aa r X i v : . [ m a t h . L O ] J u l BACK AND FORTH SYSTEMS OF CONDENSATIONS
Miloˇs S. Kurili´c Abstract If L is a relational language, an L -structure X is condensable to an L -structure Y , we write X c Y , iff there is a bijective homomorphism (condensation)from X onto Y . We characterize the preorder c , the corresponding equiv-alence relation of bi-condensability, X ∼ c Y , and the reversibility of L -structures in terms of back and forth systems and the corresponding games.In a similar way we characterize the P ∞ ω -equivalence (which is equiva-lent to the generic bi-condensability) and the P -elementary equivalence of L -structures, obtaining analogues of Karp’s theorem and the theorems ofEhrenfeucht and Fra¨ıss´e. In addition, we establish a hierarchy between thesimilarities of structures considered in the paper. Applying these results weshow that homogeneous universal posets are not reversible and that a count-able L -structure X is weakly reversible (that is, satisfies the Cantor-Schr¨oder-Bernstein property for condensations) iff its P ∞ ω ∪N ∞ ω -theory is countablycategorical.2010MSC: 03C07, 03C75, 03C50, 03E40, 06A06.Keywords: condensation, bi-condensability, reversibility, back and forth,Karp’s theorem, Ehrenfeucht–Fra¨ıss´e games, infinitary languages In this paper we continue the investigation of the condensational preorder c onthe class Mod L of structures of a relational language L , defined by X c Y iff thereexists a bijective homomorphism (condensation) from X onto Y . We also considersome naturally related relations and properties: first, the equivalence relation of bi-condensability , defined by X ∼ c Y iff X c Y and Y c X , second, the structures X ∈ Mod L with the property that Y ∼ c X implies Y ∼ = X , for all Y ∈ Mod L ,(i.e., satisfying the Cantor-Schr¨oder-Bernstein property for condensations) called weakly reversible , and, in particular, the reversible structures (that is, the structures X having the property that each self-condensation of X is an automorphism).At first sight, the relations between structures and the properties of structuresmentioned above are more of set-theoretical than of model-theoretical character;for example, reversibility is not preserved under bi-definability and elementaryequivalence [12, 13]. But, on the other hand, reversibility is an invariant of some Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovi´ca4, 21000 Novi Sad, Serbia, e-mail: [email protected] L ∞ ω sentences, as well as the structures simply definable in linearorders are reversible [14, 12]).It turns out that, in the investigation of condensability and related phenom-ena, restricting our consideration to the class of sentences naturally correspondingto condensations we obtain a possibility to use several basic concepts and meth-ods of model theory. So, in Section 3, modifying (essentially Cantor’s) theoremsaying that back-and-forth equivalent countable L -structures are isomorphic, wecharacterize (bi-)condensability of structures of the same size κ ≥ ω in terms ofback and forth systems of condensations and the corresponding games. In additionwe characterize reversible structures in this way and, as an application, show thathomogeneous-universal posets (and, in particular, the countable random poset) arenon-reversible structures.The main statements of Sections 4 and 5 are “condensational analogues” ofsome well known results (concerning isomorphism). Namely, first, it is evident thatisomorphism of two L -structures implies that they satisfy the same L ∞ ω -sentences,which implies their elementary equivalence.Second, by the well known results including Karp’s theorem, the second prop-erty – the L ∞ ω -equivalence of L -structures X and Y , their back and forth equiva-lence (partial isomorphism), the existence of a winning strategy for player II in thecorresponding Ehrenfeucht-Fra¨ıss´e game of length ω , EF ω ( X , Y ) , and the genericisomorphism of structures ( V [ G ] | = X ∼ = Y , where V [ G ] is some generic extensionof the universe) are equivalent conditions (see [7], [1], [17]).Third, by the classical results of Ehrenfeucht and Fra¨ıss´e, the third property –the elementary equivalence, the finitary isomorphism of X and Y , and the existenceof winning strategies for player II in the games EF n ( X , Y ) , for all n ∈ N , areequivalent conditions in the class of models of a finite language [3, 4, 5].So, roughly speaking, the results of Sections 4 and 5 show that, replacing iso-morphism by bi-condensability, L ∞ ω -equivalence by P ∞ ω -equivalence, and ele-mentary equivalence by P -equivalence, we obtain the analogues of all aforemen-tioned classical theorems. Of course, instead of back and forth systems of partialisomorphisms, back and forth systems of partial condensations come to the scene;also, Ehrenfeucht-Fra¨ıss´e games are replaced by similar games with a differentwinning criterion. As a by-product, in Section 4 we obtain the following character-ization: a countable structure X is weakly reversible iff the theory Th P ∞ ω ∪N ∞ ω ( X ) is ω -categorical.In Section 6 we compare the similarities of structures considered in this paperand show that the implications between them are as Figure 1 describes.We note that our restriction to relational structures is not essential. By [10] allresults of this paper are in fact true for the structures of any language.ackand forth systems of condensations 3 ✏✏✏✏✏✏✏✏✏PPPPPPPPP✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏PPPPPPPPP✏✏✏✏✏✏✏✏✏ rrrrrrr rrrrrrr ∼ = ≡ ∞ ω ∧ ∼ c ≡ ∞ ω ≡ ∧ ( ≡ ∞ ω ∨ ∼ c ) ≡ ∧ ≡ P ∞ ω ≡≡ ∨ ∼ c ≡ ∧ ∼ c ∼ c ≡ ∞ ω ∨ ∼ c ≡ P ∞ ω ∧ ( ≡ ∨ ∼ c ) ≡ P ∞ ω ≡ ∨ ≡ P ∞ ω ≡ P Figure 1: Similarities of structures
Classes of formulas
Let L = h R i : i ∈ I i be a relational language, where ar( R i ) = n i , for i ∈ I , let κ be an infinite cardinal and Var = { v α : α ∈ κ } a setof variables. By At L we denote the corresponding set of atomic formulas , that is, At L = { v α = v β : α, β < κ }∪{ R i ( v α , . . . , v α ni ) : i ∈ I ∧h α , . . . , α n i i ∈ κ n i } .We recall that the class Form L ∞ ω of L ∞ ω -formulas is the closure of the set At L under negation, arbitrary conjunctions and disjunctions and finite quantification(that is, ¬ ϕ , ∀ v ϕ and ∃ v ϕ are in Form L ∞ ω , whenever ϕ ∈ Form L ∞ ω , and V F and W F are in Form L ∞ ω , for each set F ⊂
Form L ∞ ω ).The set P of R -positive first order L -formulas is the closure of the set P = At L ∪{¬ v α = v β : α, β < κ } under finite conjunctions, disjunctions and quantification (that is, ϕ ∧ ψ , ϕ ∨ ψ , ∀ v ϕ and ∃ v ϕ are in P , whenever ϕ, ψ ∈ P ; negations are not allowed).The set N of R -negative first order L -formulas is the closure of the set N := {¬ R i ( v α , . . . , v α ni ) : i ∈ I ∧ h α , . . . , α n i i ∈ κ n i } ∪{ v α = v β : α, β < κ } ∪ {¬ v α = v β : α, β < κ } Miloˇs S.Kurili´cunder finite conjunctions, disjunctions and quantification. The class P ∞ ω (resp. N ∞ ω ) of R -positive (resp. R -negative ) L ∞ ω -formulas is the closure of the set P (resp. N ) under finite quantification and arbitrary conjunctions and disjunctions.If F is a class of formulas, then Sent F will denote the class of all sentencesfrom F and F ( v , . . . , v n − ) will be the set of formulas ϕ ∈ F such that Fv( ϕ ) ⊂{ v , . . . , v n − } . If X , Y ∈ Mod L , by X ≡ F Y (resp. X ≪ F Y ) we denote that X | = ϕ iff Y | = ϕ (resp. X | = ϕ implies Y | = ϕ ), for each sentence ϕ ∈ Sent F . L ∞ ω -formulas ϕ and ψ are logically equivalent , in notation ϕ ↔ ψ , iff foreach L -structure X and any valuation ~x ∈ κ X we have: X | = ϕ [ ~x ] iff X | = ψ [ ~x ] . Back and forth systems of condensations If X and Y are L -structures, a func-tion f will be called a partial condensation from X to Y , we will write f ∈ PC( X , Y ) , iff f is a bijection which maps dom f ⊂ X onto ran f ⊂ Y and ∀ i ∈ I ∀ ¯ x ∈ (dom f ) n i (¯ x ∈ R X i ⇒ f ¯ x ∈ R Y i ) . (1) Fact 2.1 If n ∈ N , f = {h x k , y k i : k < n } ⊂ X × Y , ¯ x = h x , . . . , x n − i and ¯ y = h y , . . . , y n − i , then f ∈ PC( X , Y ) iff for each formula ϕ ∈ P ( v , . . . , v n − ) wehave X | = ϕ [¯ x ] ⇒ Y | = ϕ [¯ y ] . (2) Proof.
Clearly, f is a function (resp. an injection) iff (2) is true for all formulas v j = v j (resp. ¬ v j = v j ), where j , j < n , and f satisfies (1) iff (2) is true forall formulas R i ( v j , . . . , v j ni − ) , where i ∈ I and j k < n , for k < n i . ✷ A set Π ⊂ PC( X , Y ) will be called a back and forth system of condensations (in the sequel, shortly: back and forth system, or b.f.s.) iff Π = ∅ and(e1) ∀ f ∈ Π ∀ x ∈ X ∃ g ∈ Π ( x ∈ dom g ∧ f ⊂ g ) ,(e2) ∀ f ∈ Π ∀ y ∈ Y ∃ g ∈ Π ( y ∈ ran g ∧ f ⊂ g ) .If such a b.f.s. exists we will write X bfsc Y and X ∼ bfs c Y will denote that X bfsc Y and Y bfsc X . Games
Let X and Y be L -structures and κ a cardinal. The game G c κ ( X , Y ) isplayed in κ steps by two players, I and II, in the following way: at the α -th stepplayer I chooses one of the two structures and an element from it and then playerII chooses an element from the other structure. More precisely, either I chooses an x α ∈ X and II chooses y α ∈ Y , or I chooses an y α ∈ Y and II chooses x α ∈ X .So, each play gives a κ -sequence of pairs π κ = hh x α , y α i : α < κ i ∈ κ ( X × Y ) ;ackand forth systems of condensations 5player II wins the play if for the set of pairs ran π κ = {h x α , y α i : α < κ } ⊂ X × Y we have ran π κ ∈ PC( X , Y ) . Otherwise, player I wins.Roughly speaking, a strategy for a player determines its moves during the playon the basis of the previous moves of both players. More formally, if Σ is a strategyfor player II, π α = hh x β , y β i : β < α i is the sequence produced in the first α moves and player I chooses x α ∈ X at the α -th step, then y α := Σ( π α , x α ) ∈ Y is the response of II suggested by Σ ; otherwise, if player I chooses y α ∈ Y , then Σ suggests an x α := Σ( π α , y α ) ∈ X . A strategy Σ is a winning strategy for a playeriff that player wins each play in which follows Σ . We will write X G κ c Y , ifplayer II has a winning strategy in the game G c κ ( X , Y ) and X ∼ G κ c Y will denotethat X G κ c Y and Y G κ c X . In this section we generalize the well known fact that back-and-forth equivalentcountable L -structures are isomorphic. We recall that a partial order P = h P, ≤i iscalled κ -closed (where κ is an infinite cardinal) iff whenever γ < κ is an ordinaland h p α : α < γ i is a sequence in P satisfying ∀ α, β ∈ γ ( α < β ⇒ p β ≤ p α ) (3)there is p ∈ P such that p ≤ p α , for all α < γ . If X , Y ∈ Mod L , we will say that ab.f.s. Π ⊂ PC( X , Y ) is a κ -closed b.f.s. iff the partial order h Π , ⊃i is κ -closed. Theorem 3.1 If X and Y are L -structures of size κ ≥ ω , then we have(I) The following conditions are equivalent:(a) X c Y ,(b) There exists a κ -closed b.f.s. Π ⊂ PC( X , Y ) ,(c) Player II has a winning strategy in the game G c κ ( X , Y ) .(II) The following conditions are equivalent:(a) X ∼ c Y ,(b) There are κ -closed b.f.s. Π X , Y ⊂ PC( X , Y ) and Π Y , X ⊂ PC( Y , X ) ,(c) Player II has a winning strategy in the games G c κ ( X , Y ) and G c κ ( Y , X ) . A proof of the theorem is given after some preliminary work. We recall that, if P = h P, ≤i is a partial order, a set D ⊂ P is called dense iff for each p ∈ P thereis q ∈ D such that q ≤ p . A set Φ ⊂ P is called a filter iff Φ ∋ p ≤ q implies q ∈ Φ and for each p, q ∈ Φ there is r ∈ Φ such that r ≤ p, q . Fact 3.2 If P is a κ -closed partial order and D a family of ≤ κ dense subsets of P , then there is a filter Φ in P intersecting all D ∈ D . Miloˇs S.Kurili´c
Proof.
Let D = { D α : α < κ } be an enumeration. By recursion we construct asequence h p α : α < κ i in P such that for all α, β ∈ κ we have (i) p α ∈ D α , and(ii) α < β ⇒ p β ≤ p α . First we take p ∈ D . Suppose that < α < κ and that h p β : β < α i is a sequence satisfying (i) and (ii). Then, since P is κ -closed, thereis p ∈ P such that p ≤ p β , for all β < α , and, since D α is dense in P , there is p α ∈ D α , such that p α ≤ p , which implies that p α ≤ p β , for all β < α . Thus thesequence h p β : β ≤ α i satisfies (i) and (ii) and the recursion works. It is evidentthat Φ := { p ∈ P : ∃ α < κ p α ≤ p } is a filter in P . ✷ Proposition 3.3 If X and Y are L -structures of size κ ≥ ω and Π ⊂ PC( X , Y ) isa κ -closed b.f.s., then each f ∈ Π extends to a condensation F ∈ Cond( X , Y ) . Proof.
It is evident that the poset P := h Π f , ⊃i , where Π f := { g ∈ Π : f ⊂ g } , is κ -closed. Let A := X \ dom f and B := Y \ ran f .For a ∈ A , let D a := { g ∈ Π f : a ∈ dom g } . If h ∈ Π f \ D a , then a dom h and by (e1) there is g ∈ Π such that a ∈ dom g and h ⊂ g . Thus, since f ⊂ h wehave g ∈ Π f and, hence, g ∈ D a and g ⊃ h . So the sets D a , a ∈ A , are dense in P and, similarly, the sets ∆ b := { g ∈ Π f : b ∈ ran g } , b ∈ B , are dense in P as well.Since | A | + | B | ≤ κ , by Fact 3.2 there is a filter Φ in P intersecting all D a ’s and ∆ b ’s. Clearly we have f ⊂ F := S Φ ⊂ X × Y .If h x, y ′ i , h x, y ′′ i ∈ F there are g ′ , g ′′ ∈ Φ such that h x, y ′ i ∈ g ′ and h x, y ′′ i ∈ g ′′ and, since Φ is a filter, there is g ∈ Φ such that g ⊃ g ′ , g ′′ . Thus h x, y ′ i , h x, y ′′ i ∈ g and, since g is a function, y ′ = y ′′ . So F is a function and in the same way weshow that it is an injection.Let i ∈ I , ¯ x = h x , . . . , x n i − i ∈ (dom F ) n i and ¯ x ∈ R X i . Then, since dom F = S g ∈ Φ dom g , for each j < n i there is g j ∈ Φ such that x j ∈ dom g j .Since Φ is a filter, there is g ∈ Φ such that g ⊃ g j and, hence, dom g ⊃ dom g j ,for all j < n i . Thus ¯ x ∈ (dom g ) n i and, since g ∈ PC( X , Y ) and ¯ x ∈ R X i , wehave F ¯ x = g ¯ x ∈ R Y i . So F ∈ PC( X , Y ) .If a ∈ A , then there is g ∈ Φ ∩ D a and, hence, a ∈ dom g ⊂ dom F ; so A ⊂ dom F , which, together with dom f ⊂ dom F , implies that dom F = X .Similarly we have ran F = Y and, thus, F ∈ Cond( X , Y ) . ✷ Proof of Theorem 3.1
We prove part (I), which evidently implies part (II).(a) ⇒ (b). If X c Y and F ∈ Cond( X , Y ) , then Π := { F } ⊂ PC( X , Y ) .Since dom F = X and ran F = Y , the set Π satisfies (e1) and (e2) trivially and Π is κ -closed because each sequence in Π is a constant sequence.(b) ⇒ (a). If Π ⊂ PC( X , Y ) is a κ -closed b.f.s., then by Proposition 3.3 thereis F ∈ Cond( X , Y ) .(a) ⇒ (c). Let F ∈ Cond( X , Y ) . Let Σ be the following strategy for player IIin the game G c κ ( X , Y ) . At the α -th step, if player I chooses an x α ∈ X , then Σ ackand forth systems of condensations 7suggests y α = F ( x α ) ; if I chooses y α ∈ Y , then Σ suggests x α = F − ( y α ) . Now,if π κ = hh x α , y α i : α < κ i is a play of the game in which player II follows Σ , then ran π κ = {h x α , y α i : α < κ } ⊂ F and, hence, ran π κ ∈ PC( X , Y ) ; thus II winsthe play. So, Σ is a winning strategy for player II.(c) ⇒ (a). Let Σ be a winning strategy for player II in the game G c κ ( X , Y ) .Clearly κ = E ∪ O , where E = { γ + 2 n : γ ∈ κ ∩ (Lim ∪{ } ) ∧ n ∈ ω } and O = { γ + 2 n + 1 : γ ∈ κ ∩ (Lim ∪{ } ) ∧ n ∈ ω } are the sets of even and oddordinals < κ respectively, and we have | E | = | O | = κ . Thus there is a bijection b : κ → X ∪ Y such that b [ E ] = X and b [ O ] = Y . Let hh x α , y α i : α < κ i be the play of the game in which, at the step α , player I chooses b ( α ) and playerII follows Σ . Then F = {h x α , y α i : α < κ } ∈ PC( X , Y ) . If x ∈ X , then x = b ( α ) = x α , for some α ∈ E , and, hence, x ∈ dom F ; so dom F = X and,similarly, ran F = Y , which gives F ∈ Cond( X , Y ) . Thus X c Y indeed. ✷ Proposition 3.3 provides a useful characterization of reversible structures, whichwe prove in the sequel. We note that the class of reversible structures contains, forexample, linear orders, Boolean lattices, well founded posets with finite levels [8],tournaments, Henson graphs [14], and Henson digraphs [11]. Reversible equiva-lence relations are characterized in [16], while reversible posets representable asdisjoint unions of well orders and their inverses are characterized in [15].If X is an L -structure, instead of PC( X , X ) we write PC( X ) . A finite partialcondensation f ∈ PC( X ) will be called bad iff f F , for all F ∈ Aut( X ) . Theorem 3.4
For an L -structure X of size κ ≥ ω the following is equivalent:(a) X is not a reversible structure,(b) There exists a κ -closed b.f.s. Π ⊂ PC( X ) containing a bad condensation,(c) There exists a b.f.s. Π ⊂ PC( X ) containing a bad condensation, if κ = ω . Proof. (a) ⇒ (b). If X is not a reversible structure and F ∈ Cond( X ) \ Aut( X ) ,then there are i ∈ I and ¯ x = h x , . . . , x n i − i ∈ X n i such that ¯ x R X i and F ¯ x ∈ R X i . Let K := { x , . . . , x n i − } and f := F ↾ K . Then, clearly, f is a badcondensation, f ∈ Π := { F ↾ K : K ⊂ X } ⊂ PC( X ) and Π is a κ -closed b.f.s.(b) ⇒ (a). If Π ⊂ PC( X ) is a κ -closed b.f.s. and f ∈ Π a bad condensation,then by Proposition 3.3 there is F ∈ Cond( X ) extending f and, hence, F Aut( X ) . So, the structure X is not reversible.The equivalence (b) ⇔ (c) is true because each poset is ω -closed. ✷ Homogeneous universal posets
Since the class of posets is a J´onsson class [6],for each regular beth number κ there is a κ -homogeneous-universal poset P . As anexample of application of Theorem 3.4 we show that such posets are not reversible; Miloˇs S.Kurili´cin particular, taking κ = ω we conclude that the random poset (i.e., the uniquecountable homogeneous universal poset, see [18]) is non-reversible as well. If P isa poset and p, q ∈ P , we will write p k q iff p q ∧ q p . For A, B ⊂ P , A < B denotes that a < b , for all a ∈ A and b ∈ B ; notation A k B is defined similarly. Theorem 3.5
Let P = h P, < i be a strict partial order of size κ ≥ ω satisfying (u1) ∀ L, G ∈ [ P ] <κ \ {∅} ( L < G ⇒ ∃ x ∈ P L < x < G ) , (u2) ∀ K ∈ [ P ] <κ \ {∅} ∃ x, y, z ∈ P ( x < K ∧ y > K ∧ z k K ) ,and let L, G, K ∈ [ P ] <κ \ {∅} , where L < G . Then we have(a) |{ x ∈ P : L < x < G }| = κ ;(b) |{ x ∈ P : x < K }| = |{ x ∈ P : x > K }| = |{ x ∈ P : x k K }| = κ ;(c) If κ is a regular cardinal, then P is not a reversible structure;(d) If P is a κ -homogeneous-universal poset, it is not a reversible structure. Proof. (a) If
L, G ∈ [ P ] <κ \ {∅} , L < G and S := { x ∈ P : L < x < G } , thenby (u1) we have S = ∅ and, clearly, L < S . So, assuming that | S | < κ , by (u1)there would be y ∈ P such that L < y < S , which would imply that y S and L < y < G . But then y ∈ S and we have a contradiction. So, | S | = κ .(b) If K ∈ [ P ] <κ \ {∅} then by (u1) we have T := { x ∈ P : x < K } 6 = ∅ .Assuming that | T | < κ , by (u1) there would be x ∈ P such that x < T , whichwould imply that x T and x < K . But then x ∈ T and we have a contradiction.So, | T | = κ and, similarly, |{ x ∈ P : x > K }| = |{ x ∈ P : x k K }| = κ .(c) By (u2) there are a , a , b , b ∈ P , where a k a and b < b . Then f := {h a , b i , h a , b i} ∈ Bad( P ) , since κ is a regular cardinal the poset h Π ⊃i ,where Π := { f ∈ PC( P ) : f ⊂ f ∧ | f | < κ } , is κ -closed and, by Theorem 3.4, itremains to be shown that Π is a b.f.s.(e1) If f ∈ Π and a ∈ P \ dom f , then the sets L a := { x ∈ dom f : x < a } and G a := { y ∈ dom f : y > a } are of size < κ and we have the following cases.1. L a = ∅ and G a = ∅ . If l ∈ f [ L a ] and g ∈ f [ G a ] , then there are x ∈ L a and y ∈ G a such that l = f ( x ) and g = f ( y ) and, since x < a < y and f isa homomorphism, f ( x ) < f ( y ) , that is l < g . Thus f [ L a ] < f [ G a ] and, since | ran f | < κ , by (a) there is b ∈ P \ ran f such that f [ L a ] < b < f [ G a ] . Now g := f ∪ {h a, b i} is an injection, a ∈ dom g and f ⊂ g . If x ∈ dom f and x < a ,then x ∈ L a and, hence, f ( x ) ∈ f [ L a ] and g ( x ) = f ( x ) < b = g ( a ) . Similarly, a < y ∈ dom f implies g ( a ) < g ( y ) . So g is a homomorphism and g ∈ Π .2. L a = ∅ and G a = ∅ . Then f [ G a ] ∈ [ P ] <κ \ {∅} and, since | ran f | < κ , by(b) there is b ∈ P \ ran f such that b < f [ G a ] . Now g := f ∪{h a, b i} is an injection, a ∈ dom g , f ⊂ g and we show that g is a homomorphism. If a < y ∈ dom f ,then y ∈ G a and, hence, f ( y ) ∈ f [ G a ] and g ( a ) = b < f ( y ) = g ( y ) .3. L a = ∅ and G a = ∅ . This case is dual of case 2.ackand forth systems of condensations 94. L a = ∅ and G a = ∅ . Then a k x , for all x ∈ dom f , and choosing b ∈ P \ ran f we have g := f ∪ {h a, b i} ∈ Π .(e2) Let f ∈ Π and b ∈ D \ ran f . Since | dom f | < κ by (b) there is a ∈ P \ dom f , such that a k x , for all x ∈ dom f . Thus g := f ∪ {h a, b i} ∈ Π and (e2) is true indeed.(d) Let P be a κ -homogeneous-universal poset (of regular size κ ). Thus, eachisomorphism between < κ -sized substructures of P extends to an automorphismof P and each poset of size ≤ κ embeds in P . By (c), for a proof that P is not areversible structure it is sufficient to show that P satisfies (u1) and (u2).Let L, G ∈ [ P ] <κ \ {∅} , where L < G , and let Y be a poset with domain Y = Y L ∪ { a } ∪ Y G , where Y L < { a } < Y G , Y L ∼ = L and Y G ∼ = G . Then | Y | < κ and, by the universality of P there is an embedding e : Y ֒ → P . Thus L ∼ = e [ Y L ] < e ( a ) < e [ Y B ] ∼ = G and there are isomorphisms f L : e [ Y L ] → L and f G : e [ Y G ] → G . Clearly, f := f L ∪ f G : e [ Y L ] ∪ e [ Y G ] → L ∪ G is anisomorphism between < κ -sized substructures of P and, by the homogeneity of P there is an automorphism F ∈ Aut( P ) such that f ⊂ F , which implies that L < x := F ( e ( a )) < G . So (u1) is true and (u2) has a similar proof. ✷ Theorem 4.1 If X and Y are infinite L -structures, then we have(I) The following conditions are equivalent:(a) X c Y , in some generic extension V [ G ] of the universe,(b) There is a b.f.s. Π ⊂ PC( X , Y ) ,(c) Player II has a winning strategy in the game G c ω ( X , Y ) ,(d) X ≪ P ∞ ω Y ,(e) Y ≪ N ∞ ω X .(f) X c Y , if, in addition, the structures X and Y are countable.(II) The following conditions are equivalent:(a) X ∼ c Y , in some generic extension V P [ G ] of the universe,(b) X ∼ bfs c Y ,(c) X ∼ G ω c Y ,(d) X ≡ P ∞ ω Y ,(e) X ≡ N ∞ ω Y ,(f) X ≡ P ∞ ω ∪N ∞ ω Y ,(g) X ≪ P ∞ ω ∪N ∞ ω Y ,(h) X ∼ c Y , if, in addition, the structures X and Y are countable. L ∞ ω -formula ϕ we adjoin a formula ϕ ¬ in the following way. First, ( v α = v β ) ¬ := ¬ v α = v β and ( R i ( v α , . . . , v α ni )) ¬ := ¬ R i ( v α , . . . , v α ni ) ; if ϕ ¬ is defined for a formula ϕ ∈ Form L ∞ ω , then ( ¬ ϕ ) ¬ := ϕ , ( ∀ v α ϕ ) ¬ := ∃ v α ϕ ¬ and ( ∃ v α ϕ ) ¬ := ∀ v α ϕ ¬ ; finally, if F ⊂
Form L ∞ ω and ϕ ¬ is defined for eachformula ϕ ∈ F , then ( V F ) ¬ := W F ¬ and ( W F ) ¬ := V F ¬ , where F ¬ denotesthe set { ϕ ¬ : ϕ ∈ F } . The following statement is easily provable by induction(see [14]). Fact 4.2
Let ϕ be an L ∞ ω -formula. Then(a) ϕ ¬ ↔ ¬ ϕ ;(b) If ϕ ∈ P ∞ ω , then ϕ ¬ ∈ N ∞ ω ;(c) If ϕ ∈ N ∞ ω , then ϕ ¬ ∈ P ∞ ω . Proof of (I) of Theorem 4.1. (a) ⇒ (b). Let P be a poset, G a P -generic filter over V , and let F ∈ V P [ G ] , where V P [ G ] | = F ∈ Cond( X , Y ) . Then F = τ G , for some P -name τ and there is p ∈ G such that p (cid:13) “ τ : ˇ X → ˇ Y is a condensation ” .We prove that Π := { f ∈ PC( X , Y ) : | f | < ω ∧ ∃ q ≤ p q (cid:13) ˇ f ⊂ τ } is ab.f.s. Let f ∈ Π , a ∈ X \ dom f and let q ≤ p , where q (cid:13) ˇ f ⊂ τ . Since q ≤ p and p (cid:13) τ ∈ Cond( ˇ X , ˇ Y ) we have q (cid:13) τ ∈ Cond( ˇ X , ˇ Y ) so, if H a P -generic filterover V and q ∈ H , in the extension V P [ H ] we have: τ H is a condensation from X to Y and f ⊂ τ H . Now, for b := τ H ( a ) we have f ⊂ g := f ∪ {h a, b i} ⊂ τ H and,hence, there is q ∈ H such that q (cid:13) ˇ g ⊂ τ . Since H is a filter and q, q ∈ H ,there is q ∈ H such that q ≤ q, q . Since q ≤ q we have q (cid:13) ˇ g ⊂ τ and q ≤ q implies that q ≤ p ; thus f ⊂ g ∈ Π , a ∈ dom g and (e1) is true. In asimilar way we prove that Π satisfies (e2).(b) ⇒ (c). Let Π ⊂ PC( X , Y ) be a b.f.s. Since the closure of a b.f.s. underrestrictions is a b.f.s. too, w.l.o.g. we suppose that Π is closed under restrictions.Let < X and < Y be well orders of the domains X and Y respectively and let us pick x ∗ ∈ X and y ∗ ∈ Y . Let Σ be the strategy for player II in the game G c ω ( X , Y ) defined as follows. Let n ∈ ω , let π n = hh x k , y k i : k < n i be the sequenceproduced in the first n moves and f n = {h x k , y k i : k < n } . Now,– if player I chooses x n ∈ X at the n -th step, then Σ( π n , x n ) = min < Y { y ∈ Y : f n ∪ {h x n , y i} ∈ Π } , if f n ∪ {h x n , y i} ∈ Π , for some y ∈ Y,y ∗ , otherwise;ackand forth systems of condensations 11– if player I chooses y n ∈ Y at the n -th step, then Σ( π n , y n ) = min < X { x ∈ X : f n ∪ {h x, y n i} ∈ Π } , if f n ∪ {h x, y n i} ∈ Π , for some x ∈ X,x ∗ , otherwise.Let π ω = hh x k , y k i : k < ω i be a play in which player II follows Σ . By inductionwe show that f n := {h x k , y k i : k < n } ∈ Π , for all n ∈ ω . First, f = ∅ ∈ Π .Suppose that f n ∈ Π . If, at the n -th step, player I have picked x n ∈ X , then by (e1)there is g ∈ Π such that f n ⊂ g and x n ∈ dom g ; so y := g ( x n ) ∈ Y and, sincethe set Π is closed under restrictions, f n ∪ {h x n , y i} = g ↾ (dom f n ∪ { x n } ) ∈ Π ,which implies that for y n = Σ( π n , x n ) we have f n +1 = f n ∪ {h x n , y n i} ∈ Π . Ifplayer I have picked y n ∈ Y , then using (e2) we show that f n +1 ∈ Π again.So, f n ∈ Π ⊂ PC( X , Y ) , for all n ∈ ω , and, in addition, f ⊂ f ⊂ . . . ,which implies that f = S n ∈ ω f n = {h x k , y k i : k < ω } ∈ PC( X , Y ) . Thus playerII wins the play and Σ is a winning strategy for player II in the game G c ω ( X , Y ) .(c) ⇒ (a). Let Σ be a winning strategy for player II in the game G c ω ( X , Y ) and let Π be the set of the ranges f π = {h x k , y k i : k < n } of all finite partialplays π = hh x k , y k i : k < n i , n ∈ ω , in which player II follows Σ . We showthat the sets D x := { g ∈ Π : x ∈ dom g } , x ∈ X , are dense in the poset P := h Π , ⊃i . So if x ∈ X and f π = {h x k , y k i : k < n } ∈ Π , then, regardinga play in which π = hh x k , y k i : k < n i is the sequence of the first n moves, x can be considered as the choice x n ∈ X of player I at the ( n + 1) -st move.Then player II takes y n = Σ( π, x n ) ; so we have g := f π ∪ {h x n , y n i} ∈ Π and x ∈ dom g , thus g ∈ D x and g ⊃ f π . In a similar way we prove that the sets ∆ y := { g ∈ Π : y ∈ ran g } , y ∈ Y , are dense in P . Now, if G is a P -genericfilter over V , then, since Π ⊂ PC( X , Y ) , in the generic extension V P [ G ] we have F := S G ∈ PC( X , Y ) . In addition, since G ∩ D x = ∅ , for all x ∈ X , we have dom F = X , and, similarly, ran F = Y ; thus F ∈ Cond( X , Y ) .(b) ⇒ (d). Let Π ⊂ PC( X , Y ) be a b.f.s. By induction on the construction of P ∞ ω -formulas we show that for each formula ψ ( v , . . . , v n − ) ∈ P ∞ ω we have ∀ f ∈ Π ∀ ¯ x ∈ (dom f ) n (cid:16) X | = ψ [¯ x ] ⇒ Y | = ψ [ f ¯ x ] (cid:17) . (4)Let ψ (¯ v ) ∈ P . If f ∈ Π , ¯ x ∈ (dom f ) n and X | = ψ [¯ x ] , then f is a condensationfrom a substructure A of X onto a substructure B of Y and ¯ x ∈ A n . Since X | = ψ [¯ x ] and ψ ∈ Σ we have A | = ψ [¯ x ] , which, since f is a condensation and ψ ∈ P ∞ ω implies B | = ψ [ f ¯ x ] , so, since ψ is a Σ -formula, Y | = ψ [ f ¯ x ] and (4) is true.Let ψ ′ (¯ v ) := V F and suppose that (4) is true for all ψ (¯ v ) ∈ F . If f ∈ Π , ¯ x ∈ (dom f ) n and X | = ψ ′ [¯ x ] , then for each ψ ∈ F we have X | = ψ [¯ x ] and, by(4), Y | = ψ [ f ¯ x ] , which implies Y | = ψ ′ [¯ y ] and we are done.2 Miloˇs S.Kurili´cLet ψ ′ (¯ v ) := W F and suppose that (4) is true for all ψ (¯ v ) ∈ F . If f ∈ Π , ¯ x ∈ (dom f ) n and X | = ψ ′ [¯ x ] , then for some ψ ∈ F we have X | = ψ [¯ x ] and, by(4), Y | = ψ [ f ¯ x ] , which implies that Y | = ψ ′ [¯ y ] .Let ψ ′ (¯ v ) := ∃ v n ψ (¯ v, v n ) and let (4) be true for ψ (¯ v, v n ) , that is ∀ g ∈ Π ∀ ¯ x ∈ (dom g ) n ∀ x ∈ dom g (cid:16) X | = ψ [¯ x, x ] ⇒ Y | = ψ [ g ¯ x, gx ] (cid:17) . (5)If f ∈ Π , ¯ x ∈ (dom f ) n and X | = ψ ′ [¯ x ] , then there is x ∈ X such that X | = ψ [¯ x, x ] , by (e1) there is g ∈ Π such that x ∈ dom g and f ⊂ g and, by (5) andsince g ¯ x = f ¯ x we have Y | = ψ [ f ¯ x, gx ] , which gives Y | = ψ ′ [ f ¯ x ] .Let ψ ′ (¯ v ) := ∀ v n ψ (¯ v, v n ) , and let (5) be true for ψ (¯ v, v n ) . If f ∈ Π , ¯ x ∈ (dom f ) n and X | = ψ ′ [¯ x ] , then for each x ∈ X we have X | = ψ [¯ x, x ] . For y ∈ Y by (e2) there is g ∈ Π such that y ∈ ran g and f ⊂ g ; thus y = g ( x ) , for some x ∈ dom g . By our assumption we have X | = ψ [¯ x, x ] and by (5), since g ¯ x = f ¯ x we have Y | = ψ [ f ¯ x, y ] . This gives Y | = ψ ′ [ f ¯ x ] .Now, by (4), if ψ ∈ Sent P ∞ ω and X | = ψ , then Y | = ψ . Thus X ≪ P ∞ ω Y .(d) ⇒ (b). Let X ≪ P ∞ ω Y . For n < ω , let P n be the set of pairs h ¯ x, ¯ y i ∈ X n × Y n such that ∀ ψ ( v , . . . , v n − ) ∈ P ∞ ω (cid:16) X | = ψ [¯ x ] ⇒ Y | = ψ [¯ y ] (cid:17) . (6)Then, since f ¯ x, ¯ y := {h x k , y k i : k < n } preserves all formulas of the form v k = v l , ¬ v k = v l and R i ( v k , . . . , v k ni − ) , we have Π := { f ¯ x, ¯ y : h ¯ x, ¯ y i ∈ S n<ω P n } ⊂ PC( X , Y ) . Since X ≪ P ∞ ω Y we have f ∅ , ∅ = ∅ ∈ Π .(e1) Let f ¯ x, ¯ y ∈ Π and a ∈ X \ { x , . . . , x n − } . We show that there is b ∈ Y such that ∀ ψ ( v , . . . , v n − , v n ) ∈ P ∞ ω (cid:16) X | = ψ [¯ x, a ] ⇒ Y | = ψ [¯ y, b ] (cid:17) . (7)On the contrary, there would be formulas ψ b ( v , . . . , v n − , v n ) , b ∈ Y , such that ∀ b ∈ Y X | = ψ b [¯ x, a ] (8) ∀ b ∈ Y Y | = ¬ ψ b [¯ y, b ] (9)Now ψ ( v , . . . , v n − ) := ∃ v n V b ∈ Y ψ b ( v , . . . , v n − , v n ) ∈ P ∞ ω and X | = ψ [¯ x ] iff there is a ∈ X such that X | = ψ b [¯ x, a ] , for all b ∈ Y , which is true by (8). So,by (6) we have Y | = ψ [¯ y ] and, hence, there is b ∗ ∈ Y such that Y | = ψ b [¯ y, b ∗ ] , forall b ∈ Y . In particular, Y | = ψ b ∗ [¯ y, b ∗ ] , which is false by (9).By (7) we have h ¯ xa, ¯ yb i ∈ P n +1 so f ¯ x, ¯ y ⊂ f ¯ xa, ¯ yb ∈ Π and a ∈ dom f ¯ xa, ¯ yb .ackand forth systems of condensations 13(e2) Let f ¯ x, ¯ y ∈ Π and b ∈ Y \ { y , . . . , y n − } . We show that there is a ∈ X such that (7) holds. On the contrary, there would be formulas ψ a ( v , . . . , v n − , v n ) , a ∈ X , such that ∀ a ∈ X X | = ψ a [¯ x, a ] (10) ∀ a ∈ X Y | = ¬ ψ a [¯ y, b ] (11)Now ψ ( v , . . . , v n − ) := ∀ v n W a ∈ X ψ a ( v , . . . , v n − , v n ) ∈ P ∞ ω and X | = ψ [¯ x ] iff for each x ∈ X there is a ∈ X such that X | = ψ a [¯ x, x ] , which is, by (10), truefor a = x . So X | = ψ [¯ x ] and by (6) we have Y | = ψ [¯ y ] . Thus, for each b ∗ ∈ Y and,in particular, for b , there is a ∈ X such that Y | = ψ a [¯ y, b ] , which is false by (11).So, there is a ∈ X such that (7) holds and we have h ¯ xa, ¯ yb i ∈ P n +1 so f ¯ x, ¯ y ⊂ f ¯ xa, ¯ yb ∈ Π and b ∈ ran f ¯ xa, ¯ yb .(d) ⇔ (e). Let X ≪ P ∞ ω Y , ϕ ∈ Sent N ∞ ω and Y | = ϕ . Assuming that X | = ¬ ϕ , by Fact4.2(a) and (c) we would have X | = ϕ ¬ and ϕ ¬ ∈ P ∞ ω ; so, since X ≪ P ∞ ω Y , Y | = ϕ ¬ that is Y | = ¬ ϕ , which gives a contradiction. So X | = ϕ and, thus, Y ≪ N ∞ ω X . The converse has a symmetric proof.The equivalence (b) ⇔ (f) for countable structures follows from part (I) ofTheorem 3.1 (the equivalence (a) ⇔ (b)), because each poset is ω -closed. ✷ Proof of (II) of Theorem 4.1.
The equivalence of conditions (b), (c), (d) and(e) follows from (I). If (d) holds, we have (e) as well and, hence (f) is true. Theimplication (f) ⇒ (g) is trivial. If (g) is true, then X ≪ P ∞ ω Y and X ≪ N ∞ ω Y ,which by (I) implies that Y ≪ P ∞ ω X . Thus X ≡ P ∞ ω Y and, thus (g) implies (d).If (a) holds, then V P [ G ] | = X c Y and V P [ G ] | = Y c X ; so by (I) there areb.f.s. Π X , Y ⊂ PC( X , Y ) and Π Y , X ⊂ PC( Y , X ) and (b) is true.If (b) is true, P := h Π X , Y , ⊃i , P := h Π Y , X , ⊃i , P = P × P and G is a P -generic filter over V , then (see [9], p. 253) the generic extension V P [ G ] containsa P -generic filter over V , G , and a P -generic filter over V , G , and, hence, V P [ G ] , V P [ G ] ⊂ V P [ G ] . Thus (see the proof of the implication (b) ⇒ (a) ofpart (I)) the extension V P [ G ] contains condensations F : X → Y and G : Y → X ,which means that V P [ G ] | = X ∼ c Y and (a) is true as well. ✷ Theorem 4.3
For a countable X ∈ Mod L the following conditions are equivalent:(a) X is weakly reversible,(b) X ≡ P ∞ ω Y implies that Y ∼ = X , for each countable Y ∈ Mod L ,(c) Th P ∞ ω ∪N ∞ ω ( X ) := { ϕ ∈ Sent P ∞ ω ∪N ∞ ω : X | = ϕ } is ω -categorical. Proof. (a) ⇔ (b) follows from Theorem 4.1 (part II, (d) ⇔ (h)). Th P ∞ ω ∪N ∞ ω ( X ) is an ω -categorical theory iff for each countable Y ∈ Mod L , X ≪ P ∞ ω ∪N ∞ ω Y X ≡ P ∞ ω Y , implies Y ∼ = X . So, (b) ⇔ (c) is true. ✷ In addition, by Theorem 4.1, condition X ≡ P ∞ ω Y in (b) can be replaced by X ≡ N ∞ ω Y , X ≡ P ∞ ω ∪N ∞ ω Y , or X ≪ P ∞ ω ∪N ∞ ω Y . Example 4.4
The theory Th P ∞ ω ∪N ∞ ω ( X ) is ω -categorical, but the first order the-ory of X , Th( X ) , is not. Let X be a countable structure with one equivalencerelation, such that there are no infinite equivalence classes and for each n ∈ N there is exactly one equivalence class of size n . It is known that Th( X ) is not ω -categorical. But X is a reversible and, hence, a weakly reversible structure (see[16] for a characterization of reversible equivalence relations). So, by Theorem4.3, the theory Th P ∞ ω ∪N ∞ ω ( X ) is ω -categorical. Here we consider the relations ≪ P and ≡ P , when L is a finite relational language.We note that, if L is an infinite language and X , Y ∈ Mod L , then X ≪ P Y iff X | L ′ ≪ P L ′ Y | L ′ , for each finite L ′ ⊂ L , and the same holds for the relation ≡ P .If X and Y are L -structures we will say that X is finitely condensable to Y iffthere is a sequence h Π r : r < ω i , where, for each r < ω , ∅ 6 = Π r ⊂ PC( X , Y ) and(f1) ∀ f ∈ Π r +1 ∀ x ∈ X ∃ g ∈ Π r ( x ∈ dom g ∧ f ⊂ g ) ,(f2) ∀ f ∈ Π r +1 ∀ y ∈ Y ∃ g ∈ Π r ( y ∈ ran g ∧ f ⊂ g ) .Then we will write X fin c Y . If, in addition, Y fin c X , we will write X ∼ fin c Y . Theorem 5.1 If L is a finite language and X and Y are L -structures, then we have(I) The following conditions are equivalent:(a) There is a sequence h Π r : r < ω i satisfying (f1) and (f2),(b) For each n ∈ ω , player II has a winning strategy for G c n ( X , Y ) ,(c) X ≪ P Y ,(d) Y ≪ N X .(II) The following conditions are equivalent:(a) X ∼ fin c Y ,(b) X ∼ G n c Y , for all n ∈ ω ,(c) X ≡ P Y ,(d) X ≡ N Y ,(e) X ≡ P∪N Y ,(f) X ≪ P∪N Y . ackand forth systems of condensations 15 Proof.
We prove part I, which gives the equivalence of (a) – (d) of part II. Theproof of the rest is similar to the proof of the corresponding part of Theorem 4.1.(a) ⇒ (c). Let X fin c Y and let h Π r : r < ω i be a sequence which wit-nesses it. By induction on the complexity of formulas we prove that for each ϕ ( v , . . . , v n − ) ∈ P we have ∀ r ≥ qr( ϕ ) ∀ f ∈ Π r ∀ ¯ x ∈ (dom f ) n (cid:16) X | = ϕ [¯ x ] ⇒ Y | = ϕ [ f ¯ x ] (cid:17) . (12)If ϕ ∈ P , then qr ϕ = 0 . So, if r < ω , f ∈ Π r , ¯ x ∈ (dom f ) n and X | = ϕ [¯ x ] , weshow that Y | = ϕ [ f ¯ x ] . Since f is an injection, this is true for all formulas of theform v i = v j and v i = v j . If ϕ := R i ( v j , . . . , v j ni − ) , where j , . . . , j n i − < n i ,then h x j , . . . , x j ni − i ∈ R X i and, by (1), h f ( x j ) , . . . , f ( x j ni − ) i ∈ R Y i which,since f ¯ x = h f ( x ) , . . . , f ( x n i − ) i , gives Y | = ϕ [ f ¯ x ] .Suppose that (12) holds for all formulas from P k and that ϕ ( v , . . . , v n − ) ∈P k +1 . Let r ≥ qr( ϕ ) , f ∈ Π r and ¯ x ∈ (dom f ) n .If ϕ = ψ ∨ ψ or ϕ = ψ ∧ ψ , then ψ , ψ ∈ P k , qr( ψ ) , qr( ψ ) ≤ qr( ϕ ) and, since (12) holds for ψ and ψ , we have X | = ψ [¯ x ] ⇒ Y | = ψ [ f ¯ x ] and X | = ψ [¯ x ] ⇒ Y | = ψ [ f ¯ x ] . Now, if ϕ = ψ ∨ ψ and X | = ϕ [¯ x ] , then X | = ψ j [¯ x ] , for some j ∈ , whichimplies Y | = ψ j [ f ¯ x ] ; and, hence, Y | = ϕ [ f ¯ x ] . If ϕ = ψ ∧ ψ and X | = ϕ [¯ x ] ,then X | = ψ [¯ x ] and X | = ψ [¯ x ] , which implies Y | = ψ [ f ¯ x ] and Y | = ψ [ f ¯ x ] and,hence, Y | = ϕ [ f ¯ x ] .If ϕ = ∃ v n ψ ( v , . . . , v n − , v n ) and X | = ϕ [¯ x ] , then there is x ∈ X such that X | = ψ [¯ x, x ] , by (f1) there is g ∈ Π r − such that x ∈ dom g and f ⊂ g and, since qr( ψ ) = qr( ϕ ) − ≤ r − , by the induction hypothesis for ψ , r − , g and ¯ x a x ,from X | = ψ [¯ x, x ] it follows that Y | = ψ [ f ¯ x, g ( x )] . So there is y = g ( x ) ∈ Y suchthat Y | = ψ [ f ¯ x, y ] , which means that Y | = ϕ [ f ¯ x ] .If ϕ = ∀ v n ψ ( v , . . . , v n − , v n ) and X | = ϕ [¯ x ] , then ∀ x ∈ X X | = ψ [¯ x, x ] . (13)If y ∈ Y , then by (f2) there is g ∈ Π r − such that y ∈ ran g and f ⊂ g . If x ∈ X ,where g ( x ) = y , then by (13) we have X | = ψ [¯ x, x ] . Since qr( ψ ) = qr( ϕ ) − ≤ r − , by the induction hypothesis for ψ , r − , g and ¯ x a x , from X | = ψ [¯ x, x ] itfollows that Y | = ψ [ g ¯ x, g ( x )] , that is Y | = ψ [ f ¯ x, y ] . So for each y ∈ Y we have Y | = ψ [ f ¯ x, y ] , which means that Y | = ϕ [ f ¯ x ] . So (12) is true.Now, if ϕ ∈ Sent P and qr( ϕ ) = r , then, since Π r = ∅ , there is f ∈ Π r and by(12) (for n = 0 ) X | = ϕ implies Y | = ϕ . Thus, X ≪ P Y .6 Miloˇs S.Kurili´c(c) ⇒ (a). Let X ≪ P Y and, for r < ω , let Π r be the set of f ∈ PC( X , Y ) such that there are n ∈ ω and an n -tuple ¯ x ∈ X n such that:(i r ) dom f = { x , . . . , x n − } ,(ii r ) ∀ ϕ ∈ P ( v , . . . , v n − ) (qr( ϕ ) ≤ r ∧ X | = ϕ [¯ x ] ⇒ Y | = ϕ [ f ¯ x ]) .First we show that ∅ ∈ I r , for all r < ω . (i r ) holds trivially and, since X ≪ P Y we have X | = ϕ ⇒ Y | = ϕ , for each ϕ ∈ Sent P (satisfying qr( ϕ ) ≤ r ); so (ii r )holds as well.In order to prove that the sequence h Π r : r ∈ ω i satisfies (f1) and (f2), wesuppose that f ∈ Π r +1 , n ∈ ω and ¯ x ∈ X n , where dom( f ) = { x , . . . , x n − } .Since | L | < ω , for each n, r < ω there are, up to logical equivalence, finitelymany L -formulas ϕ such that | Fv( ϕ ) | ≤ n and qr( ϕ ) ≤ r ; see []. So, there areformulas ψ , . . . , ψ m − ∈ P ( v , . . . , v n − , v n ) such that qr( ψ j ) ≤ r and ∀ ϕ ∈ P ( v , . . . , v n − , v n ) (cid:16) qr( ϕ ) ≤ r ⇒ ∃ j < m ( ϕ ↔ ψ j ) (cid:17) . (14)(f1) Let x ∈ X and J := { j < m : X | = ψ j [¯ x, x ] } . Then X | = V j ∈ J ψ j [¯ x, x ] and, hence, X | = ( ∃ v n V j ∈ J ψ j )[¯ x ] . Since qr( ∃ v n V j ∈ J ψ j ) ≤ r +1 , ∃ v n V j ∈ J ψ j ∈P and f ∈ Π r +1 , by (ii r +1 ) we have Y | = ( ∃ v n V j ∈ J ψ j )[ f ¯ x ] , so there is y ∈ Y such that Y | = V j ∈ J ψ j [ f ¯ x, y ] . Thus ∀ j ∈ J Y | = ψ j [ f ¯ x, y ] (15)and we prove ∀ ϕ ∈ P ( v , . . . , v n − , v n ) (cid:16) qr( ϕ ) ≤ r ∧ X | = ϕ [¯ x, x ] ⇒ Y | = ϕ [ f ¯ x, y ] (cid:17) . (16)If ϕ (¯ v, v n ) ∈ P , qr( ϕ ) ≤ r and X | = ϕ [¯ x, x ] then by (14) there is j < m such that ϕ ↔ ψ j and, hence, X | = ψ j [¯ x, x ] . So we have j ∈ J and, by (15), Y | = ψ j [ f ¯ x, y ] ,that is Y | = ϕ [ f ¯ x, y ] and (16) is proved.Let g = f ∪ {h x, y i} . By (16), for each ϕ (¯ v, v n ) ∈ P we have X | = ϕ [¯ x, x ] ⇒ Y | = ϕ [ f ¯ x, y ] and, by Fact 2.1, g ∈ PC( X , Y ) . Clearly f ⊂ g and x ∈ dom( g ) = { x , . . . , x n − , x } and by (16), g satisfies (ii r ) and, hence, g ∈ Π r .(f2) Let y ∈ Y and let us define J := { j < m : Y | = ¬ ψ j [ f ¯ x, y ] } and η (¯ v ) := ∀ v n W j ∈ J ψ j (¯ v, v n ) . Assuming that X | = η [¯ x ] , since η ∈ P , qr( η ) ≤ r +1 and f ∈ Π r +1 , by (ii r +1 ) we would have Y | = η [ f ¯ x ] and, hence, for each elementof Y and, in particular, for y , there would be j ∈ J such that Y | = ψ j [ f ¯ x, y ] whichis false by the definition of J . Thus X | = ¬ η [¯ x ] and, hence, there is x ∈ X suchthat ∀ j ∈ J X | = ¬ ψ j [¯ x, x ] . (17)ackand forth systems of condensations 17Again we prove (16). If ϕ (¯ v, v n ) ∈ P , qr( ϕ ) ≤ r and X | = ϕ [¯ x, x ] then by (14)there is j < m such that ϕ ↔ ψ j and, hence, X | = ψ j [¯ x, x ] . By (17) we have j J , which means that Y | = ψ j [ f ¯ x, y ] , that is Y | = ϕ [ f ¯ x, y ] and (16) is true.Let g = f ∪ {h x, y i} . By (16), for each ϕ (¯ v, v n ) ∈ P we have X | = ϕ [¯ x, x ] ⇒ Y | = ϕ [ f ¯ x, y ] and, by Fact 2.1, g ∈ PC( X , Y ) . Clearly f ⊂ g , y ∈ ran( g ) and by(16), g satisfies (ii r ) and, hence, g ∈ Π r .(a) ⇒ (b). Let X fin c Y and let h Π r : r < ω i be a witness for that. Defining Π ↾ r := { f ∈ PC( X , Y ) : ∃ g ∈ Π r f ⊂ g } , for r < ω , we show that the sequence h Π ↾ r : r < ω i satisfies (f1) and (f2). If f ∈ Π ↾ r +1 and g ∈ Π r +1 where f ⊂ g ,then, by (f1), for x ∈ X there is g ′ ∈ Π r such that g ⊂ g ′ and x ∈ dom g ′ . So f ⊂ f ′ := f ∪ {h x, g ′ ( x ) i} ⊂ g ′ ∈ Π r , which gives f ′ ∈ Π ↾ r . So (f1) is true andthe proof of (f2) is similar. Let < X and < Y be well orderings of the sets X and Y .By induction we prove that for each n, r < ω there is a strategy Σ n,r for playerII in the game G c n ( X , Y ) such that for each play π n = hh x k , y k i : k < n i we have f n := {h x k , y k i : k < n } ∈ Π ↾ r . For n = 0 this is true, since ∅ ∈ Π ↾ r , for all r < ω .Assuming that the statement is true for n and for all r < ω we define Σ n +1 ,r as follows. In the first n moves player II follows Σ n,r +1 and obtains a partial play π n = hh x k , y k i : k < n i satisfying f n := {h x k , y k i : k < n } ∈ Π ↾ r +1 . Then- If I chooses x n ∈ X , then by (f1) there is g ∈ Π ↾ r such that f n ⊂ g and x n ∈ dom g ; thus g ( x n ) ∈ { y ∈ Y : f n ∪ {h x n , y i} ∈ Π ↾ r } and Σ n +1 ,r suggests y n = Σ n +1 ,r ( π n , x n ) = min < Y { y ∈ Y : f n ∪ {h x n , y i} ∈ Π ↾ r } ; - If I chooses y n ∈ Y , then by (f2) there is g ∈ Π ↾ r such that f n ⊂ g and y n ∈ ran g ; thus g − ( y n ) ∈ { x ∈ X : f n ∪ {h x, y n i} ∈ Π ↾ r } and Σ n +1 ,r suggests x n = Σ n +1 ,r ( π n , y n ) = min < X { x ∈ X : f n ∪ {h x, y n i} ∈ Π ↾ r } . In both cases we have f n +1 := {h x k , y k i : k < n + 1 } ∈ Π ↾ r so our claim istrue and (b) is proved.(b) ⇒ (a). Suppose that for each n ∈ ω player II has a winning strategy Σ n for G c n ( X , Y ) . For r ∈ ω , let Π r be the set of all f such that there are j < ω , ¯ x ∈ X j and ¯ y ∈ Y j such that:(i) f = {h x k , y k i : k < j } ∈ PC( X , Y ) ,(ii) There exist m ≥ r and a play π j + m = hh x k , y k i : k < j + m i of the game G c j + m ( X , Y ) in which II follows Σ j + m .(f1) Let f = {h x k , y k i : k < j } ∈ Π r +1 and x ∈ X . Let m ≥ r + 1 and let π j + m = hh x k , y k i : k < j + m i be a play of the game G c j + m ( X , Y ) in which IIfollows Σ j + m . Then m > and in the play π ′ j + m in which player I in the first j moves plays in the same way as in the play π j + m and in the moves j + 1 , . . . , j + m plays the given x , player II following Σ j + m in the moves j + 1 , . . . , j + m playssome y ∈ Y . So π ′ j + m = hh x , y i , . . . , h x j − , y j − i , h x, y i , . . . , h x, y ii , g = f ∪ {h x, y i} ∈ Π r and x ∈ dom g thus (f1) is true. The proof of (f2) is similar.8 Miloˇs S.Kurili´c(c) ⇔ (d). The proof is similar to the proof of (d) ⇔ (e) in Theorem 4.1. ✷ In this section we compare the similarities of structures considered in this paperwith the similarities ∼ = , ≡ ∞ ω and ≡ and show that the situation is as Figure 2describes. We note that we construct pairs of structures of the same size and,moreover, a pair of countable structures with the given properties, whenever it ispossible. Namely, the pairs witnessing B and F must be uncountable, because forcountable structures X and Y the conditions X ≡ ∞ ω Y , X ≡ ω ω Y and X ∼ = Y areequivalent, by Scott’s theorem. The same holds for the pairs witnessing G and H,since for countable X and Y we have X ≡ P ∞ ω Y iff X ∼ c Y ; see Theorem 4.1. A ∼ = ∼ c C D ≡ E F ≡ ∞ ω G HI ≡ P ∞ ω ≡ P B Figure 2: Pairs of structuresLet C be the class of structures X = h X, ρ i , where | X | = ω , ρ is an equivalencerelation on the set X and, if X/ρ is the corresponding partition of X ,( C X/ρ contains infinitely many singletons,ackand forth systems of condensations 19( C
2) For each n ∈ ω there is an equivalence class P ∈ X/ρ such that | P | ≥ n .Defining C fin = { X ∈ C : X/ρ ⊂ [ X ] <ω } and C ω = C \ C fin we obtain a partition {C fin , C ω } of the class C . Claim 6.1
Let X , Y ∈ C . Then(a) X ≡ P Y ;(b) X c Y ⇔ X ∈ C fin ∨ Y ∈ C ω ;(c) X ∼ c Y ⇔ X , Y ∈ C fin ∨ X , Y ∈ C ω (that is, C / ∼ c = {C fin , C ω } ). Proof. (a) Let X = h X, ρ i and Y = h Y, σ i . For n ∈ ω we construct a strategy Σ forplayer II in the game G c n ( X , Y ) . By ( C
1) we have X ′ := S (( X/ρ ) ∩ [ X ] ) ∈ [ ω ] ω ,let Y ′ ∈ Y /σ , where | Y ′ | ≥ n , and let h X ′ , < X ′ i and h Y ′ , < Y ′ i be well orders.Let l < n , let π l = hh x k , y k i : k < l i be the sequence of the first l moves and f l = {h x k , y k i : k < l } . If in the l + 1 -st move player I picks1. x l = x k , for some k < l , then y l = Σ( π l , x l ) = y k ;2. y l = y k , for some k < l , then x l = Σ( π l , y l ) = x k ;3. x l ∈ X \ dom f l , then y l = Σ( π l , x l ) = min < Y ′ ( Y ′ \ ran f l ) ;4. y l ∈ Y \ ran f l , then x l = Σ( π l , y l ) = min < X ′ ( X ′ \ dom f l ) .Let π n = hh x k , y k i : k < n i be a play of the game in which player II follows Σ . Since in cases 3 and 4 we have x l dom ϕ l and y l ran ϕ l , case 1 ensuresthat f n = {h x k , y k i : k < n } is a function and case 2 ensures that f n is aninjection. Let h x k , x k ′ i ∈ ρ , where k, k ′ < n . If x k = x k ′ , then y k = y k ′ and, hence, h f n ( x k ) , f n ( x k ′ ) i ∈ σ . If x k = x k ′ , then, since h x k , x k ′ i ∈ ρ , wehave x k , x k ′ X ′ , which means that, at the step when the pairs h x k , y k i and h x k ′ , y k ′ i were chosen for the first time we had case 3; so y k , y k ′ ∈ Y ′ and, hence, h f n ( x k ) , f n ( x k ′ ) i = h y k , y k ′ i ∈ σ . Thus f n ∈ PC( X , Y ) and player II wins.(b) ( ⇒ ) On the contrary, suppose that X c Y , X ′ ∈ ( X/ρ ) ∩ [ X ] ω and Y /σ ⊂ [ Y ] <ω . Then there would be f ∈ Cond( X , Y ) and, hence, f [ X ′ ] ⊂ Y ′ , forsome Y ′ ∈ Y /σ , which is impossible.( ⇐ ) Let X/ρ = {{ s n } : n < ω } ∪ { X i : i ∈ ω } , where | X i | > , for i < ω .Then by ( C
2) we have | S i<ω X i | = ω .First, if there exists Y ′ ∈ ( Y /σ ) ∩ [ Y ] ω and f : X → Y is a bijection satisfying f [ S i<ω X i ] = Y ′ and f [ { s n : n < ω } ] = Y \ Y ′ , then f ∈ Cond( X , Y ) .Second, suppose that | X i | < ω , for all i < ω and that Y /σ ⊂ [ Y ] <ω . Let Y /σ = {{ t n } : n < ω } ∪ { Y j : j ∈ ω } , where < | Y j | < ω , for j < ω . Byrecursion we define a sequence h j i : i < ω i such that for each i < ω we have:(i) | X i | ≤ | Y j i | ,(ii) i ′ < i ⇒ j i ′ = j i .First, by ( C
2) there is j < ω such that | X | ≤ | Y j | . Let i < ω and supposethat h j i ′ : i ′ < i i is a sequence satisfying (i) and (ii). By ( C
2) we can choose0 Miloˇs S.Kurili´c(minimal) j i < ω such that | Y j i | > max {| X i | , | Y j | , . . . , | Y j i − |} and the sequence h j i ′ : i ′ ≤ i i satisfies (i) and (ii). The recursion works.By (i) there are injections f i : X i → Y j i , for i < ω , and, since by (ii) the sets Y j i , i < ω , are pairwise disjoint, f := S i<ω f i is an injection which maps the set X \ { s n : n ∈ ω } onto a subset Y ′ of S i<ω Y j i , which means that | Y \ Y ′ | = ω . Sothere is a bijection g : { s n : n ∈ ω } → Y \ Y ′ and, clearly, f ∪ g ∈ Cond( X , Y ) .(c) By (b), if X , Y ∈ C fin or X , Y ∈ C ω , then we have X ∼ c Y . On the otherhand, if X ∈ C ω and Y ∈ C fin , then, by (b) again, X c Y and, hence, X c Y ✷ Fact 6.2 If X is a countable equivalence relation, there is a first-order theory T fin X determining the number ( ≤ ω ) of k -sized equivalence classes, for each k ∈ N .If, in addition, for each n ∈ N there is a finite equivalence class of size ≥ n ,then the theory T fin X is complete but not ω -categorical. The following examples show that the situation is as Figure 2 describes.
Example 6.3
A: Of course here we can take X ∼ = X , for any structure X .B: X ∼ c Y , X ≡ ∞ ω Y , and X = Y . Let { A, B } ∪ { C α : α < ω } be a partitionof ω , where A, B ∈ [ ω ] ω and C α ∈ [ ω ] ω , for α < ω , and let X = h ω , ρ i and Y = h ω , σ i , where ρ and σ are the equivalence relations on ω corresponding tothe partitions { A ∪ B }∪{ C α : α < ω } and { A, B }∪{ C α : α < ω } , respectively.It is easy to see that X ∼ c Y and X = Y (see Example 3.13 from [13]). Inaddition if V [ G ] is a generic extension of the universe by the collapsing algebra Col( ω , ω ) , then, in V [ G ] , X and Y are countable structures with one equivalencerelation, having ω -many equivalence classes and all of them are infinite. Thus V [ G ] | = X ∼ = Y and, hence, X ≡ ∞ ω Y .E: X ≡ Y and X P ∞ ω Y . Let X and Y be linear orders, where X ∼ = ω and Y ∼ = ω + ζ ( ζ = ω ∗ + ω is the order type of the integers). It is well known that X ≡ Y . Assuming that X ≡ P ∞ ω Y , by Theorem 4.1 we would have X ∼ c Y and,since all linear orders are reversible and, hence, weakly reversible, X ∼ = Y , whichis false.D: X ∼ c Y and X Y . Let X = h X, ρ i be the Rado graph (see [2]), let x, y ∈ X , where x ρ y , that is, h x, y i , h y, x i ∈ ρ and let Y = h X, σ i , where σ = ρ \ {h x, y i} . Then, since the relation σ is not symmetric, we have X Y . Inaddition, the structure X = h X, ρ i , where ρ = σ \ {h y, x i} is a graph obtainedfrom X by deleting an edge and X ∼ = X (see [2]). Now we have ρ ∼ = ρ ⊂ σ ⊂ ρ ,which implies that X c Y c X and, thus, X ∼ c Y .I: X ≡ P Y , X P ∞ ω Y and X Y . Let X ∈ C fin and Y ∈ C ω (see thebeginning of this section), where X/ρ = {{ s n } : n < ω } ∪ { X i : i ≥ } and | X i | = i , for i ≥ , and Y /σ = {{ t n } : n < ω } ∪ { Y ′ } , where | Y ′ | = ω . Byackand forth systems of condensations 21Claim 6.1(a) we have X ≡ P Y and by (c) we have X c Y , which, together withTheorem 4.1 gives X P ∞ ω Y . For example we have X | = ¬ ϕ and Y | = ϕ wherethe sentence ϕ ∈ P ∞ ω says that there is an infinite equivalence class: ϕ := ∃ v V n ∈ N ∃ v , . . . , v n − (cid:16) V i This research was supported by the Ministry of Education andScience of the Republic of Serbia (Project 174006). References [1] J. Barwise, Back and forth through infinitary logic, Studies in model theory, pp. 5–34. MAAStudies in Math., Vol. 8, Math. Assoc. Amer., Buffalo, N.Y., 1973.[2] P. J. Cameron, The random graph, The mathematics of Paul Erd¨os II, Algorithms Combin. 14(Springer, Berlin, 1997) 333–351.[3] A. Ehrenfeucht, An application of games to the completeness problem for formalized theories,Fund. Math., 49 (1961) 129–141.[4] R. Fra¨ıss´e, Sur quelques classifications des syst`emes de relations, Publ. Sci. Univ. Alger. S´er.A. 1 (1954) 35–182.[5] R. Fra¨ıss´e, Sur quelques classifications des relations, bas´ees sur des isomorphismes restreints.II. Application aux relations d’ordre, et construction d’exemples montrant que ces classifica-tions sont distinctes, Publ. Sci. Univ. Alger. S´er. A. 2 (1955) 273–295.[6] B. J´onsson, Homogeneous universal relational systems, Math. Scand. 8 (1960) 137–142.[7] C. R. Karp, Finite-quantifier equivalence, 1965 Theory of Models (Proc. 1963 Internat. Sym-pos. Berkeley) pp. 407–412 North-Holland, Amsterdam.[8] M. Kukiela, Reversible and bijectively related posets, Order 26,2 (2009) 119–124.[9] K. Kunen, Set theory. An introduction to independence proofs, Studies in Logic and the Foun-dations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York, 1980.[10] M. S. Kurili´c, Condensations of structures of arbitrary languages, (to appear).[11] M. S. Kurili´c, Retractions of reversible structures, J. Symb. Log. 82,4 (2017) 1422–1437.[12] M. S. Kurili´c, Reversibility of definable relations, (to appear).[13] M. S. Kurili´c, N. Moraˇca, Condensational equivalence, equimorphism, elementary equivalenceand similar similarities, Ann. Pure Appl. Logic 168,6 (2017) 1210-1223.[14] M. S. Kurili´c, N. Moraˇca, Reversibility of extreme relational structures, Arch. Math. Logic, (toappear). http://arxiv.org/abs/1803.09619[15] M. S. Kurili´c, N. Moraˇca, Reversible disjoint unions of well orders and their inverses, Order(to appear). http://arxiv.org/abs/1711.07053[16] M. S. Kurili´c, N. Moraˇca, Reversible sequences of cardinals, reversible equivalence relations,and similar structures, (to appear). https://arxiv.org/abs/1709.09492[17] M. Nadel, J. Stavi, L ∞ λλ