On elementary equivalence in fuzzy predicate logics

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models.

which have some feature that interests us. Classifying a class of structures means grouping the structures into subclasses in a useful way, and then proving that every structure in the collection does belong in just one of the subclasses. The most basic classification in classical model theory is given by the relations of elementary equivalence and isomorphism. Our purpose in the present article is to characterize the relation of elementary equivalence between two structures of a fuzzy predicate language in terms of elementary extensions.
The basic notion of elementary equivalence between models is due to Tarski (see [23]) and the fundamental results on elementary extensions and elementary chains were introduced by Tarski and Vaught in [24]. For a general survey on the subject and an historical overview we refer the reader to [3]. In the context of fuzzy predicate logics, elementarily equivalent structures were defined in [16] (Definition 10), our starting point is the research done in this article. There the authors presented a characterization of conservative extension theories using the elementary equivalence relation (see Theorems 6 and 11 of [16]). Future work will be devoted to analise the relationship of our investigation with other approaches, for instance the one presented in [20], where a notion of elementary equivalent models in a degree d was introduced (Definition 4.33).
Hájek and Cintula proved in Theorem 6 of [16] that, in core fuzzy logics, a theory T 2 is a conservative extension of another theory T 1 if and only if each exhaustive model of T 1 is elementarily embedded in a model of T 2 . Then, they conjectured the same result to be true for arbitrary structures (Conjectures 1 and 2 of [16]). In this paper we present a counterexample to Conjectures 1 and 2, using a refinement of the method of diagrams developed in [8].
Special attention is devoted to witnessed and quasi-witnessed elementary extensions. The interest for the study of this kind of structure has grown recently (cf. [2,14,15] and [16]). The aforementioned papers show the importance of witnessed and quasi-witnessed models for applications in logic-based knowledge representation in artificial intelligence. In this article we prove that these classes of models have some good model-theoretic properties, allowing us to generalize some results of Section 7 of [16] and of Section 4 of [2] to non-exhaustive models.
This paper is a revised and extended version of the contribution [9] of the authors to the IPMU'10 Conference. The article is structured as follows: Sect. 2 is devoted to preliminaries on fuzzy predicate logics. In Sect. 3 we study the basic properties of elementary extensions and we present an analog to the Tarski-Vaught Test in the fuzzy context. In Sect. 4 we introduce some known definitions and basic facts on canonical models (see section 4 and 5 of [16]) and of the method of diagrams developed in [8]. Later on in this section, we prove some new propositions related to canonical models and diagrams. Section 5 is devoted to the study of witnessed and quasi-witnessed extensions, and in Sect. 6 we present a counterexample to Conjectures 1 and 2 of [16], using the results of Sect. 4. Finally, in Sect. 7 we present a characterization theorem of elementary equivalence in fuzzy predicate logics. We conclude the article with a section of work in progress and future work.

Preliminaries
In his seminal book [12], Hájek considered the problem of finding a common base for the most important fuzzy logics, namely Łukasiewicz, Gödel and product logics. There, he introduced a logic, named BL, and he proposed it for the role of basic fuzzy logic. Hájek's proposal was greatly supported by the proof that BL is the logic of all continuous t-norms of their residua (see [4]). But in [11] the authors observed that the minimal condition for a t-norm to have a residuum, and therefore to determine a logic, is left-continuity (continuity is not necessary). There, they proposed a weaker logic, called MTL (monoidal t-norm based logic), and conjectured that MTL is the logic of left-continuous t-norms and their residua. This conjecture was proved in [17].
In the literature of t-norm based logics, one can find not only a number of axiomatic extensions of MTL but also extensions by means of expanding the language with new connectives such as the Δ connective or with an involutive negation. All of MTL extensions and most of its expansions defined elsewhere share the property of being complete with respect to a corresponding class of linearly ordered algebras. To encompass all these logics and prove general results common to all of them, Cintula introduced in [5] the notion of core fuzzy logics (he also defines the class of Δ-core fuzzy logics to capture all expansions having the Δ connective).
Our study of the model theory of fuzzy predicate logics is focused on the basic fuzzy predicate logic MTL∀ and some of its expansions based on propositional (Δ-)core fuzzy logics. We start by introducing the definition of propositional (Δ-)core fuzzy logic: Definition 1 A propositional logic L is a core fuzzy logic iff L satisfies: -(LDT ) Local Deduction Theorem: for each theory T and formulas φ, ϕ: T, ϕ L φ iff there is a natural number n such that T L ϕ n → φ. and we say that L is a Δ-core fuzzy logic if -(DT ) Delta Deduction Theorem: for each theory T and formulas φ, ϕ: So defined (Δ-)core fuzzy logics are axiomatic extensions of MTL (of MTL , respectively). We state now, without proof, a useful property of (Δ-)core fuzzy logics that we will use later on (for a proof of this result see Theorem 1 of [6]).
Following [12] we introduce the syntax of fuzzy predicate logics. A predicate language Γ is a triple (P,F,A) where P is a non-empty set of predicate symbols, F is a set of function symbols and A is a mapping assigning to each predicate and function symbol a natural number called the arity of the symbol. The function symbols F for which A(F) = 0 are called the object constants. The predicate symbols P for which A(P) = 0 are called the truth constants.
Formulas of the predicate language Γ are built up from the symbols in (P,F,A), the connectives and truth constants of a fixed (Δ-)core fuzzy logic L, the logical symbols ∀ and ∃, variables and punctuation. From now on, the formulas of a predicate language Γ will be called Γ -formulas. A Γ -sentence is a Γ -formula without free variables.
Throughout the paper we consider the equality symbol as a binary predicate symbol not as a logical symbol, we work in equality-free fuzzy predicate logics. That is, the equality symbol is not necessarily present in all the languages and its interpretation is not fixed. Given a propositional (Δ-)core fuzzy logic L, we denote by L∀ the corresponding fuzzy predicate logic.
Let L be a fixed (Δ-)core fuzzy logic, we introduce now an axiomatic system for the predicate logic L∀: (P) the axioms resulting from the axioms of L by the substitution of the propositional variables by the Γ -formulas.
The deduction rules are those of L (modus ponens and, in the case where L is a Δ-core fuzzy logic, necessitation for the Baaz-Monteiro connective Δ: from φ infer φ) and generalization: from φ infer ∀xφ. By Σ L∀ α we denote that the formula α follows from the set of formulas Σ in the axiomatic system of the fuzzy predicate logic L∀. When it is clear by the context we omit the subscript L∀.
Let L be a fixed propositional (Δ-)core fuzzy logic, we introduce now the semantics for the fuzzy predicate logic L∀: Given an L-algebra B and a B-structure M, an M-evaluation of the object variables is a mapping v which assigns to each variable an element of M. Let B be an L-algebra, M be a B-structure and v be an M-evaluation, we define the values of the terms and truth values of the formulas as follows: for each variable x, each object constant c ∈ Γ , each n-ary function symbol F ∈ Γ for n > 0 and Γ -terms t 1 , . . . , t n , respectively.
for each n-ary predicate P ∈ Γ , for each n-ary connective δ ∈L and Γ -formulas φ 1 , . . . , φ n . Finally, for the quantifiers, provided the infimum/supremum exists; otherwise the truth value of the formula in question is undefined. Remark that, since the L-algebras we work with are not necessarily complete, the above suprema and infima could be not defined in some cases. It is said that a B-structure is safe if such suprema and infima are defined for all the formulas. From now on we assume that all our structures are safe. Safe models were introduced in [21] under the name of interpretations and in [10] as completely valued models. B-structures were introduced in [22] under the name of realizations and in [10] under the name of fuzzy models. The formalization of this concept that we use at present is due to [12]. If v is an evaluation such that for each 0 Definition 5 Let T ∪ {φ} be a set of Γ -sentences. We say that φ is a semantical consequence of T (denoted by T | φ) iff for every L-algebra B and every B-structure M, if M is a model of T , then M is also a model of φ.
From now on, given an L-algebra B, we say that (B, M) is a Γ -structure instead of saying that M is a B-structure for a predicate language Γ . Definition 6 Let (B, M) be a Γ -structure, by Alg(B, M) we denote the subalgebra of B whose domain is the set are elementarily equivalent, that is, that they are models of exactly the same Γ -sentences. The definition of elementarily equivalent classical structures is due to A. Tarski (see [23]). In the fuzzy setting was introduced in [16] (Definition 10).
From now on and throughout the article, we will assume that B is always an L-chain. In this section we have presented only a few definitions and notation, a detailed introduction to the syntax and semantics of fuzzy predicate logics can be found in [12] and [6].

Weak homomorphisms, σ -mappings and homomorphisms
There are several ways to extend classical model theory to fuzzy logic. The concepts of embedding and elementary embedding can be extended in several ways to fuzzy logic. Our choice for the counterparts of these notions in fuzzy logic (weak homomorphisms and elementary mappings) has been motivated, not only for the desire to find extensions of the corresponding notions of classical predicate logics but also to try to encompass the most commonly used definitions in the literature of predicate fuzzy logic. Different definitions have been used so far for basic model-theoretic operations on structures. For instance, the notion of elementary submodel, morphism (see [10]), elementary embeddings and submodels(see [16]), fuzzy submodel, elementary fuzzy submodel and isomorphism of structures of first-order fuzzy logic with graded syntax (see [20]), complete morphism in languages with a similarity predicate (see [1]) and the notion of σ -embedding (see [7]). Taking as our starting point all these works, we have defined these model-theoretic notions as general as possible. We want to contemplate the possibility that our languages contain function symbols and also the equality symbol but not as a logical symbol (for instance, interpreted as a similarity). For some mathematical purposes, such as algebraic applications, this could seem to be useless. But in other scientific disciplines, such as computer science or artificial intelligence, it is important to have fuzzy predicate logics able to deal with similarities, and at the same time with functions. In this section we recall the notions of weak homomorphism and of homomorphism as introduced in [8].
In fuzzy predicate languages, since we often do not work with crisp equalities, we can find mappings that preserve all quantifier-free formulas but are not homomorphisms (in the classical sense) between the algebraic reducts of the models (that is, between the interpretations of the function symbols). Since these two notions do not coincide, unlike in classical first-order logic, we define both notions, recall their differential properties and their relationship to basic constructions on model theory.
Moreover, if in addition: -Condition 3 above holds for every Γ 1 -formula, (g, f ) is said to be an elementary mapping. -g preserves the existing infima and suprema, (g, f ) is said to be a σ -mapping.
-For each n-ary function symbol F ∈ Γ 1 and elements d 1 , . . . , d n ∈ M 1 , If (g, f ) is both a σ -mapping and a homomorphism we would say that (g, f ) is a σ -homomorphism. Moreover, we would say that (g, f ) is an embedding when (g, f ) is a homomorphism and both g and f are one-to-one, and we denote by (B, M) ∼ = (A, N) when these two structures are isomorphic (that is, there is an embedding (g, f ) from (B, M) into (A, N) with g and f onto). Homomorphisms that in addition are elementary mappings will be called elementary homomorphisms. Remark that, unlike [1], homomorphisms are crisp when restricted to the algebraic reducts of the models. Observe also that, working with predicate languages without function symbols, the notions of weak homomorphism and homomorphism coincide, but it is not the case for arbitrary structures.
In the recent development of model theory for fuzzy predicate languages (see for instance [6] and [16]), elementary mappings have been used to study fundamental notions such as elementary equivalence or the notion of conservative extension of a theory. However, it is customary in classical model theory to use the notion of isomorphism instead. The main interest for using a weaker notion comes from the fact that, working in equality-free languages, it is possible to find mappings that preserve all the formulas of the language, yet not capturing the structural properties of the models (and this suffices for some of our purposes). Remark that this does not happen in classical predicate logic, being the isomorphisms the only mappings satisfying both conditions (preserving all the formulas and, at the same time, preserving the structure of the models).
Note that, by definition, weak homomorphisms are not always σ -mappings and homomorphisms are not always σ -homomorphisms (as are in [1] or [10]). We will see now that σ -mappings enjoy some good properties. The following proposition is a reformulation of Propositions 6.1 and 6.2 in [10]: The previous statement shows us that there is a close relationship between elementary mappings and σ -mappings (σ -homomorphisms in particular) with f onto. However, as shown in [8], these two notions do not coincide.

Elementary extensions
In this section we introduce the notion of elementary substructure and we present an analog to the Tarski-Vaught Test for fuzzy predicate logics.  of (B, M). Remember that we have assumed that all our structures are safe. It is easy to check, by induction on the complexity of the formulas, that, for every substructure, condition ( ) holds restricted to quantifier-free formulas. The transitivity of the notion of elementary mapping was stated in [16]. Now we introduce the notion of definable set of elements of an L-algebra. (A, N) be a Γ -structure, K ⊆ N , e 1 , . . . , e n ∈ K and φ(x, y 1 , . . . , y n ) be a Γ -formula. We denote by X (A,N) φ,e 1 ,...,e n ,K the following subset of A: . . . , e n ∈ N and a Γ -formula φ(x, y 1 , . . . , y n

Definition 10 Let
The following result is a fuzzy version of the Tarski-Vaught Test for elementary extensions. The Tarski-Vaught criterion is a necessary and sufficient condition for a substructure to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure. The proof of this result follows easily by induction on the complexity of the formulas by using the definition introduced above. (A, N) and (B, M) be two Γ -structures. Then the following are equivalent:

Diagrams and canonical models
The method of diagrams, due to L. A. Henkin and A. Robinson, has proved to be a useful tool for model theory. During the 1950s the maps between structures came to play a deeper role in model theory, not just as possible research fields but as essential tools of the subject. For a general reference on the method of diagrams in classical predicate logic see [3].
For the special case of structures of first-order fuzzy logic with graded syntax and languages with the equality symbol, diagrams are presented in [20]. For fuzzy predicate logics in general, diagrams are used in the proofs of Lemma 4 in [16], and full diagrams can be found in the proof of Theorem 5.9 in [7]. Finally, the diagram technique was developed for arbitrary predicate core fuzzy logics in [8].

The method of diagrams in fuzzy predicate logics
Certain classes of mappings preserve all formulas with certain syntactic forms. Conversely, we can classify mappings by means of the formulas they preserve. In this subsection we recall some definitions and characterizations of mappings in fuzzy predicate logics developed in [8]. Later on we prove some new propositions related to canonical models and diagrams.  (a 1 , . . . , a k ), where δ, are L- terms and a 1 , . . . , a k , b 1 , . . . , b n (g, f ) from (B, M) into (A, N).

There is an elementary mapping
Moreover, g is one-to-one iff for every sentence ψ ∈ NEQ(B), the expansion of (A, N) in condition 1. is not a model of ψ.

Moreover, g is one-to-one iff for every sentence of Γ M , ψ / ∈ EDIAG 0 (B, M), the expansion of (A, N) in condition 1. is not a model of ψ.
Remark that, as pointed out in [8], the mapping f of Proposition 14 and of Corollary 15 is not necessarily one-to-one, because we do not work with a crisp equality.

Canonical models
Now we recall some definitions and basic facts on canonical models of fuzzy predicate logics (cf. section 4 and 5 of [16]).

Definition 16 A Γ -theory T is linear iff for each pair of
there is a Γ -sentence χ ∈ Ψ such that both φ → χ and ψ → χ are probable.

Definition 18
Let Γ and Γ be predicate languages such that Γ ⊆ Γ and let T be a Γ -theory. We say that T is Γ -Henkin if for each formula ψ(x) ∈ Γ such that T ∀xψ, there is a constant c ∈ Γ such that T ψ(c). And we say that T is ∃-Γ -Henkin if for each formula ψ(x) ∈ Γ such that T ∃xψ, there is a constant c ∈ Γ such that T ψ(c). Finally, a Γ -theory is called doubly-Γ -Henkin if it is both Γ -Henkin and ∃-Γ -Henkin. In case that Γ = Γ , we say that T is Henkin (∃-Henkin, doubly Henkin, respectively).
Theorem 19 (Theorem 2.20 of [6]) Let T 0 be a Γ -theory and Ψ a directed set of Γ -sentences such that T 0 Ψ . Then, there is a linear doubly Henkin theory T ⊇ T 0 in a predicate language Γ ⊇ Γ such that T Ψ .

Definition 20 Let T be a Γ -theory. The canonical model of T, denoted by (Lind T , CM(T )), where Lind T is the Lindenbaum algebra of T (that is, the L-algebra of classes of T-equivalent Γ -sentences) is defined as follows: the domain of CM(T )
is the set of closed Γ -terms, for every n-ary function symbol F ∈ Γ , F(t 1 . . . t n ) and for each n-ary predicate symbol P ∈ Γ , P (Lind T ,CM(T )) (t 1 From now on we will use a shorter notation and write CM(T ) instead of (Lind T , CM(T )).

Refinements of the method of diagrams
Now we prove some new facts on diagrams and elementary extensions, using canonical models. The following proposition can be regarded as an improvement of Proposition 32 of [8] in two main aspects. On the one hand, the technique obtains now an elementary extension (a canonical model) which is well-known for us and has some good model-theoretic properties. On the other hand, we have an elementary mapping (g, f ) with g and f one-to-one, fact that does not hold in general for arbitrary structures. Observe that Theorem 19 does not hold for Δ-core fuzzy logics (for an explanation see [16]). Theorem 23 (Theorem 2.21 of [6]) Let L be a Δ-core fuzzy logic, T 0 a Γ -theory and Ψ a directed set of Γ -sentences such that T 0 Ψ . Then, there is a linear Henkin theory T ⊇ T 0 in a predicate language Γ ⊇ Γ such that T Ψ .
Nevertheless, imitating the proof of Proposition 22 and using Theorem 23, it is possible to obtain a version of the previous result for Δ-core fuzzy logics: L be a Δ-core fuzzy logic, (B, M)

Witnessed and quasi-witnessed models
In this section we focus on extensions of witnessed and quasi-witnessed models. We will show a direct application of Proposition 22, giving a generalization of Lemmas 3.3 of [2] and 5 of [16] for non-exhaustive models.
Notice that every model of classical predicate logic is witnessed, as well as all the models of finitely-valued logics. However, when we move to infinitely valued logics, this is not always the case. The infimum or supremum of a set of truth values might not be included in this set, and thus we can find models in which some quantified formula has no witness. Following these ideas, Hájek introduced in [14,15] the aforementioned notion of witnessed model and proved that this is an important property because it implies a limited form of finite model property for certain fragments of predicate fuzzy logic (see [13]). In [16] the following axiom schemes, originally introduced by Baaz, are discussed: (C∀) ∃x(φ(x) → ∀yφ(y)) and (C∃) ∃x(∃yφ(y) → φ(x)). Cintula and Hájek showed in [16] that adding the witnessed axioms (C∀) and (C∃) to any first order core fuzzy logic, we obtain a logic complete with respect this kind of models. Subsequently, they proved that these axioms are derivable in Łukasiewicz First Order Logic, showing that Ł∀ is complete with respect to witnessed models (we will say that Ł∀ has the witnessed model property). Nevertheless, they proved that neither G∀ nor ∀ share this property (the witnessed axioms are not theorems of these logics). In fact, no other first order logic of a continuous t-norm enjoys this property. This characteristic is strongly related to the continuity of the truth functions, a property that only Łukasiewicz logic has. Now we show a direct application of Proposition 22 to obtain witnessed models, giving a generalization of Lemma 5 of [16] to non-exhaustive structures.

Proposition 27 Let T be a Γ -theory and T its extension with axioms C∀ and C∃.
Then every Γ -structure model of T is elementarily one-to one mapped into a witnessed model of T .
Proof Let (B, M) be a Γ -structure model of T . We consider the theory T 0 = FDI-AG (M, B). Now let Ψ be the closure of NEQ(B) under disjunctions. Clearly Ψ is a directed set. We show that T 0 Ψ : it is enough to prove that for every α, β ∈NEQ(B), α∨β / ∈ T 0 . Assume the contrary, since B is an L-chain, we have that either α → β ∈ T 0 or β → α ∈ T 0 . Then, since L is a core fuzzy logic, by Theorem 2, we will have either that α ∈ T 0 or β ∈ T 0 , which is absurd, by definition of NEQ(B).
Then, by Proposition 22, since T 0 ⊇ EDIAG(B, M) and Ψ ⊇ NEQ(B), there is a linear doubly Henkin theory T * ⊇ T 0 such that T * Ψ and (B, M) elementarily one-to one mapped into CM(T * ).
Now we see that CM(T ) is witnessed: let φ(y, x 1 , . . . , x n ) be a formula and t 1 , . . . , t n closed terms elements of CM(T ), assume that the constants that occur in these terms are c 1 , . . . , c k . Consider now the formula φ (y, c 1 , . . . , c k ) obtained from φ(y, x 1 , . . . , x n ) by substituting the variables x 1 , . . . , x n for the terms t 1 , . . . , t n . By assumption, T (C∃), we have then that T ∃z(∃yφ (y, c 1 , . . . , c k ) → φ (z, c 1 , . . . , c k )) and thus, since T is ∃-Henkin, there is a constant d such that , c 1 , . . . , c k ). Then, by definition of CM(T ), we have that The proof for the universal step is analogous by using axiom (C∀).
In [18] it is proved that Product Predicate Logic ∀ enjoys a property weaker than the witnessed model property, the so-called quasi-witnessed model property.
Quasi-witnessed models are models in which universally quantified formulas taking truth-values greater than 0 have witnesses, while existentially quantified formulas are always witnessed. In [2] the authors introduced the so-called quasi-witnessed axioms: and they proved that the axiomatic extension of any strict core fuzzy logic 1 with the quasi-witnessed axioms is complete w.r.t. quasi-witnessed models. In particular they proved that in ∀ these axioms are deducible, and thus ∀ is complete w.r.t. quasiwitnessed models (result already proved directly in [18]). Finally it is also proved that no other logic of a continuous t-norm except Ł∀ and ∀ satisfy the quasi-witnessed axioms. Notice that we have taken the name coined in [2] for this kind of models instead of the original one of [18], which was strictly closed models. The main reason is that the new name seems more informative about the properties of the described model. Moreover, with this notation we avoid possible confusions with other usages of the name closed in mathematics and logic. Imitating the proof of Proposition 27 and using the method of diagrams we can generalize Lemma 19 of [2] to non-exhaustive models.

Proposition 28 Let T be a Γ -theory and T its extension with axioms C∀ and C∃.
Then every Γ -structure model of T is elementary one-to one mapped into a quasiwitnessed model of T .

Counterexample to conjectures 1 and 2 of [16]
Given two theories T 1 ⊆ T 2 in the respective predicate languages Γ 1 ⊆ Γ 2 , it is said that T 2 is a conservative extension of T 1 if and only if each Γ 1 -formula provable in T 2 is also provable in T 1 .
Hájek and Cintula proved in Theorem 6 of [16] that, in core fuzzy logics, a theory T 2 is a conservative extension of another theory T 1 if and only if each exhaustive model of T 1 can be elementarily one-to one mapped into some model of T 2 . In Theorem 7 of [16], they conjectured the same result to be true for arbitrary structures, showing that the following two conjectures were equivalent: Conjecture 1 of [16]: Let P be a truth constant symbol and for i ∈ {1, 2}, T i be a Γ i -theory, and T + i be a Γ i ∪ {P}-theory such that T + i = T i (i.e. P is added to the language but no new axioms are added). If T 2 is a conservative extension of T 1 , then T + 2 is a conservative extension of T + 1 . Conjecture 2 of [16]: A theory T 2 is a conservative extension of another theory T 1 if and only if each model of T 1 can be elementarily one-to one mapped into some model of T 2 .
We present here a counterexample to Conjecture 2 (and thus to Conjecture 1). Let L be the logic that has as equivalent algebraic semantics the variety generated by the union of the classes of Łukasiewicz and Product chains. L is an axiomatic extension of BL, to find an axiomatization we refer to [4] where it is proved also that the set of chains of the variety coincide with the union of the sets of Łukasiewicz and Product chains). Let now ({0, 1}, M) be a classical first-order structure in a predicate language Γ , and let B 1 = [0, 1] and B 2 = [0, 1] Ł be the canonical Product and Łukasiewicz chains, respectively.
Remark that the structure ({0, 1}, M) can also be regarded as a Γ -structure over both B 1 A, N). Consequently, there is an L-embedding k from [0, 1] into A and at the same time there is an L-homomorphism h from [0, 1] Ł into A (not necessarily one-to-one). If A is an L-chain, it is clear that this is not possible. We show now that, for any arbitrary L-algebra A, this fact leads to a contradiction. 2 If such embeddings k and h exist, and c and b are the images of 1/2 under h and k respectively, we have b = ¬b (because h is an L-homomorphism), c < 1 and ¬c = 0 (because k is an L-embedding and the negation in [0, 1] is the Gödel negation). If we decompose A as a subdirect product of an indexed family of subdirectly irreducible BL-chains, say (A i : i ∈ I ), every such A i is either a Łukasiewicz, or a Product chain (for a reference see [12] and [4]). Therefore, if we take an index i such that the i-component, c i , satisfies 0 < c i < 1, we will have at the same time ¬c i = 0 and for the i-component b i , b i = ¬b i , which is absurd, because A i can not be, at the same time, a Łukasiewicz and a Product chain.
We show now that T 0 Ψ . Otherwise, if for some α ∈ Ψ , T 0 α, since the set EDIAG 0 (B 1 , M 1 ) is closed under conjunction and the proof is finitary, there is ψ ∈ EDIAG 0 (B 1 , M 1 ) such that EDIAG 0 (B 2 , M 2 ), ψ α. Then, by the same kind of argument we used to show that T 0 is consistent, we would obtain that α ∈ EDI-AG 0 (B 1 , M 1 ), which is absurd.
Then, by Corollary 25, there is a linear doubly Henkin theory T ⊇ T 0 in a predicate language Γ ⊇ Γ such that T Ψ and an elementary mapping (g, f ) from (B 1 , M 1 ) into CM(T ), with g and f one-to-one. Moreover, since CM(T ) is also a model of EDIAG 0 (B 2 , M 2 ), by Corollary 15, (B 2 , M 2 ) is elementarily mapped into CM(T ). Finally, by Lemma 21, Lind T is an L-chain.
By the analogue of Corollary 25 for Δ-core fuzzy logics, if we substitute in the first line of the last paragraph of the previous proof the expression 'linear doubly Henkin theory' by 'linear Henkin theory', then Theorem 29 holds also for Δ-core fuzzy logics.
Remark that Theorem 29 can not be generalized to arbitrary structures. If we take the structures of the counterexample to Conjectures 1 and 2 of Sect. 7, we have

Future work
Work in progress is devoted to find characterizations of the notion of elementary equivalence using other model-theoretic constructions such as ultraproducts. In our future research we plan to analise the relationship of our study with other approaches, for instance the one presented in [20], where a notion of elementary equivalent models in a degree d was introduced (cf. Definition 4.33).