∀a) = ∀∧ ϕ(a) for every a ∈ A. This function induces an embedding ψ : An → BX×{1,...,n} given by ψ(a1 , . . . , an )(x, i) = ϕ(ai )(x) for x ∈ X, 1 ≤ i ≤ n. We claim that ψ(∃(a1 , . . . , an )) = ∃∨ ψ(a1 , . . . , an ) and ψ(∀(a1 , . . . , an )) = ∀∧ ψ(a1 , . . . , an ) for every (a1 , . . . , an ) ∈ An . We prove the first equation, the other one being analogous. We have

39

ψ(∃(a1 , . . . , an ))(x, i) = ϕ((∃(a1 , . . . , an ))(i))(x)

37 38

40 41 42 43 44 45 46

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= ϕ(∃(a1 ∨ . . . ∨ an ))(x)

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= ∃∨ (ϕ(a1 ) ∨ . . . ∨ ϕ(an ))(x)

48

49

= sup{ϕ(a1 )(x) ∨ . . . ∨ ϕ(an )(x) : x ∈ X}

49

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47

= sup{ϕ(ai )(x) : x ∈ X, 1 ≤ i ≤ n} = (∃∨ ψ(a1 , . . . , an ))(x, i).



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Remark 4.8. In the previous proof we could have taken B to be the standard Gödel algebra [0, 1]G and the set X to be N×{1,...,n} , ∃∨ , ∀∧ ), which, in turn, is isomorphic N. Thus we get that (A, ∃, ∀)n is isomorphic to a subalgebra of ([0, 1]G N to ([0, 1]G , ∃∨ , ∀∧ ), since N × {1, . . . , n} is countably infinite.

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Proof. Let (A, ∃, ∀) be a finite subdirectly irreducible monadic Gödel algebra. Thus  ∃A is a finite Gödel chain. By Theorem 1.4, there are finite Gödel chains A1 , . . . , An and an embedding ϕ : A → ni=1 Ai such that each πi ◦ ϕ is one-one on ∃A, where πi is the projection map on the i-th coordinate. Put ∃A = {c0 , c1 , . . . , ck } where 0 = c0 < c1 < . . . < ck = 1. Without loss of generality we may assume that Ai ∩ Aj = ∃A for i = j . Put Ai,j = {x ∈ Ai : cj −1 < x < cj } for 1≤ i ≤ n, 1 ≤ j ≤ k. Let B = ni=1 Ai and consider the Gödel algebra B with universe B and operations defined so that the induced ordering is the following: given b1 , b2 ∈ B, b1 ≤B b2 if and only if one of the following is true:

7

≤Ai

b2 ; • b1 , b2 ∈ Ai for some i and b1 • b1 ∈ Ai1 ,j1 , b2 ∈ Ai2 ,j2 , i1 = i2 , j1 < j2 ; • b1 ∈ Ai1 ,j1 , b2 ∈ Ai2 ,j2 , i1 < i2 , j1 = j2 .

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It is clear that Ai ≤ B for 1 ≤ i ≤ n. Moreover, B is an amalgam of the system of inclusion maps ∃A → Ai , 1 ≤ i ≤ n. Thus we may consider ϕ as an embedding from A into Bn . It is immediate to check that ∃A is an m-relative complete subalgebra of B; hence we can define quantifiers over B so that their image is ∃A. Therefore, we may apply the construction described before Lemma 4.7 to produce a monadic Gödel algebra (B, ∃, ∀)n . We claim that ϕ is an embedding from (A, ∃, ∀) into (B, ∃, ∀)n . Indeed, since ϕ is an embedding of Gödel algebras, for every a ∈ A: ϕ(∃a) = ϕ(min{c ∈ C : a ≤ c}) = min{ϕ(c) : ϕ(a) ≤ ϕ(c)} = ∃ϕ(a) and analogously ϕ(∀a) = ∀ϕ(a). Finally, by the last lemma, same is true for (A, ∃, ∀). 

(B, ∃, ∀)n

is isomorphic to a G-functional algebra, so the

Remark 4.10. From the proof of the previous theorem and the remarks following Lemmas 4.5 and 4.7, we actually get the following stronger result: Every finite subdirectly irreducible monadic Gödel algebra is a subalgebra of ([0, 1]N G , ∃∨ , ∀∧ ). Using now Corollary 4.4 and Theorem 2.1 we get the following results. Corollary 4.11. MG is generated, as a quasivariety, by its finite G-functional members. Corollary 4.12. MG is generated, as a quasivariety, by

6

15

([0, 1]N G , ∃∨ , ∀∧ ).

Theorem 4.13 (Strong completeness for monadic Gödel logic). For any formula ϕ and any set of formulas , we have  S5(G ) ϕ if and only if  S5(G ) ϕ.

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4.3. General functionality

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Theorem 4.9. Every finite subdirectly irreducible monadic Gödel algebra is isomorphic to a G-functional algebra.

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1

The notion of functional algebras that we use in this article is crafted to fit the way the models of basic logic were defined by Hájek in [18]. That is the reason for insisting that the base algebra A be a chain. However, the broader notion of functionality whereby A need not be totally ordered has also been considered in the literature (see e.g. [20,22,5]). It is worth noting that if A is a non totally ordered complete BL-algebra, the algebra (AX , ∃∨ , ∀∧ ) need not be a monadic BL-algebra. The same happens even if A is a complete Gödel algebra. For example, if A is the Gödel algebra whose lattice reduct is 1 ⊕ (2 × N), where N = (N, ≥), it may be checked that (AN , ∃∨ , ∀∧ ) does not satisfy condition

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(M4). However, in this section we show that for every monadic Gödel algebra (B, ∃, ∀) there is a Gödel algebra A and a Gödel embedding g : B → Aω such that g(∃b) = ∃∨ g(b) and g(∀b) = ∀∧ g(b) for b ∈ B. We follow a method indicated in [5] for monadic Heyting algebras, but with a slight modification of the proof. Given a V-formation of Gödel algebras (A, A1 , A2 , α1 , α2 ), an amalgam (B, β1 , β2 ) of the V-formation is a superamalgam if for every a1 ∈ A1 , a2 ∈ A2 and 1 ≤ i = j ≤ 2, βi (ai ) ≤ βj (aj ) implies there exists a ∈ A with βi (ai ) ≤ (βi ◦ αi )(a) = (βj ◦ αj )(a) ≤ βj (aj ).

8

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A2

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Maksimova proved in [21] that amalgamation and superamalgamation are equivalent conditions for varieties of Heyting algebras, and showed precisely which subvarieties of Heyting algebras enjoy both properties. The result we need now is the following.

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ψ1

A1

ψ2

A2

ψ3

···

35

B

λ0

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A0

B

λ1 ψ1

A1

ψ2

χ0

49

A2

···

χ2

52

32

37

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We skip the proof of the following result because it is the same as [5, Lemma 3.5] and [5, Lemma 3.3]. Lemma 4.15. For any a ∈ B, the set {(χn ◦ λn )(a) : n ∈ ω} has both a greatest lower bound and a least upper bound in A. Furthermore, for each k ∈ ω,

 (χn ◦ λn )(a) = (χk ◦ λk )(∀a) and (χn ◦ λn )(a) = (χk ◦ λk )(∃a). n∈ω

n∈ω

If a ∈ ∃B and m, n ∈ ω, then (χm ◦ λm )(a) = (χn ◦ λn )(a).

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45

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Let A, with the maps χn : An → A, be the direct limit of the family in the category associated with the variety G. Note that each χn is an embedding.

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Consider now the directed family generated by

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• For n = 0, let (A0 , ψ0 , λ0 ) be a superamalgam of the V-formation (∃B, B, B, ι, ι), and let α0 := ψ0 ◦ ι = λ0 ◦ ι : ∃B → A0 . • For n > 0, let (An , ψn , λn ) be a superamalgam of the V-formation (∃B, An−1 , B, αn−1 , ι), and let αn := ψn ◦ αn−1 = λn ◦ ι : ∃B → An .

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Let (B, ∃, ∀) be a monadic Gödel algebra. We know that ∃B is a subalgebra of B, so the inclusion map ι : ∃B → B is an embedding. For each n ≥ 0, recursively define algebras An and embeddings ψn , λn and ϕn as follows:

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Theorem 4.14 ([21, Theorem 1 and Proposition 6]). Every V-formation in G has a superamalgam in G.

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Theorem 4.16. Every monadic Gödel algebra is isomorphic to a functional monadic Gödel algebra (in the broader sense).

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Proof. Let (B, ∃, ∀) be a monadic Gödel algebra and let us consider the direct limit A constructed as indicated above. Define the map g : B → Aω by g(a)(n) = (χn ◦ λn )(a). Since χn ◦ λn is a Gödel embedding, g is a Gödel embedding. Furthermore, from Lemma 4.15, for each a ∈ G and k ∈ ω,

(χn ◦ λn )(a) = (χk ◦ λk )(∀a) = g(∀a)(k), (∀∧ g(a)) (k) = n∈ω

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and

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(χn ◦ λn )(a) = (χk ◦ λk )(∃a) = g(∃a)(k).

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This shows that (B, ∃, ∀) is isomorphic to a functional algebra. 

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4.3.1. Concluding remarks and future work In this article we presented a way of proving the completeness of certain monadic logical calculi by means of algebraic representation theorems. We would like to pursue this route even deeper in order to prove the completeness for monadic basic logic. The key fact we used to prove the representation theorems in the cases addressed in this article was the amalgamation property. However, this is not enough in general. We already have some partial results that point in the right direction, but have not been able to encompass all cases. For example, we are currently exploring monadic product logic, which has turned out to be more complex. In cases where completeness holds, a deeper study of the equivalent algebraic semantics is in order. Monadic MValgebras have been already extensively studied (starting from Rutledge in [24] and continuing with [16,1,12–15]), but research in monadic Gödel algebras has only been done in the last few years, see [9,7,8]; for a more algebraic setting regarding monadic Heyting algebras see [2–5]. We are currently studying monadic Gödel algebras in depth. Acknowledgements We would like to thank the anonymous referees for their useful remarks that improved the clarity of this article. References

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