Combinations related to classes of finite and countably categorical structures and their theories
aa r X i v : . [ m a t h . L O ] J a n Combinations related to classesof finite and countably categorical structuresand their theories ∗ Sergey V. Sudoplatov † Abstract
We consider and characterize classes of finite and countably cate-gorical structures and their theories preserved under E -operators and P -operators. We describe e -spectra and families of finite cardinalitiesfor structures belonging to closures with respect to E -operators and P -operators. Key words: finite structure, countably categorical structure, el-ementary theory, E -operator, P -operator, e -spectrum. We continue to study structural properties of E -combinations and P -combinations of structures and their theories [1, 2, 3, 4, 5] applying thegeneral context to the classes of finitely categorical and ω -categorical theories.Approximations of structures by finite ones as well as correspondent ap-proximations of theories were studied in a series of papers, e.g. [6, 7, 8]. Weconsider these approximations in the context of structural combinations.We consider and describe e -spectra and families of finite cardinalities forstructures belonging to closures with respect to E -operators and P -operators. ∗ Mathematics Subject Classification: † [email protected] Preliminaries
Throughout the paper we use the following terminology in [1, 2] as well asin [9, 10].Let P = ( P i ) i ∈ I , be a family of nonempty unary predicates, ( A i ) i ∈ I bea family of structures such that P i is the universe of A i , i ∈ I , and thesymbols P i are disjoint with languages for the structures A j , j ∈ I . Thestructure A P ⇋ S i ∈ I A i expanded by the predicates P i is the P -union of thestructures A i , and the operator mapping ( A i ) i ∈ I to A P is the P -operator . Thestructure A P is called the P -combination of the structures A i and denotedby Comb P ( A i ) i ∈ I if A i = ( A P ↾ A i ) ↾ Σ( A i ), i ∈ I . Structures A ′ , whichare elementary equivalent to Comb P ( A i ) i ∈ I , will be also considered as P -combinations.Clearly, all structures A ′ ≡ Comb P ( A i ) i ∈ I are represented as unions oftheir restrictions A ′ i = ( A ′ ↾ P i ) ↾ Σ( A i ) if and only if the set p ∞ ( x ) = {¬ P i ( x ) | i ∈ I } is inconsistent. If A ′ = Comb P ( A ′ i ) i ∈ I , we write A ′ =Comb P ( A ′ i ) i ∈ I ∪{∞} , where A ′∞ = A ′ ↾ T i ∈ I P i , maybe applying Morleyzation.Moreover, we write Comb P ( A i ) i ∈ I ∪{∞} for Comb P ( A i ) i ∈ I with the emptystructure A ∞ .Note that if all predicates P i are disjoint, a structure A P is a P -combinationand a disjoint union of structures A i . In this case the P -combination A P is called disjoint . Clearly, for any disjoint P -combination A P , Th( A P ) =Th( A ′ P ), where A ′ P is obtained from A P replacing A i by pairwise disjoint A ′ i ≡ A i , i ∈ I . Thus, in this case, similar to structures the P -operatorworks for the theories T i = Th( A i ) producing the theory T P = Th( A P ), be-ing P -combination of T i , which is denoted by Comb P ( T i ) i ∈ I . In general, fornon-disjoint case, the theory T P will be also called a P -combination of thetheories T i , but in such a case we will keep in mind that this P -combinationis constructed with respect (and depending) to the structure A P , or, equiv-alently, with respect to any/some A ′ ≡ A P .For an equivalence relation E replacing disjoint predicates P i by E -classeswe get the structure A E being the E -union of the structures A i . In thiscase the operator mapping ( A i ) i ∈ I to A E is the E -operator . The structure A E is also called the E -combination of the structures A i and denoted byComb E ( A i ) i ∈ I ; here A i = ( A E ↾ A i ) ↾ Σ( A i ), i ∈ I . Similar above, structures A ′ , which are elementary equivalent to A E , are denoted by Comb E ( A ′ j ) j ∈ J ,where A ′ j are restrictions of A ′ to its E -classes. The E -operator works for2he theories T i = Th( A i ) producing the theory T E = Th( A E ), being E -combination of T i , which is denoted by Comb E ( T i ) i ∈ I or by Comb E ( T ), where T = { T i | i ∈ I } .Clearly, A ′ ≡ A P realizing p ∞ ( x ) is not elementary embeddable into A P and can not be represented as a disjoint P -combination of A ′ i ≡ A i , i ∈ I .At the same time, there are E -combinations such that all A ′ ≡ A E can berepresented as E -combinations of some A ′ j ≡ A i . We call this representabilityof A ′ to be the E -representability .If there is A ′ ≡ A E which is not E -representable, we have the E ′ -representability replacing E by E ′ such that E ′ is obtained from E addingequivalence classes with models for all theories T , where T is a theory of arestriction B of a structure A ′ ≡ A E to some E -class and B is not elementaryequivalent to the structures A i . The resulting structure A E ′ (with the E ′ -representability) is a e -completion , or a e -saturation , of A E . The structure A E ′ itself is called e -complete , or e -saturated , or e -universal , or e -largest .For a structure A E the number of new structures with respect to thestructures A i , i. e., of the structures B which are pairwise elementary non-equivalent and elementary non-equivalent to the structures A i , is called the e -spectrum of A E and denoted by e -Sp( A E ). The value sup { e -Sp( A ′ )) |A ′ ≡ A E } is called the e -spectrum of the theory Th( A E ) and denoted by e -Sp(Th( A E )).If A E does not have E -classes A i , which can be removed, with all E -classes A j ≡ A i , preserving the theory Th( A E ), then A E is called e -prime ,or e -minimal .For a structure A ′ ≡ A E we denote by TH( A ′ ) the set of all theoriesTh( A i ) of E -classes A i in A ′ .By the definition, an e -minimal structure A ′ consists of E -classes with aminimal set TH( A ′ ). If TH( A ′ ) is the least for models of Th( A ′ ) then A ′ iscalled e -least . Definition [2]. Let T be the class of all complete elementary theoriesof relational languages. For a set T ⊂ T we denote by Cl E ( T ) the set ofall theories Th( A ), where A is a structure of some E -class in A ′ ≡ A E , A E = Comb E ( A i ) i ∈ I , Th( A i ) ∈ T . As usual, if T = Cl E ( T ) then T is saidto be E -closed .The operator Cl E of E -closure can be naturally extended to the classes T ⊂ T as follows: Cl E ( T ) is the union of all Cl E ( T ) for subsets T ⊆ T .For a set T ⊂ T of theories in a language Σ and for a sentence ϕ with3( ϕ ) ⊆ Σ we denote by T ϕ the set { T ∈ T | ϕ ∈ T } . Proposition 1.1 [2]. If T ⊂ T is an infinite set and T ∈ T \ T then T ∈ Cl E ( T ) ( i.e., T is an accumulation point for T with respect to E -closure Cl E ) if and only if for any formula ϕ ∈ T the set T ϕ is infinite. Theorem 1.2 [2]. If T ′ is a generating set for a E -closed set T then thefollowing conditions are equivalent: (1) T ′ is the least generating set for T ; (2) T ′ is a minimal generating set for T ; (3) any theory in T ′ is isolated by some set ( T ′ ) ϕ , i.e., for any T ∈ T ′ there is ϕ ∈ T such that ( T ′ ) ϕ = { T } ; (4) any theory in T ′ is isolated by some set ( T ) ϕ , i.e., for any T ∈ T ′ there is ϕ ∈ T such that ( T ) ϕ = { T } . Definition [2]. For a set
T ⊂ T we denote by Cl P ( T ) the set of alltheories Th( A ) such that Th( A ) ∈ T or A is a structure of type p ∞ ( x ) in A ′ ≡ A P , where A P = Comb P ( A i ) i ∈ I and Th( A i ) ∈ T are pairwise distinct.As above, if T = Cl P ( T ) then T is said to be P -closed .Using above only disjoint P -combinations A P we get the closure Cl dP ( T )being a subset of Cl P ( T ).The closure operator Cl d,rP is obtained from Cl dP permitting repetitions ofstructures for predicates P i .Replacing E -classes by unary predicates P i (not necessary disjoint) beinguniverses for structures A i and restricting models of Th( A P ) to the set ofrealizations of p ∞ ( x ) we get the e -spectrum e -Sp(Th( A P )), i. e., the numberof pairwise elementary non-equivalent restrictions of M | = Th( A P ) to p ∞ ( x )such that these restrictions are not elementary equivalent to the structures A i . Definition [9, 10]. A n -dimensional cube , or a n -cube (where n ∈ ω ) isa graph isomorphic to the graph Q n with the universe { , } n and such thatany two vertices ( δ , . . . , δ n ) and ( δ ′ , . . . , δ ′ n ) are adjacent if and only if thesevertices differ exactly in one coordinate. The described graph Q n is calledthe canonical representative for the class of n -cubes.Let λ be an infinite cardinal. A λ -dimensional cube , or a λ -cube , is agraph isomorphic to a graph Γ = h X ; R i satisfying the following conditions:(1) the universe X ⊆ { , } λ is generated from an arbitrary function f ∈ X by the operator h f i attaching, to the set { f } , all results of substitutionsfor any finite tuples ( f ( i ) , . . . , f ( i m )) by tuples (1 − f ( i ) , . . . , − f ( i m ));42) the relation R consists of edges connecting functions differing exactlyin one coordinate (the ( i -th ) coordinate of function g ∈ { , } λ is the value g ( i ) correspondent to the argument i < λ ).The described graph Q ⇋ Q f with the universe h f i is a canonical repre-sentative for the class of λ -cubes.Note that the canonical representative of the class of n -cubes (as well asthe canonical representatives of the class of λ -cubes) are generated by anyits function: { , } n = h f i , where f ∈ { , } n . Therefore the universes ofcanonical representatives Q f of n -cubes like λ -cubes, is denoted by h f i . ω -categorical theories Remind that a countable complete theory T is ω -categorical if T has exactlyone countable model up to isomorphisms, i.e. I ( T, ω ) = 1. A countabletheory T is n -categorical , for natural n ≥
1, if T has exactly one n -elementmodel up to isomorphisms, i.e. I ( T, n ) = 1. A countable theory T is finitelycategorical if T is n -categorical for some n ∈ ω \ { } .The classes of all finitely and ω -categorical theories will be denoted by T fin and T ω, , respectively.Let T be a set (class) of theories in T fin ∪ T ω, , T be a theory in T . ByRyll-Nardzewski theorem, S n ( T ) is finite for any n . Then, for any n , classes ⌈ ϕ (¯ x ) ⌉ = { ϕ ′ (¯ x ) | ϕ ′ (¯ x ) ≡ ϕ (¯ x ) } of T -formulas with n free variables and ⌈ ϕ (¯ x ) ⌉ ≤ ⌈ ψ (¯ x ) ⌉ ⇔ ϕ (¯ x ) ⊢ ψ (¯ x ) form a finite Boolean algebra B n ( T ) with2 m n elements, where m n is the number of n -types of T .The algebra B n ( T ) can be interpreted as a m n -cube C m n ( T ), whose ver-tices form the universe B n ( T ) of B n ( T ), edges [ a, b ] link vertices a and b suchthat a -covers b or b -covers a , and each vertex a is marked by some u a ⇋ ⌈ ϕ (¯ x ) ⌉ , where a b ⇔ u a ≤ u b . The label 0 is used for the vertexcorresponding to ⌈¬ ¯ x ≈ ¯ x ⌉ and 1 — for the vertex corresponding to ⌈ ¯ x ≈ ¯ x ⌉ .Obviously, the sets ⌈ ϕ (¯ x ) ⌉ and the relation ≤ depend on the theory T but we omit T if the theory is fixed or it is clear by the context.Clearly, algebras B n ( T ) and B n ( T ), for T , T ∈ T , may be not coordi-nated: it is possible ⌈ ϕ (¯ x ) ⌉ < ⌈ ψ (¯ x ) ⌉ for T whereas ⌈ ψ (¯ x ) ⌉ < ⌈ ϕ (¯ x ) ⌉ for T . If ⌈ ϕ (¯ x ) ⌉ < ⌈ ψ (¯ x ) ⌉ for T and ⌈ ϕ (¯ x ) ⌉ < ⌈ ψ (¯ x ) ⌉ for T , we say that T witnesses that ⌈ ϕ (¯ x ) ⌉ < ⌈ ψ (¯ x ) ⌉ for T (and vice versa).5t the same time, if a countable theory T does not belong to T fin ∪ T ω, then for some n ≥ B n ( T ) is infinite and therefore there is a formula ϕ (¯ x ), for instance (¯ x ≈ ¯ x ), such that for the label u = ⌈ ϕ (¯ x ) ⌉ there is aninfinite decreasing chain ( u k ) k ∈ ω of labels: u k +1 < u k < u , witnessed bysome formulas ϕ k (¯ x ). In such a case, if T ∈ Cl E ( T ), then by Proposition1.1 for any finite sequence ( u l , . . . , u , u ) there are infinitely many theoriesin T witnessing that u l < . . . < u < u . In particular, cardinalities m n forBoolean algebras B n ( T ) and for cubes C m n ( T ) are unbounded for T : distances ρ n,T (0 , u ) are unbounded for the cubes C m n ( T ), i.e., sup { ρ n,T (0 , u ) | T ∈T } = ∞ . It is equivalent to take (¯ x ≈ ¯ x ) for ϕ (¯ x ) and to get sup { ρ n,T (0 , | T ∈ T } = ∞ .Thus we get the following Theorem 2.1.
Let T be a class of theories in T fin ∪ T ω, . The followingconditions are equivalent: (1) Cl E ( T )
6⊆ T fin ∪ T ω, ; (2) for some natural n ≥ , Boolean algebras B n ( T ) , T ∈ T , have un-bounded cardinalities and, moreover, there is an infinite decreasing chain ( u k ) k ∈ ω of labels for some formulas ϕ k (¯ x ) such that any finite sequence ( u l , . . . , u ) with u l < . . . < u is witnessed by infinitely many theories in T ; (3) the same as in (2) with u = 1 . Corollary 2.2.
A class
T ⊆ T fin ∪ T ω, does not generate, using the E -operator, theories, which are neither finitely categorical and ω -categorical,if and only if for the Boolean algebras B n ( T ) , T ∈ T , there are no infinitedecreasing chains ( u k ) k ∈ ω of labels for some formulas ϕ k (¯ x ) such that anyfinite sequence ( u l , . . . , u ) with u l < . . . < u is witnessed by infinitely manytheories in T . Remark 2.3.
Corollary 2.2 together with Proposition 1.1 allow to deter-mine E -closed classes of finitely categorical and ω -categorical theories. Here,since finite sets of theories are E -closed, it suffices to consider infinite sets.Considering a set T of theories with disjoint languages, for the E -closenessit suffices to add theories of the empty language describing cardinalities, in ω + 1, of universes if these cardinalities meet infinitely many times in T .In such a case we obtain relative closures [4] and have the following as-sertions. Proposition 2.4.
A class T of theories of pairwise disjoint languages is E -closed if and only if the following conditions hold: for any n ∈ ω \ { } whenever T contains infinitely many theories with n -element models then T contains the theory T n of the empty language andwith n -element models; (ii) if T contains theories with unbounded finite cardinalities of models,or infinitely many theories with infinite models, then T contains the theory T ∞ of the empty language and with infinite models. Corollary 2.5.
A class
T ⊂ T fin ∪ T ω, of theories of pairwise disjointlanguages is E -closed if and only if the conditions (i) and (ii) hold. Corollary 2.6.
A class
T ⊂ T fin of theories of pairwise disjoint lan-guages is E -closed if and only if the condition (i) holds and there is N ∈ ω such T does not have n -categorical theories for n > N . Corollary 2.7.
A class
T ⊂ T ω, of theories of pairwise disjoint lan-guages is E -closed if and only if the condition (ii) holds. Corollary 2.8.
For any class
T ⊂ T fin ∪ T ω, of theories of pairwisedisjoint languages, Cl E ( T ) is contained in the class ⊂ T fin ∪ T ω, , moreover, Cl E ( T ) ⊆ T ∪ { T λ | λ ∈ ( ω \ { } ) ∪ {∞}} . Remark 2.9.
Using relative closures [4] the assertions 2.4–2.8 also holdif languages are disjoint modulo a common sublanguage Σ such that allrestrictions of n -categorical theories in T ∩ T fin to Σ have isomorphic (fi-nite) models M n and all restrictions of theories in T ∩ T ω, have isomorphiccountable models M ω . In such a case, the theories T n should be replaced byTh( M n ) and T ∞ — by Th( M ω ).It is also permitted to have finitely many possibilities for each M n andfor M ω .The following example shows that (even with pairwise disjoint languages) ω -categorical theories T with unbounded ρ n,T (0 ,
1) do not force theories out-side the class of ω -categorical theories. Example 2.10.
Let T n be a theory of infinitely many disjoint n -cubeswith a graph relation R (2) n , R m = R n for m = n . For the set T = { T n | n ∈ ω } we have Cl E ( T ) = T ∪ { T ∞ } . All theories in Cl E ( T ) are ω -categoricalwhereas ρ ,T n (0 ,
1) = n + 2 that witnessed by formulas describing distances d ( x, y ) ∈ ω ∪ {∞} between elements. 7imilarly, taking for each n ∈ ω exactly one n -cube with a graph relation R (2) n , we get a set T of theories such that Cl E ( T ) ⊂ T fin ∪ T ω, . Remark 2.11.
Assertions 2.1 – 2.5 and 2.7 – 2.9 hold for the operatorsCl dP and Cl d,rP replacing E -closures by P -closures. As non-isolated typesalways produce infinite structures, Corollary 2.6 holds only for Cl dP withfinite sets T of theories. Definition [6]. An infinite structure M is pseudofinite if every sentence truein M has a finite model. Definition (cf. [11]). A consistent formula ϕ forces the infinity if ϕ doesnot have finite models.By the definition, an infinite structure M is pseudofinite if and only if M does not satisfy formulas forcing the infinity.We denote the class T \ T fin by T inf . Proposition 3.1.
A theory T ∈ T inf belongs to some E -closure of theo-ries in T fin if and only if T does not have formulas forcing the infinity. Proof.
If a formula ϕ forces the infinity then T ϕ ⊂ T inf for any T ⊆ T .Thus, having such a formula ϕ ∈ T , T can not be approximated by theoriesin T fin and so T does not belong to E -closures of families T ⊆ T fin .Conversely, if any formula ϕ ∈ T does not force the infinity then, since T / ∈ T fin , ( T fin ) ϕ is infinite using unbounded finite cardinalities and we canchoose infinitely many theories in ( T fin ) ϕ , for each ϕ ∈ T , forming a set T ⊂ T fin such that T ∈ Cl E ( T ). ✷ Note that, in view of Proposition 1.1, Proposition 3.1 is a reformulationof Lemma 1 in [7].
Corollary 3.2.
If a theory T ∈ T inf belongs to some E -closure of theoriesin T fin then T is not finitely axiomatizable. Proof. If T is finitely axiomatizable by some formula ϕ then |T ϕ | ≤ T ⊆ T and ϕ forces the infinity. Thus, in view of Proposition 3.1,8 can not be approximated by theories in T fin , i. e., T does not belong to E -closures of families T ⊂ T fin . ✷ In fact, in view of Theorem 1.2, the arguments for Corollary 3.3 showthat Cl E ( T ), for a family T of finitely axiomatizable theories, has the leastgenerating set T and does not contain new finitely axiomatizable theories.Note that Proposition 3.1 admits a reformulation for Cl dP repeating theproof. At the same time theories in T fin can not be approximated by theoriesin T inf with respect to Cl E (in view of Proposition 1.1) whereas each theoryin T fin can be approximated by theories in T inf with respect to Cl dP : Proposition 3.3.
For any theory T ∈ T fin there is a family T ⊂ T inf such that T belongs to the Σ( T ) -restriction of Cl dP ( T ) . Proof.
It suffices to form T by infinitely many theories of structures A i , i ∈ I , with infinitely many copies of models M | = T forming E i -classes forequivalence relations E i , where E j is either equality or complete for j = i .Considering disjoint unary predicates P i for A i we get the nonprincipal 1-type p ∞ ( x ) isolated by the set {¬ P i ( x ) | i ∈ I } which can be realized by theset M with the structure M witnessing that T belongs to the restriction ofCl dP ( T ) removing new relations E i . ✷ Remark 3.4.
We have a similar effect removing all relations E j in thestructures A i and obtaining isomorphic structures A ′ i : by compactness the P -combination of structures A ′ i (where disjoint A ′ i form unary predicates P i ) has the theory with a model, whose p ∞ -restriction forms a structureisomorphic to M . In this case we have Cl d,rP ( { Th( A ′ i ) } ). ✷ Remark 3.5.
As in the proof of Proposition 3.1 theories in T can bechosen consistent modulo cardinalities of their models we can add that e -Sp( T ) = 1 for the E -combination T of the theories in T .As the same time e -Sp( T ′ ) is infinite for the P -combination T ′ of A i inthe proof of Proposition 3.3, since p ∞ ( x ) has infinitely many possibilities forfinite cardinalities of sets of realizations for p ∞ ( x ). ✷ e -spectra for finitely categoricaland ω -categorical theories We refine the notions of e -spectra e -Sp( A E ) and e -Sp( T ) for the theories T = Th( A E ) restricting the class of possible theories to a given class T in9he following way.For a structure A E the number of new structures with respect to thestructures A i , i. e., of the structures B with Th( B ) ∈ T , which are pairwiseelementary non-equivalent and elementary non-equivalent to the structures A i , is called the ( e, T ) -spectrum of A E and denoted by ( e, T )-Sp( A E ). Thevalue sup { ( e, T )-Sp( A ′ )) | A ′ ≡ A E } is called the ( e, T ) -spectrum of thetheory Th( A E ) and denoted by ( e, T )-Sp(Th( A E )).The following properties are obvious.1. (Monotony) If T ⊆ T then ( e, T )-Sp(Th( A E )) ≤ ( e, T )-Sp(Th( A E ))for any structure A E .2. (Additivity) If the class T of all complete elementary theories of rela-tional languages is the disjoint union of subclasses T and T then for anytheory T = Th( A E ), e -Sp( T ) = ( e, T )-Sp( T ) + ( e, T )-Sp( T ) . We divide a class T of theories into two disjoint subclasses T fin and T inf having finite and infinite non-empty language relations, respectively. Moreprecisely, for functions f : ω → λ f , where λ f are cardinalities, we divide T into subclasses T f of theories T such that T has f ( n ) n -ary predicate symbolsfor each n ∈ ω .For the function f we denote by Supp( f ) its support , i.e., the set { n ∈ ω | f ( n ) > } .Clearly, the language of a theory T ∈ T f is finite if and only if ρ f ⊂ ω and Supp( f ) is finite.Illustrating ( e, T )-spectra for the class T of all cubic theories and takingthe class T fin0 ⊂ T of all theories of finite cubes we note that for an E -combination T of theories T i in T fin0 , ( e, T )-Sp( T ) is positive if and only ifthere are infinitely many T i . In such a case, ( e, T )-Sp( T ) = 1 and new theory,which does not belong to T fin0 , is the theory of ω -cube.The class T fin is represented as disjoint union of subclasses T fin , n of the-ories having n -element models, n ∈ ω \ { } . For N ∈ ω , the class S n ≤ N T fin ,n is denoted by T fin , ≤ N . Proposition 4.1.
For any
T ⊂ T , Cl E ( T ) \ T fin = ∅ if and only if forany natural N , T 6⊂ T fin , ≤ N . roof. If T contains a theory with infinite models, the assertion isobvious. If T ⊂ T fin , then we apply Compactness and Proposition 1.1. ✷ The following obvious proposition is also based on Proposition 1.1.
Proposition 4.2. If T ⊂ T fin ,n ( respectively T ⊂ T fin , ≤ N ) then Cl E ( T ) ⊂T fin ,n (Cl E ( T ) ⊂ T fin , ≤ N ) . For any theory T = Th( A E ) , where all E -classes have theories in T , e - Sp( T ) = ( e, T fin ,n ) - Sp(Th( A E )) ( e - Sp( T ) =( e, T fin , ≤ N ) - Sp(Th( A E ))) . If, additionally, T is the set of theories in a finitelanguage then T is finite ( and so E -closed ) . In particular, for any theory T = Th( A E ) in a finite language, where all E -classes have theories in T , e - Sp( T ) = 0 . Remark 4.3.
In fact, the conclusions of Proposition 4.2 follow implyingthe following fact. If all theories in T contain a formula ϕ then all theoriesin Cl E ( T ) contain ϕ . For (1) we take a formula ϕ “saying” that models haveexactly n elements, and for (2) — a formula ϕ “saying” that models haveat most N elements. If the language is finite there are only finitely manypossibilities for isomorphism types on n -element sets and these possibilitiesare formula-definable.Similarly Proposition 4.2 we have Proposition 4.4. If T ∩ T fin = ∅ then Cl E ( T ) ∩ T fin = ∅ . Definition [3]. A theory T in a predicate language Σ is called languageuniform , or a LU -theory if for each arity n any substitution on the set ofnon-empty n -ary predicates (corresponding to the symbols in Σ) preserves T . The LU-theory T is called IILU -theory if it has non-empty predicatesand as soon as there is a non-empty n -ary predicate then there are infinitelymany non-empty n -ary predicates and there are infinitely many empty n -arypredicates.Since for any finite cardinality n there are IILU-theories with n -elementmodels, repeating the proof of [3, Proposition 12] and [3, Proposition 13] weget Proposition 4.5. (1)
For any n ∈ ω \ { } and µ ≤ ω there is an E -combination T = Th( A E ) of IILU -theories T i ∈ T fin ,n in a language Σ of thecardinality ω such that T has an e -least model and e - Sp( T ) = µ . (2) For any uncountable cardinality λ there is an E -combination T =Th( A E ) of IILU -theories T i ∈ T fin ,n in a language Σ of the cardinality λ such that T has an e -least model and e - Sp( T ) = λ . roposition 4.6. For any n ∈ ω \ { } and infinite cardinality λ thereis an E -combination T = Th( A E ) of IILU -theories T i ∈ T fin ,n in a language Σ of cardinality λ such that T does not have e -least models and e - Sp( T ) ≥ max { ω , λ } . Proposition 4.7.
For any n ∈ ω \ { } and infinite cardinality λ there isan E -combination T = Th( A E ) of LU -theories T i ∈ T fin ,n in a language Σ of cardinality λ such that T does not have e -least models and e - Sp( T ) = 2 λ . Proof.
Let Σ be a language consisting, for some natural m , of m -arypredicate symbols R i , i < λ . For any Σ ′ ⊆ Σ we take a structure A Σ ′ of the cardinality n such that R i = ( A Σ ′ ) m for R i ∈ Σ ′ , and R i = ∅ for R i ∈ Σ \ Σ ′ . Clearly, each structure A Σ ′ has a LU-theory and A Σ ′
6≡ A Σ ′′ for Σ ′ = Σ ′′ . For the E -combination A E of the structures A Σ ′ we obtainthe theory T = Th( A E ) having a model of the cardinality λ . At the sametime A E has 2 λ distinct theories of the E -classes A Σ ′ . Thus, e -Sp( T ) = 2 λ .Finally we note that T does not have e -least models by Theorem 1.2 andarguments for [2, Proposition 9]. ✷ Remark 4.8.
Considering countable LU-theories for the assertions abovewe can assume that these theories belong to a class T f , where f ∈ ω ω andSupp( f ) is infinite. Note also that Propositions 4.5–4.7 hold replacing theclasses T fin ,n by T ω, .Replacing E -classes by unary predicates P i (not necessary disjoint) beinguniverses for structures A i and restricting models of Th( A P ) to the set ofrealizations of p ∞ ( x ) we get the ( e, T ) -spectrum ( e, T )-Sp(Th( A P )), i. e.,the number of pairwise elementary non-equivalent restrictions N of M | =Th( A P ) to p ∞ ( x ) such that Th( N ) ∈ T . Proposition 4.9.
If the structures A i have pairwise disjoint languageswith disjoint predicates P i then for any natural n ≥ , ( e, T fin ,n ) - Sp(Th( A P )) ≤ , and ( e, T \ T fin ) - Sp(Th( A P )) ≤ . Proof.
Clearly, if the structures A i have pairwise disjoint languages withdisjoint predicates P i then structures for p ∞ ( x ) do not contain realizations oflanguage predicates, i. e., have theories T λ . Now ( e, T fin ,n )-Sp(Th( A P )) ≤ e, T fin ,n )-Sp(Th( A P )) = 1 if and only if there are infinitely many indexes i and Th( A i ) = T n for any i . Similarly, ( e, T \ T fin )-Sp(Th( A P )) ≤ e, T \ T fin )-Sp(Th( A P )) = 1 if and only if there are infinitely many indexes i and Th( A i ) = T ∞ for any i . ✷ P -combinations with( e, T fin ,n )-Sp(Th( A P )) = 1, for n ∈ ω \{ } , and ( e, T \T fin )-Sp(Th( A P )) = 1.Comparing approximations in Section 3 and proofs for [1, Propositions4.12, 4.13] we get Proposition 4.10.
For any infinite cardinality λ there is a theory T =Th( A P ) being a P -combination of theories in T fin and of a language Σ suchthat | Σ | = λ and e - Sp( T ) = 2 λ . Definition.
A theory T in a predicate language Σ is called almost languageuniform , or a ALU -theory if for each arity n with n -ary predicates for Σthere is a partition for all n -ary predicates, corresponding to the symbols inΣ, with finitely many classes K such that any substitution preserving theseclasses preserves T , too. The ALU-theory T is called IIALU -theory if it hasnon-empty predicates and as soon as there is a non-empty n -ary predicatein a class K then there are infinitely many non-empty n -ary predicates in K and there are infinitely many empty n -ary predicates.By the definition any LU-theory is an ALU-theory and any IILU-theoryis an IIALU-theory as well.Since any finite structure can have only finitely many distinct predicatesfor each arity n we get the following Proposition 5.1.
Any theory T ∈ T fin is an ALU -theory.
Replacing LU- and IILU- by ALU- and IIALU- and the proofs in Propo-sitions 4.5–4.7 we get analogs for these assertions attracting expansions ofarbitrary theories in T fin ,n . Thus any theory in T fin ,n can be used obtainingdescribed e -spectra. Let T be a nonempty family of theories in T . We denote by c E ( T ) (re-spectively, c P ( T ), c dP ( T ), c d,rP ( T )) the set of finite cardinalities for models of13heories in Cl E ( T ) (Cl P ( T ), Cl dP ( T ), Cl d,rP ( T )) and by ¯ c E ( T ) (respectively,¯ c P ( T ), ¯ c dP ( T ), ¯ c d,rP ( T )) the set of finite cardinalities for models of theoriesin Cl E ( T ) (Cl P ( T ), Cl dP ( T ), Cl d,rP ( T )) which are not cardinalities for modelsof theories in T . Additionally, for Cl P ( T ), Cl dP ( T ) and Cl d,rP ( T ) we denoteby ˆ c P ( T ), ˆ c dP ( T ), ˆ c d,rP ( T ), respectively, the set of finite cardinalities for mod-els of theories being restrictions for corresponding P -combinations to sets ofrealizations of types p ∞ ( x ). Remark 6.1.
Since E -closures preserve finite cardinalities for modelsof theories in families in T , i.e., c E ( T ) consists of these cardinalities for T , then ¯ c E ( T ) ≡ ∅ . Thus we can use the notation c E ( T ) for the set offinite cardinalities for models of theories in T , or, equivalently, for models oftheories in Cl E ( T ). Remark 6.2. If T is finite, or corresponding p ∞ ( x ) is consistent andthere are no models with finitely many realizations for p ∞ ( x ), then c P ( T ) = c dP ( T ) = c E ( T ) and ¯ c P ( T ) = ¯ c dP ( T ) = ˆ c P ( T ) = ˆ c dP ( T ) = ∅ .Examples of families of theories in the empty language Σ witness thatthe cardinalities for sets of realizations of p ∞ ( x ) can vary arbitrarily and forfinite T we have c d,rP ( T ) = ˆ c d,rP ( T ) = Z + and ¯ c d,rP ( T ) = Z + \ c E ( T ).Having an infinite family T in the language Σ , similarly we get c P ( T ) = c dP ( T ) = c d,rP ( T ) = ˆ c P ( T ) = ˆ c dP ( T ) = ˆ c d,rP ( T ) = Z + and ¯ c P ( T ) = ¯ c dP ( T ) =¯ c d,rP ( T ) = Z + \ c E ( T ). The latter formula shows that ¯ c P ( T ), ¯ c dP ( T ), and¯ c d,rP ( T ) can be arbitrary subsets of Z + with infinite complements. Thus wehave the following Proposition 6.3.
For any infinite set Y ⊆ Z + there is a family T suchthat ¯ c P ( T ) = ¯ c dP ( T ) = ¯ c d,rP ( T ) = Z + \ Y . Example 6.4.
If the language Σ consists of the symbol E k of the equiv-alence relation whose each class has k ∈ ω elements then p ∞ ( x ) can form anarbitrary structure with k -element equivalence classes and for a finite family T k we have c d,rP ( T k ) = ˆ c d,rP ( T k ) = k Z + and ¯ c d,rP ( T k ) = k Z + \ c E ( T k ). If thefamily T k is infinite then, similarly, c P ( T k ) = c dP ( T k ) = c d,rP ( T k ) = ˆ c P ( T k ) =ˆ c dP ( T k ) = ˆ c d,rP ( T k ) = k Z + and ¯ c P ( T k ) = ¯ c dP ( T k ) = ¯ c d,rP ( T k ) = k Z + \ c E ( T ).More generally, collecting the families of theories with distinct E k , k ∈ K , K ⊂ ω , we obtain nonempty values for c P , c dP , c d,rP , ˆ c P , ˆ c dP , ˆ c d,rP as unions S k ∈ K k Z + .Now we have to show that all possible nonempty values for ˆ c dP and ˆ c d,rP U k ∈ K k Z + (unions with finite sums for numbers in k Z + ) whereas values for ˆ c P may differ. Theorem 6.5.
For any nonempty family T there is K ⊂ ω such that ˆ c d,rP ( T ) = U k ∈ K k Z + . Proof.
Recall that for P -combinations with respect to Cl d,rP there areno links between disjoint predicates P i with structures A i being models oftheories in T . Therefore if p ∞ ( x ) can produce finite structures then structures A i with 1-types approximating p ∞ ( x ), define (partial) definable equivalencerelations with bounded finite classes E ( a ) and without definable extensions,for the approximations and for p ∞ ( x ). So there are no links between theclasses E ( a ) and having k elements in E ( a ) we produce, by compactness, aseries of 1 , , . . . , n, . . . E -classes for p ∞ ( x ) since p ∞ ( x ) is not isolated. Thuswe get a series k Z + for ˆ c d,rP ( T ). Varying finite cardinalities for the classes E ( a ) we obtain the required formula ˆ c d,rP ( T ) = U k ∈ K k Z + for some set K ⊂ ω witnessing these cardinalities. If p ∞ ( x ) can produce finite structures then weset K ⇋ ∅ . ✷ Theorem 6.6.
For any infinite family T there is K ⊂ ω such that ˆ c dP ( T ) = U k ∈ K k Z + . Proof repeats the proof of Theorem 6.5 using structures A i which pair-wise are not elementary equivalent. ✷ Remark 6.7.
1. In Theorems 6.5 and 6.6, if we have minimal K with | K | > p ∞ ( x ) is not complete. Indeed, taking, for sets ofrealizations of p ∞ ( x ), maximal definable equivalence relations E and E for k = k ∈ K we can not move, by automorphisms, elements of E -classes toelements of E -classes.2. Clearly, having E -classes and E -classes of same cardinalities withnon-isomorphic structures we again can not connect elements of these classesby automorphisms. Thus, | K | = 1 is a necessary but not sufficient conditionfor the completeness of p ∞ ( x ).3. The least cardinality | K | , with positive ˆ c d,rP or ˆ c dP , gives a lowerbound for independent equivalence relations with respect to their realizabil-ity/omitting for restrictions of models to sets of realizations of p ∞ ( x ). ✷ Remark 6.8.
Finite structures A ∞ for maximal definable equivalencerelations for p ∞ ( x ) with respect to Cl dP and to Cl d,rP can be isomorphic if and15nly if they are represented in some A i for Cl d,rP and infinitely many timesfor Cl dP , or approximated both for Cl d,rP and for Cl dP . Hence, for any infinitefamily T , ˆ c dP ( T ) = ˆ c d,rP ( T ) if and only if each n -element class for maximaldefinable equivalence relations for p ∞ ( x ) with respect to Cl d,rP has n -elementclasses for correspondent definable equivalence relations in infinitely manypairwise elementary non-equivalent structures A i , with respect to Cl dP . ✷ Definition [12]. Let M be a model of a theory T , ¯ a and ¯ b tuples in M , A a subset of M . The tuple ¯ a semi-isolates the tuple ¯ b over the set A if thereexists a formula ϕ (¯ a, ¯ y ) ∈ tp(¯ b/A ¯ a ) for which ϕ (¯ a, ¯ y ) ⊢ tp(¯ b/A ) holds. Inthis case we say that the formula ϕ (¯ x, ¯ y ) (with parameters in A ) witnessesthat ¯ b is semi-isolated over ¯ a with respect to A .If p ∈ S ( T ) and M | = T then SI M p denotes the relation of semi-isolation(over ∅ ) on the set of all realizations of p :SI M p ⇋ { (¯ a, ¯ b ) | M | = p (¯ a ) ∧ p (¯ b ) and ¯ a semi-isolates ¯ b } . The following definition generalizes the previous one for a family of 1-types, in particular, for incomplete p ∞ ( x ). Definition [13]. Let T be a complete theory, M | = T . We consider closed nonempty sets (under the natural topology) sets p ( x ) ⊆ S ( ∅ ), i. e.,sets p ( x ) such that p ( x ) = T i ∈ I [ ϕ p ,i ( x )], where [ ϕ p ,i ( x )] ⇋ { p ( x ) ∈ S ( ∅ ) | ϕ p ,i ( x ) ∈ p ( x ) } for some formulas ϕ p ,i ( x ) of T .For closed sets p ( x ) , q ( y ) ⊆ S ( ∅ ) of types, realized in M , we con-sider ( p , q ) -preserving ( p , q ) -semi-isolating , ( p → q ) - , or ( q ← p ) -formulas ϕ ( x, y ) of T , i. e., formulas for which if a ∈ M realizes a type in p ( x ) thenevery solution of ϕ ( a, y ) realizes a type in q ( y ).If p ( x ) = q ( y ) then ( p , q )-preserving formulas are called p -preserving or p -semi-isolating and we define, similarly to SI M p , the generalized relation SI M p of semi-isolation for the set of realizations of types in p ( x ):SI M p ⇋ { ( a, b ) | M | = p ( a ) ∧ p ′ ( b ) ∧ ϕ ( a, b )for p, p ′ ∈ p and a p -preserving formula ϕ ( x, y ) } . If ( a, b ) ∈ SI M p we say that a semi-isolates b with respect to p .Thus, a semi-isolates b (in sense of [12]) if and only if a semi-isolates b with respect to { tp( a ) , tp( b ) } . 16 emark 6.9. Since there are no links between structures A i with respectto Cl dP and Cl d,rP , the set p ∞ of all completions q ( x ) of p ∞ ( x ) has symmetricSI M p ∞ . Thus, the relations SI M p ∞ form equivalence relations. Positive valuesfor ˆ c dP and ˆ c d,rP imply that these equivalence relations have finite classes.Cardinalities of these classes define formulas in Theorems 6.5 and 6.6. ✷ Now we consider the general case, with the operator Cl P . One can hardlyexpect productive descriptions considering arbitrary links of structures withrespect to arbitrary links of predicates P i , in contrast to the disjoint pred-icates when, obviously, there are no links between the structures. So wewill fix a P -combination A P (and its theory T = Th( A P )) and consider theset ˆ c P ( T ) of values of finite cardinalities for p ∞ ( x ) with respect to given P -combination T , instead of the set ˆ c P ( T ) of values for all finite values for allpossible P -combinations. In other words we argue to describe sets of finitecardinalities for sets of realizations of a nonprincipal, not necessary complete,1-type p ∞ ( x ).We note the following obvious observations. Remark 6.10.
1. If any n ∈ ω realizations of a type p ∞ ( x ) force infinitelymany realizations of p ∞ ( x ) then it is true for any m > n .2. If a and b are realizations of a type p ∞ ( x ) and a does not semi-isolate b with respect to p ∞ then there are no formulas ϕ ( x, y ) with | = ϕ ( a, b ) andforcing finitely or infinitely many realizations for the type q = tp( b/a ), i. e.,the set of realizations of q can be empty and infinite, depending on a model.The first observation shows that having n which forces infinity, we getˆ c P ( T ) ⊂ n . The second one implies that realizations of p ∞ , which are notconnected by the relation of semi-isolation, contribute to ˆ c P ( T ) indepen-dently on the binary level. Moreover, these contributions by realizations a and b can generate distinct series, as in Theorems 6.5 and 6.5, only iftp( a ) = tp( b ).The following example shows that there is a theory T with ˆ c P ( T ) = { } clarifying that contributions above on the binary level deny by the ternarylevel. Example 6.11.
Consider a coloring Col: M → ω ∪ {∞} of an infiniteset M such that each color λ ∈ ω ∪ {∞} has infinitely many elements in M , i. e., each Col n = { a ∈ M | Col( a ) = n } is infinite as well as there areinfinitely many elements of the infinite color. We put P i = M \ S j
We modify Example 6.11 replacing elements by E k -classes, where each class contains k elements, and repeat the generic con-struction satisfying the following conditions:2) if aE k a ′ then Col( a ) = Col( a ′ );2) if ( a, b ) ∈ Q , aE k a ′ , bE k b ′ , then ( a ′ , b ′ ) ∈ Q .The theory T k of resulting generic structure M k satisfies ˆ c P ( T k ) = { k } since each realization a of p ∞ ( x ) forces k realizations of p ∞ ( x ) consistingof E k ( a ) and any two realizations of p ∞ ( x ) belonging to distinct E k -classesforces infinitely many E k -classes with elements satisfying p ∞ ( x ).Combining structures M k with distinct k we obtain a generic structurewhose theory T satisfies ˆ c P ( T ) = K for a given set K ⊆ Z + . Here sets ofrealizations of p ∞ ( x ) are divided into E k -classes for k ∈ K .Thus we have the following theorem asserting that values ˆ c P ( T ) can bearbitrary. Theorem 6.13.
For any set K ⊆ Z + there is a P -combination T suchthat ˆ c P ( T ) = K . Now we argue to modify the generic construction and Theorem 6.13 usingtransitive arrangements of algebraic systems similar to [9, 19], and obtaining18 similar result with complete p ∞ ( x ) describing possibilities for ˆ c P ( T ).For this aim we fix a nonempty set K ⊆ Z + claiming for ˆ c P ( T ) = K withsome P -combination T . Note that if 1 / ∈ K then either any realization a of p ∞ ( x ) forces infinitely many realizations or a belongs to the maximal finitedefinable E -class with some k > p ∞ ( x ), any finite set of realizations of p ∞ ( x ) forces that infinity andtherefore K = ∅ contradicting the condition K = ∅ . At second case, againby completeness of p ∞ ( x ), we have K ⊆ k Z + . Replacing elements by their E -classes we reduce the problem of construction of T with ˆ c P ( T ) = K to thecase 1 ∈ K .Example 6.11 witnesses the possibility for ˆ c P ( T ) = { } . So below weassume that 1 ∈ K and | K | ≥ k ∈ K \ { } we introduce a ternary relation R k defining afree (acyclic) precise pseudoplane P k [9] with infinitely many lines containingany fixed point and exactly k points belonging to any fixed line such that P k has infinitely many connected components. Then we combine these freepseudoplanes P k allowing that each point belongs to each pseudoplane P k and the union of sets of lines does not form cycles. We embed copies of thatcombination P of the pseudoplanes into unary predicates Col n as well as tothe structure of p ∞ ( x ).Modifying Example 6.11 we introduce a binary predicate Q such that;1) if ( a, b ) ∈ Q and ( a, c ) ∈ Q then a , b , c belong to pairwise distinctconnected components of P , the same is satisfied for Q − (as in Exampledescribed in [15, Section 1.3] and in [20]);2) elements a , . . . , a m , m >
1, realizing p ∞ ( x ) and belonging to a com-mon line l force all elements of l and do not force elements outside l ;3) if a and b are realizations of p ∞ ( x ) which do not have a commonline then a and b force infinitely many realizations of p ∞ ( x ) by the formula Q ( a, y ) ∧ Q ( b, y ).The resulted generic structure M of the language h Col n , Q, R k i n ∈ ω,k ∈ K \{ } and its theory T satisfy the following properties:i) any realization of p ∞ ( x ) does not force new realizations of p ∞ ( x ) wit-nessing 1 ∈ ˆ c P ( T );ii) any at least two distinct realizations of p ∞ ( x ) in a line l belonging to P k force exactly the set l witnessing k ∈ ˆ c P ( T ) for k ∈ K ;iii) any two distinct realizations of p ∞ ( x ) which do not have common linesforce infinitely many realizations of p ∞ ( x ) witnessing k ′ / ∈ ˆ c P ( T ) for k ′ / ∈ K .Thus we get ˆ c P ( T ) = K . 19ollecting the arguments above we have the following Theorem 6.14. (1) If T is a P -combination with a type p ∞ ( x ) isolating acomplete -type then ˆ c P ( T ) is either empty or contains k such that ˆ c P ( T ) ⊆ k Z + . (2) For any set K ⊆ k Z + , being empty or containing k , there is a P -combination T with a type p ∞ ( x ) isolating a complete -type such that ˆ c P ( T ) = K . References [1]
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