Syntactic Characterizations of Classes of First-Order Structures In

Soft Computing (2019) 23:2177–2186 https://doi.org/10.1007/s00500-019-03850-6

FOCUS

Syntactic characterizations of classes of ﬁrst-order structures in mathematical fuzzy logic

Guillermo Badia1,2 · Vicent Costa3 · Pilar Dellunde3,4,5 · Carles Noguera6

Published online: 22 February 2019 © The Author(s) 2019

Abstract This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their ﬁrst-order axiomatization. We focus on classes given by universal and universal–existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a ﬁnite MTL-algebra, from which analogues of the Ło´s–Tarski and the Chang–Ło´s–Suszko preservation theorems follow.

Keywords Graded model theory · Mathematical fuzzy logic · Universal classes · Universal-existential classes · Amalgamation theorems · Preservation theorems

1 Introduction

Graded model theory is the generalized study, in mathematical fuzzy logic (MFL), of the construction and classification of graded structures. The field was properly started in Hájek This paper is dedicated to Lluís Godo in the occasion of his 60th birthday. and Cintula (2006) and has received renewed attention in recent years (Badia and Noguera 2018a; Bagheri and Moniri Communicated by L. Spada. 2013; Cintula and Metcalfe 2013; Cintula et al. 2015;Costa and Dellunde 2017; Dellunde 2011, 2014). Part of the pro- B Guillermo Badia [email protected] gramme of graded model theory is to find non-classical analogues of results from classical model theory (e.g. Hodges Vicent Costa [email protected] 1993; Sacks 1972; Chang and Keisler 1973). This will not only provide generalizations of classical theorems but will Pilar Dellunde [email protected] also provide insight into what avenues of research are particular to classical first-order logic and do not make sense in a Carles Noguera [email protected] broader setting. On the other hand, classical model theory was developed 1 Department of Knowledge-Based Mathematical Systems, together with the analysis of some very relevant math- Johannes Kepler University Linz, Linz, Austria ematical structures. In consequence, its principal results 2 School of Historical and Philosophical Inquiry, University provided a logical interpretation of such structures. Thus, if of Queensland, Brisbane, Australia we want model theory’s idiosyncratic interaction with other 3 Department of Philosophy, Universitat Autónoma de disciplines to be preserved, the redefinition of the funda- Barcelona, Bellaterra, Catalonia, Spain mental notions of graded model theory cannot be obtained 4 Artificial Intelligence Research Institute (IIIA – CSIC), from directly fuzzifying every classical concept. Quite the Bellaterra, Catalonia, Spain contrary, the experience acquired in the study of different 5 Barcelona Graduate School of Mathematics, Barcelona, structures, the results obtained using specific classes of struc- Catalonia, Spain tures, and the potential overlaps with other areas should 6 Institute of Information Theory and Automation, Czech determine the light the main concepts of graded model theory Academy of Sciences, Prague, Czech Republic 123 2178 G. Badia et al. have to be defined in. It is in this way that several fundamen- We end with some concluding remarks and suggestions for tal concepts of the model theory of mathematical fuzzy logic lines of further research. have already appeared in the literature. The goal of this paper is to give syntactic characterizations of classes of graded structures; more precisely, 2 Preliminaries we want to study which kinds of formulas can be used to axiomatize certain classes of structures based on finite In this section, we introduce the syntax and semantics (expansions of) MTL-chains. Traditional examples of such of fuzzy predicate logics, and recall the basic results on sort of results are preservation theorems in classical model diagrams we will use in the paper. We use the notation theory, which, in general, can be obtained as consequences and definitions of the Handbook of Mathematical Fuzzy of certain amalgamation properties (cf. Hodges 1993). We Logic (Cintula et al. 2011). provide some amalgamation results using the technique of diagrams which will allow us to establish analogues of the Definition 1 (Syntax of Predicate Languages)Apredicate P , , Ło´s–Tarski preservation theorem (Hodges 1993, Theorem language is a triple PredP FuncP ArP , where PredP 6.5.4) and the Chang–Ło´s–Suszko theorem (Hodges 1993, is a non-empty set of predicate symbols, FuncP is a set Theorem 6.5.9). of function symbols (disjoint from PredP ), and ArP repre- This is not the first work that addresses a model-theoretic sents the arity function, which assigns a natural number to study of the preservation and characterization of classes of each predicate symbol or function symbol. We call this nat- fuzzy structures. Indeed, Bagheri and Moniri (2013)have ural number the arity of the symbol. The predicate symbols obtained results for the particular case of continuous model with arity zero are called truth-constants, while the function symbols whose arity is zero are named object constants theory by working over the standard MV-algebra [0, 1]Ł and with a predicate language enriched with a truth-constant for (constants for short). each element of [0, 1] . In that context, they characterize Ł P-terms, P-formulas, ∀n and ∃n P-formulas, and the universal theories in terms of the preservation under sub- notions of free occurrence of a variable, open formula, substructures (Bagheri and Moniri 2013, Prop. 5.1) and prove stitutability, and sentence are defined as in classical predicate versions of the Tarski–Vaught theorem (Bagheri and Moniri logic. A theory is a set of sentences. When it is clear from 2013, Prop. 4.6) and of the Chang–Ło´s–Suszko theorem the context, we will refer to P-terms and P-formulas simply (Bagheri and Moniri 2013, Prop. 5.5). as terms and formulas. The connection between classical model theory and the Let MTL stand for the monoidal t-norm based logic intro- study of classes of fuzzy structures needs to be clarified. duced by Esteva and Godo (2001). Throughout the paper, we Namely, as explained and developed in previous papers (Cin- consider the predicate logic MTL∀ [for a definition of the tula et al. 2009; Dellunde et al. 2016, 2018), there is a axiomatic system for MTL∀ we refer the reader to Cintula et translation of fuzzy structures into classical many-sorted al. (2011, Def. 5.1.2, Ch. I)]. Let us recall that the deduction structures, more precisely, two-sorted structures with one sort rules of MTL∀ are those of MTL and the rule of general- for the first-order domain and another accounting for truth- ization: from ϕ infer (∀x)ϕ. The definitions of proof and values in the algebra. Such connection certainly allows to provability are analogous to the classical ones. We denote by directly import to the fuzzy setting several classical results, Φ MTL∀ ϕ the fact that ϕ is provable in MTL∀ from the set but, as already noted in the mentioned papers, it does not go of formulas Φ. For the sake of clarity, when it is clear from a long way. Indeed, the translation does not preserve the syn- the context we will write to refer to MTL∀. The algebraic tactical complexity of sentences (regarding quantifiers), and semanticsofMTL∀ are based on MTL-algebras (Esteva and hence, it cannot be used for syntactically sensitive results, Godo 2001). such as those studied in the present paper. A is called an MTL-chain if its underlying lattice is lin- The paper is structured as follows: in Sect. 1,weintro- early ordered. Since it is customary to consider fuzzy logics duce the syntax and semantics of fuzzy predicate logics. in languages expanding that of MTL, henceforth, we will In Sect. 2, several fuzzy model-theoretic notions such as confine our attention to algebras which are expansions of homomorphisms or the method of diagrams are presented. MTL-chains of such kind and just call them chains. In Sect. 3, we study the preservation of universal formulas, obtain an existential form of amalgamation and derive Definition 2 (Semantics of Predicate Fuzzy Logics [Cintula from it an analogue of the Ło´s–Tarski theorem. In Sect. 4, et al. 2011, Def. 5.2.1, Ch. I]) Consider a predicate language we study classes given by universal-existential sentences by P =PredP , FuncP , ArP and let A be a chain. We define showing that such formulas are preserved under unions of an A-structure M for P as a pair M =A, M where chains, obtaining another corresponding amalgamation result and a version of Chang–Ło´s–Suszko preservation theorem. M =M,(PM )P∈Pred,(FM )F∈Func, 123 Syntactic characterizations of classes of first-order structures… 2179 where M is a non-empty domain, PM is an n-ary fuzzy rela- important properties: they are witnessed (the values of the tion for each n-ary predicate symbol, i.e. a function from quantifiers are maxima and minima achieved in particular Mn to A, identified with an element of A if n = 0; and instances) and have the compactness property, both for sat- n FM is a function from M to M, identified with an ele- isfiability and for consequence (see e.g. Dellunde 2014). ment of M if n = 0. As usual, if M is an A-structure for P,anM-evaluation of the object variables is a mapping Proposition 1 Let A be a fixed finite chain. For every set of Σ ∪{α} v assigning to each object variable an element of M.The sentences , the following holds: set of all object variables is denoted by Var.Ifv is an M- Σ ⊆ Σ , Σ evaluation, x is an object variable and d ∈ M, we denote 1. If every finite subset 0 has a model A M 0 , then Σ , by v[x → d] the M-evaluation so that v[x → d](x) = d has a model A N . Σ | α Σ ⊆ Σ and v[x → d](y) = v(y) for y an object variable such that 2. If A , then there is a finite subset 0 such Σ | α y = x.IfM is an A-structure and v is an M-evaluation, we that 0 A . define the values of terms and the truth-values of formulas in M for an evaluation v recursively as follows: From now on, we refer to A-structures simply as structures (or as P-structures if we need to specify the language). For the remainder of the article, let us assume that we have a crisp x A = v(x); M,v identity ≈ in the language. ( ,..., ) A = ( A ,..., A ) F t1 tn M,v FM t1 M,v tn M,v ,for F ∈ Func; Definition 3 Let P be a predicate language, A, M and ( ,..., ) A = ( A ,..., A ) P t1 tn M,v PM t1 M,v tn M,v ,for B, N structures for P, f a mapping from A to B and g a P ∈ Pred; mapping from M to N. The pair f , g is said to be a strong (ϕ ,...,ϕ ) A =◦A( ϕ A ,..., ϕ A ) c 1 n M,v 1 M,v n M,v ,for homomorphism from A, M to B, N if f is an algebraic ◦∈L; homomorphism and for every n-ary function symbol F ∈ P (∀ )ϕ A = { ϕ A | ∈ } x M,v inf≤A M,v[x→d] d M ; and d1,...,dn ∈ M, (∃ )ϕ A = { ϕ A | ∈ } x M,v sup≤A M,v[x→d] d M . g(FM(d1,...,dn)) = FN(g(d1),...,g(dn)) Φ Φ A = For a set of formulas , we write M,v 1, if and for every n-ary predicate symbol P ∈ P and d1,...,dn ∈ ϕ A = 1 for every ϕ ∈ Φ. We denote by ϕ A = 1 M,v M M, ϕ A = v the fact that M,v 1 for all M-evaluations .Wesay that A, M is a model of a set of formulas Φ,if ϕ A = 1 ( ( ..., ) A ) = ( ( ), . . . , ( )) B . −→ M f P d1 dn M P g d1 g dn N for any ϕ ∈ Φ. Sometimes, we will denote by x a sequence −→ of variables x1,...,xn (and the same with sequences d of A strong homomorphism f , g is said to be elemen- elements of the domain). Given a structure A, M and a tary if we have, for every P-formula ϕ(x ,...,x ) and −→ −→ −→ 1 n formula ϕ( x ), we say that d ⊆ M satisfies ϕ( x ) (or that d1,...,dn ∈ M, −→ −→ −→ ϕ( ) ϕ( )A −→ = A x is satisfied by d )if x ,v[−→→ ] 1 for any M x d ( ϕ( ..., ) A ) = ϕ( ( ), . . . , ( ) B . −→ A A f d1 dn M g d1 g dn N M-evaluation v (also written ϕ[ d ] = 1 ); for the sake M −→ of clarity, we will use also the notation A, M|ϕ[ d ] Let f , g be a strong homomorphism from A, M to when is needed. Two theories T and U are said to be 1- B, N, we say that f , g is an embedding from A, M to equivalent if a structure is a model of T if it is also a model B, N if both functions f and g are injective, and we say that of U (in the case where T and U are singletons of formulas, f , g is an isomorphism from A, M to B, N if f , g is we say that these formulas are 1-equivalent). an embedding and both functions f and g are surjective. For Given a set of sentences Σ, and a sentence φ, we denote by a general study of different kinds of homomorphisms and the Σ | A φ the fact that every A-model of Σ is also an A-model formulas they preserve, we refer to Dellunde et al. (2016). of φ. We focus on classes of structures over a fixed finite Later in the article, we will use diagram techniques. We chain A whose set of elements is denoted by {a1,...,ak}. present here some corollaries of the results obtained in Del- Such restriction is due to the fact that dropping finiteness lunde (2011). Given a language P, we start by introducing can cause to lose compactness, which is an essential element three different expansions adding either a new truth-constant of our proofs. However, the results will still be quite encom- for each element of the algebra, or new object constants. passing in practice. Indeed, for instance, prominent examples For any element a of A, we will use the truth-constant a to A A of weighted structures in computer science are valued over denote it. When a = 1 or a = 0 , then a = 1ora = 0, finite chains. Structures over a fixed finite chain A have two respectively. 123 2180 G. Badia et al.

Definition 4 Given a predicate language P, we expand it by (1) A is a subalgebra of B; adding an individual constant symbol cm for every m ∈ M, (2) M ⊆ N; and denote it by PM.IfA, M is a PM-structure, we denote (3) for any n-ary function symbol F ∈ P and elements M by A, M the expansion of the structure A, M to P , d1,...,dn ∈ M,wehave ∈ ( ) = where for every m M, cm M m. FM(d1,...,dn) = FN(d1,...,dn); Definition 5 Given a predicate language P, we expand it by adding a truth-constant symbol a for every a ∈ A, and denote (4) for any n-ary predicate symbol P ∈ P and elements PA PA it by . When we expand the language further by adding d1,...,dn ∈ M,wehave an individual constant symbol cm for every m ∈ M, we will PA,M denote it by . PM(d1,...,dn) = PN(d1,...,dn). Definition 6 Let P be a predicate language and A, M a P- , Remark that A, M is a substructure of B, N if and structure. We define the following sets of P A M -sentences: only if conditions (1)–(3) are satisfied and, instead of (4), ElDiag(A, M) ={σ ↔ a | σ is a sentence of PM, a ∈ A the following condition holds: for every quantifier-free formula ϕ(x ,...,x ) and any elements d ,...,d ∈ M, σ A = }, 1 n 1 n and M a A B ϕ(d1,...,dn) = ϕ(d1,...,dn) . whereas Diag(A, M) is the subset of ElDiag(A, M) contain- M N σ ↔ σ ing all formulas a where is quantifier-free. With this notion at hand, we can define a corresponding Following the same lines of the proof of Dellunde (2011, closure property for classes of structures. Prop. 32), we can obtain a characterization of strong and ele- Definition 8 (Class Closed Under Substructures)LetK be mentary embeddings between two P-structures over a chain aclassofP-structures. We say that K is closed under sub- A. structures if, for any structure A, M∈K, Corollary 2 Let A, M and A, N be two P-structures for if B, N is a substructure of A, M, then B, N∈K. PA. The following are equivalent: Since our characterizations will be based on axioma- , 1. There is an expansion of A N that is a model of tizability of classes, we need to recall the definition of ( , ) ( , ) Diag A M (ElDiag A M , respectively). elementary class of structures. 2. There is a mapping g : M → N such that IdA, g is a strong (elementary, respectively) embedding from A, M Definition 9 [Elementary Class (Burris and Sankappanavar K P into A, N. 1981, Def. 2.15)] A class of -structures is an elementary class (or a first-order class)ifthereisasetΣ of sentences such that for every A, M, 3 Universal classes A, M∈K if and only if A, M|Σ. In this section, we prove a result on existential amalgama- In this case, K is said to be axiomatized (or defined) by Σ. tion (Proposition 4) from which we extract a Ło´s–Tarski preservation theorem for universal theories (Theorem 5) Using a predicate language with only one binary relation and a characterization of universal classes of structures R, the class of weighted undirected graphs is axiomatized by (Theorem 6). Relevant structures in computer science are the following set of universal sentences: axiomatized by sets of universal formulas; one prominent example is the class of weighted graphs. Particular versions {(∀x)(R(x, x) → 0), (∀x)(∀y)(R(x, y) → R(y, x))}. of the above-mentioned results appeared for Ł∀ in Spada (2009). In the context of fuzzy logic programming, Gerla Notice that the notion of induced weighted undirected (2005) studied universal formulas with relation to Herbrand subgraph corresponds to the model-theoretic notion of sub- interpretations. structure used in MFL. For the upcoming results, we need to recall the notion of Definition 10 Let P be a predicate language. We say that a substructure. P-formula ϕ(x1,...,xn) is preserved under substructures P , , Definition 7 (Substructure)LetA, M and B, N be P- if for any -structure A M and any substructure B N , ϕ( ,..., ) A = A ,..., ∈ structures. We say that A, M is a substructure of B, N if d1 dn M 1 for some d1 dn N, then ϕ( ,..., ) B = B if: d1 dn N 1 . 123 Syntactic characterizations of classes of first-order structures… 2181

The following lemma can be easily proved by induction Quantifying away the new individual constants, we obtain ∗( , ) on the complexity of universal formulas. a set of formulas Diag0 A M2 such that: ϕ( ,..., ) Lemma 3 Let x1 xn be a universal formula. Then, −→ ∗ ElDiag(A, M ) (∃ x ) Diag (A, M ) → a. ϕ(x1,...,xn) is preserved under substructures. 1 0 2

In classical model theory, amalgamation properties are −→ −→ Since A, M , d ∃ A, M , d , then often related in elegant ways to preservation theorems (see 2 1 1 e.g. Hodges 1993). We will try an analogous approach to , | (∃−→) ∗( , ) , obtain our desired preservation result. The importance of A M2 x Diag0 A M2 this idea is that the problem of proving a preservation result reduces then to finding a suitable amalgamation counterpart. which is a contradiction. Note, moreover, that if This provides us with proofs that have a neat common struc- −→ −→ ture (such as those of the main results in this section and the , , , , , A M2 d ∃1 A M1 d next one). −→ −→ We will write A, M , d ∃ A, M , d if for any 2 −→ n 1 −→ we also have that whenever ϕ(x¯) is quantifier-free formula of ∃ ϕ , |ϕ[ ] , |ϕ[ ] −→ −→ n-formula , A M2 d only if A M1 d . A P , A, M2|ϕ[ d ] iff A, M1|ϕ[ d ]. Left-to-right is , clear; the contrapositive of the right-to-left direction follows Proposition 4 (Existential amalgamation) Let A M1 and −→ −→ A , | ϕ[ ] , | ϕ ↔ [ ] A, M2 be two structures for P with a common part A, M easily: if A M1 d , then A M1 a d for −→ A −→ a = A, M |ϕ ↔ a[ d ] with domain generated by a sequence of elements d.More- some 1 ,so−→ 2 , which means that over, suppose that A, M2 | ϕ[ d ]. −→ −→ Observe that the proof can be similarly carried out, mutatis , , , , . , , A M2 d ∃1 A M1 d mutandi, when A M1 and A M2 have no common part as well. Then, there is a structure A, N into which A, M2 can A be strongly embedded by f , g, while A, M1 is P - Now, we have the elements to establish an exact analogue elementarily strongly embedded (taking isomorphic copies, of Theorem 5 from Ło´s(1955), Ło´s–Tarski preservation the- A we may assume that A, M1 is just a P -elementary orem. substructure). The situation is described by the following Theorem 5 (Ło´s–Tarski preservation theorem) Let T be a picture: −→ P A-theory and Φ( x ) a set of formulas in P A. Then, the A, N following are equivalent: f , g , ⊆ , −→ ∃ −→ (i) For any models of T, A M A N , we have: if 1 , |Φ , |Φ A, M2, d A, M1, d A N , then A M . P A Θ(−→) ⊆ ⊆ (ii) There is a set of universal -formulas x such that: T ,Φ Θ and T ,Θ Φ. A, M Proof Let us prove the difficult direction (the converse direc- −→ Moreover, the result is also true when A, M and A, M tion is clear by Lemma 3). Consider (T ∪ Φ( x ))∀ ,the 1 2 −→1 have no common part. collection of all ∀ logical consequences of T ∪ Φ( x ).We 1 −→ need to establish that the only models of (T ∪ Φ( x ))∀ Proof ( , ) ∪ 1 It is not a difficult to show that ElDiag A M1 among the models of T are the substructures of models of ( , ) −→ −→ Diag A M2 (where we let the elements of the domain serve Φ( x ).LetA, M be a model of (T ∪ Φ( x ))∀ .Allwe 1 −→ as constants to name themselves) has a model, which suffices need to do is find a model A, N of the theory T ∪ Φ( x ) for the purposes of the result. Suppose otherwise, that is, for such that A, M ∃ A, N and then quote the existential ( , ) ⊆ ( , ) 1 some finite Diag0 A M2 Diag A M2 , we have that amalgamation theorem. Let U be all ∃1-formulas that hold in A, M. We claim ( , ) ( , ) → −→ ElDiag A M1 Diag0 A M2 a then that T ∪ Φ( x ) ∪ U has a model. Otherwise, by compactness, for for the immediate predecessor a in the lattice order of A of A −→ −→ −→ −→ 1 . {(∃x0 )φ0(x0 ),...,(∃xn )φn(xn )}⊆U 123 2182 G. Badia et al. −→ we have that in all models of T ∪ Φ( x ), it holds that Then, one would expect to reduce the left side of the inequal- ity to a finite set Ψ such that −→ −→ −→ −→ (∃ x 0)φ0(x0 ) ∧···∧(∃xn )φn(xn ) → a, A A A inf≤A { ψ | ψ ∈ Ψ }≤ φ . A M M where a is the immediate predecessor of 1 , and by basic manipulations, However, this reduction would come from compactness in the −→ −→ −→ −→ usual argument, but it does not in this one. This is because (∃x0 ,...,xn )(φ0(x0 ) ∧···∧φn(xn )) → a, compactness is about consequence as opposed to implica- tion, which are different in a setting without a deduction which is just equivalent to theorem such as this. In fact, in Spada (2009) similar results in the framework Łukasiewicz logic are obtained only for (∀−→,...,−→)(φ (−→) ∧···∧φ (−→) → ). x0 xn 0 x0 n xn a 1-equivalence as well. −→ Needless to say, the previous results, in particular, allow (T ∪ Φ( x ))∀ The latter formula must be in 1 then, which is to conclude that a class of P-structures (that is, structures for a contradiction. a language without additional truth-constants) closed under PA Following a similar proof, we can obtain an algebraic char- substructures can be axiomatized by universal -sentences. acterization equivalent to Theorem 5 (as long as we have One might wonder, of course, if it is really necessary to resort truth-constants around). a universal axiomatization in the expanded language. Let us present a counterexample showing that, in general, A Theorem 6 Let K beaclassofP -structures. Then, the fol- the base language P does not suffice for Theorem 6.LetP lowing are equivalent: be the language with only one monadic predicate P and take two structures over the standard Gödel chain, [0, 1]G, M K (i) is closed under isomorphisms, substructures, and and [0, 1]G, N. The domain in both cases is the set of all ultraproducts. natural numbers N, and the interpretation of the predicate (ii) K is axiomatized by a set of universal PA-sentences. ( ) = 3 ( ) = 1 is, respectively, defined as: PM n 4 , and PN n 2 ,for every n ∈ N. First, we show that [0, 1]G, M≡[0, 1]G, N, The following corollary can be obtained because in our that is, that for any P-sentence ϕ, [0, 1]G, M|ϕ iff setting, two forms of compactness (that are generally dis- [0, 1]G, N|ϕ.Take f as any non-decreasing bijection tinct, in, say, Łukasiewicz logic) collapse, namely (1) the [ , ] [ , ] ( 3 ) = 1 ( ) = ( ) = from 0 1 to 0 1 such that f 4 2 , f 1 1, f 0 0. compactness of the consequence relation and (2) the com- It is easy to check that f is a G-homomorphism preserving pactness of the satisfiability relation. (1) clearly implies (2) suprema and infima. Then, we can consider the σ -mapping in the presence of 0 in our language. To see the converse, say f , Id and apply (Dellunde et al. 2018, Lemma 11) to obtain ϕ ∪{ϕ → } that T , which amounts to say that T a (where that [0, 1]G, M≡[0, 1]G, N. Consider now the finite A a [ , ] { , 1 , 3 , } is the predecessor of 1 ) does not have a model. Hence, by subalgebra A of 0 1 G generated by the subset 0 2 4 1 . ⊆ ∪{ϕ → } (2), there is a finite T0 T such that T0 a has no Clearly, the structures [0, 1]G, M and [0, 1]G, N can be model, so, in fact, T0 ϕ. regarded as structures over A. Thus, we have Corollary 7 Let T ∪{ϕ} be a set of P A-sentences. Then, ϕ is A, M≡A, N. preserved under substructures of models of T if, and only if, ϕ P A is 1-equivalent to a universal -sentence modulo T . (∀ ) ( ) A = 3 (∀ ) ( ) A = 1 Observe that x P x M 4 and x P x N 2 . Proof Apply Theorem 5 for Φ ={ϕ}. Consequently, ϕ is Consider the expanded language PA obtained by adding a axiomatized by a set of universal P A-sentences. Then, bring constant symbol a for every element a ∈ A.LetK be the class it down to a single such formula using A-compactness for of P-structures valued on A, whose natural expansion to PA consequence. (that is, the expansion in which every constant a is interpreted as the corresponding element a) satisfies the sentence A natural question is whether Corollary 7 can be strength- ↔ ened to strong equivalence in terms of , that is, whether one 3 can find a universal formula that agrees with φ on each value → (∀x)P(x). A 4 in every structure (not just on value 1 ). Following the lines of the above proof, this would require to show something Clearly, K is closed under substructures and A, M∈K. , , ∈/ K (∀ ) ( ) A = 1 like, for an arbitrary model A M , However, A N , because x P x N 2 . There- fore, K cannot be axiomatized by P-sentences, and a fortiori, ψ A ≤A φ A ( ψ φ → ψ). P , M M for all s.t. by a set of universal -sentences, since it contains A M 123 Syntactic characterizations of classes of first-order structures… 2183 but not the elementary equivalent A, N. Hence, we have Next, we recall a useful theorem that has been established produced an example of a class of P-structures closed and used to construct saturated models in the context of math- under substructures (and, obviously, under isomorphisms and ematical fuzzy logic in Badia and Noguera (2018b). ultraproducts) which is not axiomatizable with universal P- Theorem 8 (Badia and Noguera 2018b) (Tarski–Vaught) Let sentences. A, M be the union of the elementary chain {A, M |i < Whether constants are necessary for Theorem 5 in general −→ i γ } is an open question, we conjecture that they are. . Then, for every sequence d of elements of Mi and for- −→ A −→ A mula ϕ, ϕ( d ) = ϕ( d ) . Moreover, if the chain is M Mi −→ A −→ A 4 Universal–existential classes not elementary, we still have that ϕ( d ) = ϕ( d ) M Mi for every quantifier-free formula. This section runs quite parallel to the previous one. We recall the notion of elementary chain of structures and its corre- Therefore, unions of elementary chains preserve the val- sponding Tarski–Vaught theorem and prove that universal– ues of all formulas. It is also interesting to consider formulas existential formulas are preserved under unions of chains that are preserved by all unions of chains.

(Lemma 9). After that, we prove a result on existential– Definition 11 We say that a formula ϕ(x1,...,xn) is pre- universal amalgamation (Proposition 10) and derive from it served under unions of chains if whenever we have a chain a Chang–Ło´s–Suszko preservation theorem (Theorem 11). { , | <γ} of models A Mi i such that for every i, K −→ A −→ Consider the class of all structures in a signature with ϕ( ) = A( <γ) · −1 d 1 i for some sequence d of ele- a binary function symbol , a unary function symbols ,an Mi individual constant 1, and a unary predicate G satisfying the −→ A A ments of M0, then ϕ( d ) = 1 , where A, M is the following axioms: M union of the chain. A (∀x)(∃y)(yn ≈ x) for each n 2. Let a be the element of A immediately above 0 . (∀x, y)((x · y) · z ≈ x · (y · z)). Lemma 9 ∀ -formulas are preserved under unions of chains. (∀x)(x · 1 ≈ x). 2 (∀ )( · −1 ≈ ) −→ −→ x x x 1 . Proof Let (∀ x )(∃ y )φ be a ∀2-formula, A, M be the (∀ , )( · ≈ · ) −→ x y x y y x . union of a chain {A, Mi |i <γ}, and c some (∀ , )(( ∧ ) → ( )) <γ x y Gx Gy G xy . sequence of elements of M0. Assume that for every i , −1 −→ −→ −→ A A −→ (∀x)(Gx → G(x )). (∀ x )(∃ y )φ( c ) = 1 .Let d ∈ M, we show that Mi −→ −→ −→ A A This is the class of divisible Abelian groups with a fuzzy (∃ y )φ( d , c ) = 1 .Let j <γbe such that M subgroup defined by the predicate G (following the definition −→ −→ −→ −→ A A d ∈ M j . Since (∀ x )(∃ y )φ( c ) = 1 ,wehave of Rosenfeld 1971). By our Chang–Ło´s–Suszko preservation M j −→ −→ −→ A A theorem below, K is a class closed under unions of chains. (∃ y )φ( d , c ) = 1 . Since A, M j is ∃-witnessed, M Another example of such class be provided by the class j −→ −→ −→ −→ A A of all weighted graphs where the formula there are e ∈ M j such that φ( d , e , c ) = 1 . M j −→ −→ −→ A A (∀x)(∃y, z)(y ≈ z ∧ Rxy ∧ Rxz) Therefore, φ( d , e , c ) = 1 , because extensions M preserve quantifier-free formulas, and then clearly holds, that is, every vertex has at least two incident edges. −→ −→ −→ A A This axiomatizes the class of graphs where every vertex has (∃ y )φ( d , c ) = 1 . degree 2. M Given an ordinal γ , a sequence {A, Mi |i <γ} of −→ ∈ models is called a chain when for all i < j <γwe have that We can conclude that for every d M, A, Mi is a substructure of A, M j . If, moreover, these sub- −→ −→ −→ A A structures are elementary, we speak of an elementary chain. (∃ y )φ( d , c ) = 1 , M The union of the chain {A, Mi |i <γ} is the structure A, M where M is defined by taking as its domain M , −→ −→ −→ A A i<γ i and hence, (∀ x )(∃ y )φ( c ) = 1 . interpreting the constants of the language as they were inter- M preted in each Mi and similarly with the relational symbols Next, we provide the amalgamation result that will allow of the language. Observe as well that M is well defined given us to prove a version of Chang–Ło´s–Suszko theorem for that {A, Mi |i <γ} is a chain. graded model theory. 123 2184 G. Badia et al. −→ −→ ∗ Proposition 10 (∃2-amalgamation) Let A, M1 and A, M2 A, M , d | (∃ x ) Diag (A, M ) , 2 ∀10 2 be two structures for PA with a common part A, M with −→ domain generated by a sequence of elements d . Moreover, which is a contradiction. suppose that Now, we are ready to prove the promised analogue −→ −→ , , , , . of Robinson (1959, Theorem 1.2). A M2 d ∃2 A M1 d Theorem 11 (Chang–Ło´s–Suszko preservation theo.) Let T −→ A Then, there is a structure A, N into which A, M2 can be a theory and Φ( x ) a set of formulas in P . Then, the be strongly embedded by f , g preserving all ∀1-formulas, following are equivalent: A while A, M1 is P -elementarily strongly embedded (taking A −→ isomorphic copies, we may assume that A, M1 is just a P - (i) Φ( x ) is preserved under unions of chains of models of elementary substructure). The situation is described by the T. −→ following picture: (ii) Φ( x ) is 1-equivalent modulo T to a set of ∀2-formulas.

A, N Proof Once more, we only deal with the non-trivial direction −→ of the equivalence. Consider (T ∪Φ( x ))∀ . We want to show f , g 2 that −→ ∃ −→ , , 2 , , A M2 d A M1 d −→ −→ T ∪ (T ∪ Φ( x ))∀ Φ( x ), ⊆ ⊆ 2

A, M which will suffice to establish the theorem. The strategy is ∪ ( ∪ Φ(−→)) establish that any model of T T x ∀2 has an elementary extension which is a union of ω-many models of Moreover, the result is also true when A, M and A, M −→ −→ 1 2 Φ( x ), so by hypothesis, Φ( x ) will hold there, and hence have no common part. ∪ ( ∪ Φ(−→)) back in our original model of T T x ∀2 . Proof ( , ) ∀ So, we start with A, M0 being an arbitrary model of Let Diag∀1 A M2 be the collection of all 1-formulas −→ ∪ ( ∪ Φ( ))∀ , in the language of the diagram of A, M (where we let the T T x 2 . Now, assuming that we have A Mi 2 , elements of the domain serve as constants to name them- which is an elementary extension of A M0 . We first need , ∪ Φ(−→) to find a model A Mi of the theory T x such that selves) that hold in said structure. It is not difficult to show A, M ∃ A, M and then quote the ∃ -amalgamation that ElDiag(A, M ) ∪ Diag∀ (A, M ) (where again we let i 2 i 2 1 1 2 , ∪ Φ(−→) the elements of the domain serve as constants to name them- theorem to obtain a model A Ni of T x into which , selves) has a model, which suffices for the purposes of the A Mi can be strongly embedded in such a way that all ∀ result. For suppose otherwise, that is, for some finite 1-formulas are preserved by such strong embedding. Let U be all ∃ -formulas that hold in A, M . We claim 2−→ i ( , ) ⊆ ( , ), then that T ∪ Φ( x ) ∪ U has a model. Otherwise by com- Diag∀ 0 A M2 Diag∀ A M2 1 1 pactness, for we have that {(∃−→ )(∀−→ )φ (−→ , −→ ),...,(∃−→ )(∀−→ )φ (−→ , −→ )}⊆ x 0 y 0 0 x 0 y 0 x n y n n x n y n U ( , ) ( , ) → ElDiag A M1 Diag∀ 0 A M2 a 1 we have that in all models of T , it holds that

= A −→ −→ −→ −→ for some a 1 (the supremum of all the values taken by (∃ x 0)(∀ y 0)φ0( x 0, y 0) ∧···∧ Diag∀ (A, M2) in A). Quantifying away the new individ- −→ −→ −→ −→ 10 (∃ x n)(∀ y n)φn( x n, y n) → a ual constants, A −→ ∗ where a is the immediate predecessor of 1 and by basic ElDiag(A, M ) (∀ x ) Diag∀ (A, M ) → a , 1 10 2 manipulations, so −→ −→ −→ −→ −→ −→ (∃ x 0,..., x n)(∀ y 0,..., y n)(φ0( x 0, y 0) ∧···∧ −→ −→ −→ ∗ φ ( x , y )) → a. ElDiag(A, M ) (∃ x ) Diag (A, M ) → a. n n n 1 ∀10 2 −→ ( ∪ Φ( ))∀ −→ −→ The latter formula must be in T x 2 then, which is , , , , Since A M2 d ∃2 A M1 d , then a contradiction. 123 Syntactic characterizations of classes of ﬁrst-order structures… 2185 −→ Now,A, Ni is also such that for a listing d of all the ele- Acknowledgements Open access funding provided by Austrian Sci- −→ −→ ence Fund (FWF). The authors are indebted to two anonymous referees ments of A, M , A, N , d ∃ A, M , d . To prove i i 1 i and to the editor for their critical and interesting remarks that have the contrapositive, suppose that helped improving the presentation the paper. Costa, Dellunde, and Noguera received funding from the European Union’s Horizon 2020 −→ −→ , , |(∃−→)ϕ(−→, ) → research and innovation programme under the Marie Curie Grant Agree- A Mi d x x d a ment No. 689176 (SYSMICS project). Badia is supported by the Project I 1923-N25 (New perspectives on residuated posets)oftheAustrian A where a is the immediate predecessor of 1 in the linear Science Fund (FWF). Costa is also supported by the Grant for the recruitment of early-stage research staff (FI-2017) from the Generalitat order of A. But then de Catalunya. Dellunde is also partially supported by the Project RASO −→ −→ TIN2015-71799-C2-1-P, CIMBVAL TIN2017-89758-R, and the Grant −→ −→ 2017SGR-172 from the Generalitat de Catalunya. The research leading A, Mi , d |(∀ x )(ϕ( x , d ) → a), to these results has received funding from AppPhil-RecerCaixa. Finally, Noguera is also supported by the Project GA17-04630S of the Czech so indeed, Science Foundation (GACR).ˇ

−→ −→ −→ −→ A, Ni , d |(∀ x )(ϕ( x , d ) → a), Compliance with ethical standards and hence, Conﬂict of interest The authors declare they have no conﬂict of interest.

−→ −→ −→ −→ Human and animal rights This article does not contain any studies with A, Ni , d |(∃ x )ϕ( x , d ) → a. human participants or animals performed by any of the authors.

Open Access This article is distributed under the terms of the Creative Now, using the existential amalgamation theorem we can Commons Attribution 4.0 International License (http://creativecomm obtain a structure A, Mi+1 as an elementary extension of ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, , , and reproduction in any medium, provided you give appropriate credit A Mi into which ANi can be strongly embedded. Now, just take the union A, M =A, N and apply to the original author(s) and the source, provide a link to the Creative i∈ω i i∈ω i Commons license, and indicate if changes were made. Theorem 8.

As a consequence, we can again obtain a result for single formulas, using the compactness of the consequence relation. References

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