Relative e-spectra and relative closures 7 1 for families of theories∗ 0 2 n Sergey V. Sudoplatov† a J 1 ] O Abstract L We define the notions of relative e-spectra, with respect to E- . h operators, relative closures, and relative generating sets. We study t a properties connected with relative e-spectra and relative generating m sets. [ Keywords: E-operator,combinationoftheories,relativee-spectrum, 1 disjoint families of theories, relative closure, relative generating set. v 6 We continue to study structural properties of combinations of structures 0 2 and their theories [1, 2, 3] generalizing the notions of e-spectra, closures and 0 generating sets to relative ones. Properties of relative e-spectra and relative 0 . generating sets are investigated. 1 0 7 1 1 Preliminaries : v i X Throughout the paper we use the following terminology in [1, 2]. r Let P = (P ) , be a family of nonempty unary predicates, (A ) be a i i∈I i i∈I a family of structures such that P is the universe of A , i ∈ I, and the i i symbols P are disjoint with languages for the structures A , j ∈ I. The i j structure A ⇋ A expanded by the predicates P is the P-union of the P i i i∈I S ∗Mathematics Subject Classification: 03C30, 03C50,54A05. The research is partially supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1) and by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No. 0830/GF4). †[email protected] 1 structuresA ,andtheoperatormapping(A ) toA istheP-operator. The i i i∈I P structure A is called the P-combination of the structures A and denoted P i by Comb (A ) if A = (A ↾ A ) ↾ Σ(A ), i ∈ I. Structures A′, which P i i∈I i P i i are elementary equivalent to Comb (A ) , will be also considered as P- P i i∈I combinations. Clearly, all structures A′ ≡ Comb (A ) are represented as unions of P i i∈I their restrictions A′ = (A′ ↾ P ) ↾ Σ(A ) if and only if the set p (x) = i i i ∞ {¬P (x) | i ∈ I} is inconsistent. If A′ 6= Comb (A′) , we write A′ = i P i i∈I Comb (A′) , where A′ = A′ ↾ P , maybe applying Morleyzation. P i i∈I∪{∞} ∞ i i∈I Moreover, we write Comb (A ) Tfor Comb (A ) with the empty P i i∈I∪{∞} P i i∈I structure A . ∞ NotethatifallpredicatesP aredisjoint,astructureA isaP-combination i P and a disjoint union of structures A . In this case the P-combination A i P is called disjoint. Clearly, for any disjoint P-combination A , Th(A ) = P P Th(A′ ), where A′ is obtained from A replacing A by pairwise disjoint P P P i A′ ≡ A , i ∈ I. Thus, in this case, similar to structures the P-operator i i works for the theories T = Th(A ) producing the theory T = Th(A ), i i P P being P-combination of T , which is denoted by Comb (T ) . i P i i∈I Foranequivalence relationE replacing disjoint predicates P by E-classes i we get the structure A being the E-union of the structures A . In this E i case the operator mapping (A ) to A is the E-operator. The structure i i∈I E A is also called the E-combination of the structures A and denoted by E i Comb (A ) ; hereA = (A ↾ A ) ↾ Σ(A ), i ∈ I. Similarabove, structures E i i∈I i E i i A′, which are elementary equivalent to A , are denoted by Comb (A′) , E E j j∈J where A′ are restrictions of A′ to its E-classes. The E-operator works for j the theories T = Th(A ) producing the theory T = Th(A ), being E- i i E E combinationofT , whichisdenotedbyComb (T ) orbyComb (T ), where i E i i∈I E T = {T | i ∈ I}. i Clearly, A′ ≡ A realizing p (x) is not elementary embeddable into A P ∞ P and can not be represented as a disjoint P-combination of A′ ≡ A , i ∈ I. i i At the same time, there are E-combinations such that all A′ ≡ A can be E representedasE-combinationsofsomeA′ ≡ A . Wecallthisrepresentability j i of A′ to be the E-representability. If there is A′ ≡ A which is not E-representable, we have the E′- E representability replacing E by E′ such that E′ is obtained from E adding equivalence classes with models for all theories T, where T is a theory of a restriction B of a structure A′ ≡ A to some E-class and B is not elementary E 2 equivalent to the structures Ai. The resulting structure AE′ (with the E′- representability) is a e-completion, or a e-saturation, of A . The structure E AE′ itself is called e-complete, or e-saturated, or e-universal, or e-largest. For a structure A the number of new structures with respect to the E structures A , i. e., of the structures B which are pairwise elementary non- i equivalent and elementary non-equivalent to the structures A , is called the i e-spectrum of A and denoted by e-Sp(A ). The value sup{e-Sp(A′)) | E E A′ ≡ A } is called the e-spectrum of the theory Th(A ) and denoted by E E e-Sp(Th(A )). E If A does not have E-classes A , which can be removed, with all E- E i classes A ≡ A , preserving the theory Th(A ), then A is called e-prime, j i E E or e-minimal. For a structure A′ ≡ A we denote by TH(A′) the set of all theories E Th(A ) of E-classes A in A′. i i By the definition, an e-minimal structure A′ consists of E-classes with a minimal set TH(A′). If TH(A′) is the least for models of Th(A′) then A′ is called e-least. Definition [2]. Let T be the class of all complete elementary theories of relational languages. For a set T ⊂ T we denote by Cl (T) the set of E all theories Th(A), where A is a structure of some E-class in A′ ≡ A , E A = Comb (A ) , Th(A ) ∈ T . As usual, if T = Cl (T) then T is said E E i i∈I i E to be E-closed. The operator Cl of E-closure can be naturally extended to the classes E T ⊂ T as follows: Cl (T ) is the union of all Cl (T ) for subsets T ⊆ T . E E 0 0 For a set T ⊂ T of theories in a language Σ and for a sentence ϕ with Σ(ϕ) ⊆ Σ we denote by T the set {T ∈ T | ϕ ∈ T}. ϕ Proposition 1.1 [2]. If T ⊂ T is an infinite set and T ∈ T \ T then T ∈ Cl (T ) (i.e., T is an accumulation point for T with respect to E-closure E Cl ) if and only if for any formula ϕ ∈ T the set T is infinite. E ϕ Theorem 1.2 [2]. For any sets T ,T ⊂ T , Cl (T ∪ T ) = Cl (T ) ∪ 0 1 E 0 1 E 0 Cl (T ). E 1 Definition [2]. Let T be a closed set in a topological space (T ,O (T )), 0 E where O (T ) = {T \ Cl (T ′) | T ′ ⊆ T }. A subset T′ ⊆ T is said to be E E 0 0 generating if T = Cl (T ′). The generating set T ′ (for T ) is minimal if T′ 0 E 0 0 0 0 does not contain proper generating subsets. A minimal generating set T ′ is 0 least if T′ is contained in each generating set for T . 0 0 3 Theorem 1.3 [2]. If T′ is a generating set for a E-closed set T then the 0 0 following conditions are equivalent: (1) T′ is the least generating set for T ; 0 0 (2) T′ is a minimal generating set for T ; 0 0 (3) any theory in T ′ is isolated by some set (T′) , i.e., for any T ∈ T′ 0 0 ϕ 0 there is ϕ ∈ T such that (T ′) = {T}; 0 ϕ (4) any theory in T ′ is isolated by some set (T ) , i.e., for any T ∈ T′ 0 0 ϕ 0 there is ϕ ∈ T such that (T ) = {T}. 0 ϕ 2 Relative e-spectra and their properties Definition. For a structure A and a class K of structures, the number of E new structures with respect to the structures A and to the class K, i. e., i of the structures B forming E-classes of models of Th(A ) such that B are E pairwise elementary non-equivalent and elementary non-equivalent to the structures A in A as well as to the structures in K, is called the relative i E e-spectrum of A with respect to K and denoted by e -Sp(A ). The value E K E sup{e -Sp(A′)) | A′ ≡ A } is called the relative e-spectrum of the theory K E Th(A ) with respect to K and denoted by e -Sp(Th(A )). E K E Similarly for a class T of theories and for a theory T = Th(A ) we E denote by e -Sp(T) the value e -Sp(T), where K = K(T ) is the class of all T K structures, each of which is a model of a theory in T . The value e -Sp(T) is T called the relative e-spectrum of the theory T with respect to T. Remark 2.1. 1. the class K(T), in the definition above, can be replaced by any subclass K′ ⊆ K(T) such that any structure in K(T) is elementary equivalent to a structure in K′. 2. if K ⊆ K then e -Sp(T) ≥ e -Sp(T), and if T ⊆ T then e - 1 2 K1 K2 1 2 T1 Sp(T) ≥ e -Sp(T). T2 3. The value e -Sp(T) is equal to the supremum |T \ T | for theories T 1 0 of E-classes of models of T such that T consists of all these theories and 1 T ⊆ T with Cl (T ) = T . 0 1 E 0 1 Definition. Two theories T and T of a language Σ are disjoint modulo 1 2 Σ , where Σ ⊆ Σ, or Σ -disjoint if T and T are do not have common 0 0 0 1 2 nonempty predicates for Σ\ Σ . If T and T are ∅-disjoint, these theories 0 1 2 are called simply disjoint. 4 Families T , j ∈ J, of theories in the language Σ are disjoint modulo j Σ , or Σ -disjoint if T and T are Σ -disjoint for any T ∈ T , T ∈ T , 0 0 j1 j2 0 j1 j1 j2 j2 j 6= j . If T and T are disjoint for any T ∈ T , T ∈ T , j 6= j , then 1 2 j1 j2 j1 j1 j2 j2 1 2 the families T , j ∈ J, are disjoint too. j The following properties are obvious. 1. Any families of theories in a language Σ are Σ-disjoint. 2. (Monotony) If Σ ⊆ Σ ⊆ Σ then disjoint families modulo Σ , in the 0 1 0 language Σ, are disjoint modulo Σ . 1 3. (Monotony) If families T and T are Σ -disjoint then any subfamilies j1 j2 0 T′ ⊆ T and T′ ⊆ T are Σ -disjoint too. j1 j1 j2 j2 0 Below we denote by K the class of all structures in languages containing Σ Σ such that all predicates outside Σ are empty. Similarly we denote by T Σ the class of all theories of structures in K . Σ Theorem 2.2. (Relative additivity for e-spectra) If T , j ∈ J, are Σ - j 0 disjoint families then for the E-combination T = Comb (T ) of {T | i ∈ E i i∈I i I} = T and for the E-combinations T = Comb (T ), j ∈ J, j j E j j∈J S e -Sp(T) = (e -Sp(T )). (1) TΣ0 TΣ0 j j∈J X Proof. Denote by T the set of theories for E-classes of models of T. Since the families T are Σ -disjoint, applying Proposition 1.1 we have that j 0 a theory T∗ belongs to Cl (T∗), where T ∗ ⊆ T , if and only if some of the E following conditions holds: 1) T∗ ∈ T ∗; 2) for any formula ϕ ∈ T∗ without predicate symbols in Σ\Σ , or with 0 predicate symbols in Σ \ Σ and saying that corresponding predicates are 0 empty, there are infinitely many theories in T ∈ T ∗ containing ϕ; 3) for any formula ϕ ∈ T∗, saying that some predicates in Σ\Σ which 0 used in ϕ are nonempty, there are infinitely many theories in T ∈ T∗ ∩ T , j for some j, containing ϕ; moreover, the theories T belong to the unique T . j Indeed, taking a formula ϕ in the language Σ we have finitely many symbols R ,...,R in Σ \ Σ , used in ϕ. Considering formulas ψ saying 1 n 0 i that R are nonempty, k = 1,...,n, we get finitely many possibilities for k n χδ1,...,δn ⇋ ϕ ∧ ψδk, δ ∈ {0,1}. Since ϕ is equivalent to χδ1,...,δn k k k=1 δ1,...,δn V W 5 and only subdisjunctions with positive ψ related to the fixed T hold, we k j can divide the disjunction to disjoint parts related to T . Since for ϕ there j are finitely many related T , we have finitely many cases for ϕ, each of which j related to the fixed T . These cases are described in Item 3. Item 2 deals j with formulas in the language Σ and with formulas for empty part in Σ\Σ . 0 0 In particular, by Proposition 1.1 these formulas define Cl (T ∗)∩T . E Σ0 Using Items 1–3 we have for T∗ that a theory T∗ belongs to T ∗ \ T Σ0 if and only if T∗ belong to (T∗ ∩T )\ T for unique j ∈ J. Thus theories j Σ0 witnessing the value e -Sp(T) are divided into disjoint parts witnessing the TΣ0 values e -Sp(T ). Thus the equality (1) holds. ✷ TΣ0 j Remark 2.3. Having positive ComLim [1] the equality (1) can fail if families T are not Σ -disjoint, even for finite sets J of indexes, producing j 0 e -Sp(T′) < (e -Sp(T )) (2) TΣ0 TΣ0 j j∈J X for appropriate T′. Theorem 2.2 immediately implies Corollary 2.4. If T , j ∈ J, are disjoint then for the E-combination j T = Comb (T ) of {T | i ∈ I} = T and for the E-combinations E i i∈I i j j∈J T = Comb (T ), j ∈ J, S j E j e -Sp(T) = (e -Sp(T )). (3) T∅ T∅ j j∈J X Definition. The theory T in Theorem 2.2 is called the Σ -disjoint E- 0 union of the theories T , j ∈ J, and the theory T in Corollary 2.4 is the j disjoint E-union of the theories T , j ∈ J. j Remark 2.5. Additivity (1) and, in particular, (3) can be failed without indexes T . Indeed, it is possible to find T with e-Sp(T ) = 0 (for instance, Σ0 j j with finite T ) while e-Sp(T) can be positive. Take, for example, disjoint j singletons T = {T }, n ∈ ω \{0}, such that T has n-element models. We n n n have e-Sp(T ) = 0 for each n while e-Sp(T) = 1, since the theory T ∈ T n ∞ ∅ with infinite models belong to Cl ({T | n ∈ ω \ {0}}). Thus, for disjoint E n families T , j ∈ J, the equality j e-Sp(T) = (e-Sp(T )) (4) j j∈J X 6 can fail. Moreover, producing the effect above for definable subsets in models of T we get j e -Sp(T) > (e -Sp(T )). TΣ0 TΣ0 j j∈J X At the same time, by Corollary 2.4 (respectively, by Theorem 2.2) the equality (4) holds for (Σ -)disjoint families T , j ∈ J, if J is finite and each 0 j T does not generate theories in T (in T ). j ∅ Σ0 Applying theequality(3)wetakeanE-combinationT withe -Sp(T ) = 0 T∅ 0 λ. Furthermore we consider disjoint copies T , j ∈ J, of T . Combining E- j 0 classes of all T we obtain a theory T such that if J is finite then e -Sp(T) = j T∅ |J|·λ. We have the same formula if |J| ≥ ω and λ > 0 since, in this case, the E-closure for theories of E-classes of models of T consists of theories of E-classes for theories T as well some theories in T . If E-classes have a fixed j ∅ finite or only infinite cardinalities, this theory has models whose cardinalities (finite or countable) areequal to the (either finite or countable) cardinality of models of T . Similarly, having theories T of languages Σ with cardinalities j λ |Σ| = λ + 1 and with e-Sp(T ) = λ > 0 [1, Proposition 4.3] and taking 0 E-combinations with their disjoint copies we get Proposition 2.6. For any positive cardinality λ there is a theory T such that E-classes of models of T form copies T , j ∈ J, of some E-combination j T with a language Σ in the cardinality λ + 1, with e -Sp(T ) = λ, and 0 T∅ 0 e -Sp(T) = |J|·λ. T∅ Remark 2.7. Since there are required theories T which do not generate 0 E-classes for T , Proposition 2.6 can be reformulated without the index T . ∅ ∅ Remark 2.8. Extending the Σ -disjoint Σ -coordinated E-union T by 0 0 definable bijections linking E-classes we can omit the additivity (1). Indeed, adding, forinstance, bijections f witnessing isomorphisms formodels ofdis- jk joint copies T and T , have we e -Sp(T ) instead of e -Sp(T )+e -Sp(T ). j j T∅ j T∅ j T∅ k Thus, bijections f allow to vary e -Sp(T) from λ to |J| · λ in terms of jk T∅ Proposition 2.6. Thus the equality (1) can fail again producing (2) for ap- propriate T′. 7 3 Families of theories with(out) least gener- ating sets BelowweapplyTheorem1.3characterizingtheexistenceofe-leastgenerating sets for Σ -disjoint families of theories. 0 The following natural questions arises: Question 1. When the existence of the least generating sets for the families T , j ∈ J, is equivalent to the existence of the least generating set j for the family T ? j j∈J S Question 2. Is it true that under conditions of Theorem 2.2 the existence of the least generating sets for the families T , j ∈ J, is equivalent to the j existence of the least generating set for the family T ? j j∈J S ConsideringQuestion2,wenotebelowthatthepropertyofthe(non)existence of the least generating sets is not preserving under expansions and extensions of families of theories. Proposition 3.1. Any E-closed family T of theories in a language Σ 0 0 can be transformed to an E-closed family T ′ in a language Σ′ ⊇ Σ such that 0 0 0 T′ consists of expansions of theories in T and T ′ has the least generating 0 0 0 set. Proof. FormingΣ′ itsufficestotakenewpredicatesymbolsR ,T ∈ T , 0 T0 0 0 such that R 6= ∅ for interpretations in the models of expansion T′ of T T0 0 0 and R = ∅ for interpretations in the models of expansion T′ of T 6= T . T0 1 1 0 Each formula ∃x¯R (x¯) isolates T′, and thus T′ has the least generating set T0 0 0 ✷ in view of Theorem 1.3. Existence of families T without least generating sets implies 0 Corollary 3.2. The property of non-existence of least generating sets is not preserved under expansions of theories. Remark 3.3. The expansion T′ of T in the proof of Proposition 3.1 0 0 produces discrete topologies for sets of theories in T ∪T ′. In fact, for this 0 0 purpose it suffices to isolate finite sets in T since any two distinct elements 0 T ,T ∈ T are separated by formulas ϕ such that ϕ ∈ T and ¬ϕ ∈ T , 0 1 0 i 1−i i = 0,1. 8 Note also that these operators of discretization transform the given set T to a set T′ with identical Cl . 0 0 E Clearly, if a set T has the discrete topology it can not be expanded to a 0 set without the least generating set. At the same time, there are expansions that transform sets with the least generating sets to sets without the least generating sets. Indeed, take Example in [3, Remark 3] with countably many disjoint copies F , q ∈ Q, of linearly ordered sets isomorphic to hω,≤i and q ordering limits J = limF by the ordinary dense order on Q such that q q {J | q ∈ Q} is densely ordered. We have a dense interval {J | q ∈ Q} q q whereas the set ∪{F | q ∈ Q} forms the least generating set T of theories q 0 for Cl (T ). Now we expand the LU-theories for F and J by new predicate E 0 q q symbol R such that R is empty for all theories corresponding to F and q ∀x¯R(x¯) is satisfied for all theories corresponding to J . The predicate R q separates the set of theories for J with respect to Cl . At the same time the q E theories for J forms the dense interval producing the set without the least q generating set in view of [3, Theorem 2]. Thus, we get the following Proposition 3.4. There is an E-closed family T of theories in a lan- 0 guage Σ and with the least generating set, which can be transformed to an 0 E-closed family T′ in a language Σ′ ⊇ Σ such that T ′ consists of expansions 0 0 0 0 of theories in T and T ′ does not have the least generating set. 0 0 Corollary 3.5. The property of existence of least generating sets is not preserved under expansions of theories. Remark 3.6. Adding the predicate R which separates theories for J q from theories for F , we get a copy for each J containing empty R. This q q effect is based on the property that separating an accumulation point J for q F wegetnewaccumulationpoint preserving formulasintheinitial language. q Introducing the predicate R together with the discretization for F , E- q closures do not generate new theories. Proposition 3.7. Any family T of theories in a language Σ, with in- 0 finitely many empty predicates for all theories in T , can be extended to a 0 family T ′ in the language Σ such that T′ does not have the least generating 0 0 set. Proof. Let Σ ⊆ Σ consist of predicate symbols which are empty for all 0 theories in T . Now we consider a family T of LU-theories such that all these 0 1 theories have empty predicates for Σ\Σ , and, using Σ as for [3, Theorem 0 0 9 2], T does not have the least generating set forming a dense interval. The 1 family T ′ = T ∪˙ T extends T and does not have the least generating set 0 0 1 0 since for any T ′′ ⊆ T ′, Cl (T′′) = Cl (T′′ ∩T )∪˙ Cl (T′′ ∩T ). ✷ 0 0 E 0 E 0 0 E 0 1 Corollary 3.8. The property of existence of least generating sets is not preserved under extensions of sets of theories. In view of Theorem 1.3 any family consisting of all theories in a given infinite language both does not have the least generating set and does not have a proper extension in the given language. Thus there are families of theories without least generating sets and without extension having least generating sets. At the same time the following proposition holds. Proposition 3.9. There is an E-closed family T of theories in a lan- 0 guage Σ and without the least generating set such that T can be extended to 0 an E-closed family T′ in the language Σ and with the least generating set. 0 Proof. It suffices to take Example in [3, Remark 3] that we used for the proof of Proposition 3.4. The theories for {J | q ∈ Q} form a family without q the least generating set whereas an extension of this family by the theories for F has the least generating set. ✷ q Corollary 3.10. The property of non-existence of least generating sets is not preserved under extensions of sets of theories. Remark 3.11. If an extension of an E-closed family T of theories trans- 0 forms T with the least generating set to an E-closed family T ′ without the 0 0 least generating set then, in view of Theorem 1.3, having the generating set in T consisting of isolated points we lose this property for T′. If an extension 0 0 of an E-closed family T of theories transforms T without the least gener- 0 0 ating set to an E-closed family T ′ with the least generating set then, again 0 in view of Theorem 1.3, we add a set of isolated theories to T generating all 0 theories in T ′. 0 Now we return to Questions 1 and 2. Clearly, for any set T of theories, Cl (T ∩T ) ⊂ T . Therefore Cl (T ) E Σ0 Σ0 E and each its generating set are divided into parts: in T and disjoint with Σ0 T . Since T , j ∈ J, are disjoint with respect to T , each T has the least Σ0 j Σ0 j generatingsetifandonlyifbothT ∩T andT \T havetheleastgenerating j Σ0 j Σ0 sets. Since under conditions of Theorem 2.2 the sets T \ T are disjoint, j Σ0 j ∈ J, we have the following proposition answering Question 1. 10