Condensational equivalence, equimorphism, elementary equivalence and similar similarities

Accepted Manuscript Condensational equivalence, equimorphism, elementary equivalence and similar similarities

Miloš S. Kurili´c, Nenad Moraˇca

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S0168-0072(16)30178-6 http://dx.doi.org/10.1016/j.apal.2016.12.001 APAL 2560

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Annals of Pure and Applied Logic

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13 August 2015 11 August 2016 7 December 2016

Please cite this article in press as: M.S. Kurili´c, N. Moraˇca, Condensational equivalence, equimorphism, elementary equivalence and similar similarities, Ann. Pure Appl. Logic (2016), http://dx.doi.org/10.1016/j.apal.2016.12.001

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CONDENSATIONAL EQUIVALENCE, EQUIMORPHISM, ELEMENTARY EQUIVALENCE AND SIMILAR SIMILARITIES Miloˇs S. Kurili´c and Nenad Moraˇca Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbia. e-mail: [email protected], [email protected] Abstract We investigate the interplay between several similarities of relational structures: the condensational equivalence (defined by X ∼c Y iff there are bijective homomorphisms f : X → Y and g : Y → X), the isomorphism, the equimorphism (bi-embedability), the elementary equivalence and the similarities of structures determined by some similarities of their self-embedding monoids. It turns out that the Hasse diagram describing the hierarchy of these equivalence relations restricted to the set ModL (κ) of all L-structures of size κ collapses significantly for a finite cardinal κ or for a unary language L, while for infinite structures of non-unary languages we have a large diversity. 2010 Mathematics Subject Classification: 03C07, 03E40, 20M20, Key words and phrases: relational structure, condensational equivalence, isomorphism, equimorphism, elementary equivalence.

1

Introduction

Let L = Ri : i ∈ I be a relational language, where ar(Ri ) = ni ∈ N, for i ∈ I, and c the pre-order on the class of all L-structures, ModL , defined by: X c Y iff there is a bijective homomorphism (condensation) f : X → Y. Defining X ∼c Y iff X c Y and Y c X we obtain an equivalence relation on the class ModL and call it the condensational equivalence. (We note that in topology continuous bijections are sometimes called condensations, while condensationally equivalent spaces are called bijectively related, see [4], [15], and the same term is used for first-order structures as well, see [7]). The aim of the paper is to investigate the interplay between the condensational equivalence and some other similarities of relational structures: the isomorphism, X ∼ = Y; equimorphism (bi-embedability), X  Y; elementary equivalence, X ≡ Y; and the similarities of structures determined by some similarities of their self-embedding monoids considered in [13]. Namely, if X is a relational structure, Emb(X) the monoid of its self-embeddings and P(X) = {f [X] : f ∈ Emb(X)} the set of copies of X inside X, then the poset P(X), ⊂ (isomorphic to the inverse 1

Miloˇs S. Kurili´c and Nenad Moraˇca

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of the right Green’s order on Emb(X)1 ) contains a certain information about X and the equality P(X) = P(Y) defines an equivalence relation on the class of all relational structures. Some coarser classifications of structures are determined by the following weaker conditions2 : P(X) ∼ = P(Y) (implied by Emb(X) ∼ = Emb(Y)), sq P(X) ∼ = sq P(Y) and P(X) ≡f orc P(Y) (the forcing equivalence of posets of copies, which is equivalent to ro sq P(X) ∼ = ro sq P(Y), see [13]). Concerning the last similarity relation we note that the forcing related properties of posets of copies were investigated for countable structures in general in [8], for equivalence relations and similar structures in [9], for ordinals in [11], for scattered and non-scattered linear orders in [10] and [14], and for several ultrahomogeneous structures in [12] and [14]. The main results of the paper are presented in Figures 1, 2, 3 and 4 and Figure 1 describes the hierarchy between several classifications of structures in general (for each language L and each domain X). So the equality gives the finest (and trivial) classification and it implies the isomorphism, which is finer than the elementary equivalence, equimorphism and condensational equivalence, which are, in general, unrelated (see Figure 2). In addition these diagrams can be regarded in the following way. If we fix an interpretation ρ of the language L with the domain X, then the diagram in Figure 1 describes the inclusions between the equivalence classes of ρ with respect to 14 similarity relations considered in the paper. For example, if L = R is the language with one binary relational symbol, X = ω and ρ ∈ IntL (ω) a linear order on ω such that the structure X = ω, ρ is isomorphic to the rational line Q = Q, <, then, by the general diagram, [ρ]= ⊂ [ρ]∼ = ⊂ [ρ]∼c ∩ [ρ] ∩ [ρ]≡ ∩ [ρ]≡f orc . But [ρ]∼ = = [ρ]≡ (since the theory Th(Q) is ω-categorical), [ρ]∼ = = [ρ]∼c (since each linear order L is a reversible structure, that is Cond(L) = Aut(L)), [ρ] is the set of all non-scattered linear orders on ω and, under CH, [ρ]≡f orc is the set of all structures on ω having the poset of copies forcing equivalent to the two-step iteration S ∗ π, where S is the Sacks prefect tree forcing and π the S-name for the poset (P (ω)/ Fin)+ in the Sacks extension (see [14]). On the other hand, if ρ = ∅, then, clearly, [ρ]= = [ρ]∼ = = [ρ]∼c = [ρ] = [ρ]≡ = {∅} and [ρ]≡f orc is the set of all structures on ω having the poset of copies forcing equivalent to the poset (P (ω)/ Fin)+ , which under CH includes all scattered linear orders and equivaThe right Green’s pre-order R on the monoid Emb(X), ◦, idX  is defined by: f R g iff f ◦ h = g, for some h ∈ Emb(X). The right Green’s equivalence relation ≈R on Emb(X), given by: f ≈R g iff f R g and g R f , determines the antisymmetric quotient Emb(X)/ ≈R , ≤R , the right Green’s order. It is easy to check that Emb(X)/≈R , ≤R  ∼ = P(X), ⊃. 2 We will abuse notation, writing P(X) instead of P(X), ⊂, whenever the context admits it. By sq P we denote the separative quotient of a poset P and by ro sq P its Boolean completion. P ≡f orc Q denotes that the posets P and Q produce the same generic extensions by forcing (see [6]). 1

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lence relations on ω (see [10] and [9]). These phenomena suggest several natural questions; for example, for each triple of the form ρ, ∼ , ∼ , where [ρ]∼  [ρ]∼ we can ask what is the size of the quotient [ρ]∼ / ∼ (of course the implication |[ρ]≡ / ∼ = | > ω ⇒ |[ρ]≡ / ∼ = | = c is the Vaught Conjecture for ρ). The paper is organized in the following way. In Section 2 we introduce notation and prove necessary technical statements. Section 3 contains an analysis of the hierarchy of the similarities between infinite structures of non-unary3 languages, where the main technical part is Theorem 3.10, which, for arbitrary non-unary language, L, establishes a correspondence between the binary structures of size λ and L-structures of size κ ≥ λ, such that all the similarities considered in the paper are preserved (in both directions). In Section 4 we consider finite structures and structures of unary languages. Terminilogy and notation from model theory is mainly standard and follows [5]. The facts from set theory used in the paper can be found in [6].

2

Preliminaries

Since X c Y (and, hence, X ∼c Y), whenever the structures X and Y are of different size, without loss of information we consider the restrictions of the relation ∼c to the sets of the form ModL (X) = {X ∈ ModL : dom(X) = X}, where X is a set (or, workingin ZFC, to the sets ModL (κ), where κ is a cardinal). Equivalently, if IntL (X) = i∈I P (X ni ) is the complete Boolean algebra of all interpretations of the language L over the domain X, then (slightly abusing notation) for elements ρ = ρi : i ∈ I and σ = σi : i ∈ I of IntL (X) we will write ρ c σ (resp. ρ ∼c σ; ρ ∼ = σ; ρ ⊂ σ) iff X, ρ c X, σ (resp. X, ρ ∼c X, σ; X, ρ ∼ = X, σ; ρi ⊂ σi , for all i ∈ I) and explore these relations on IntL (X). For convenience we introduce notation ρc := X ni \ ρi : i ∈ I. Morphisms For a function f : X → Y and an integer n ≥ 2, the mapping f n : X n → Y n , defined by f n (x1 , . . . , xn ) = f (x1 ), . . . , f (xn ), for each x1 , . . . , xn  ∈ X n , is a bijection (surjection, injection) iff f has the same property. For interpretations ρ = ρi : i ∈ I ∈ IntL (X) and σ = σi : i ∈ I ∈ IntL (Y ), let the interpretations f [ρ] ∈ IntL (Y ) and f −1 [σ] ∈ IntL (X) be defined by: f [ρ] := f ni [ρi ] : i ∈ I and f −1 [σ] := (f ni )−1 [σi ] : i ∈ I.

(1)

If X and Y are L-structures, Hom(X, Y), SHom(X, Y), Emb(X, Y), Cond(X, Y) and Iso(X, Y) will denote the sets of all homomorphisms, strong homomorphisms, 3

A language L = Ri : i ∈ I is called unary iff ar(Ri ) = 1, for all i ∈ I.

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Miloˇs S. Kurili´c and Nenad Moraˇca

embeddings, condensations, and isomorphisms f : X → Y , respectively. The monoid Emb(X, X) will be denoted shortly by Emb(X), Hom(X, X) by End(X), etc. Working with elements of IntL (X), instead of Hom(X, ρ, X, σ) we will write Hom(ρ, σ) and similarly for the other sets of morphisms. Fact 2.1 For each ρ = ρi : i ∈ I, σ = σi : i ∈ I ∈ IntL (X) we have: (a) SHom(ρ, σ) = {f ∈ X X : ρ = f −1 [σ]}; (b) Hom(ρ, σ) = {f ∈ X X : f [ρ] ⊂ σ} = {f ∈ X X : ρ ⊂ f −1 [σ]}; (c) Iso(ρ, σ) = {f ∈ Sym(X) : f [ρ] = σ}; ∼ (d) [ρ]∼ = := {σ ∈ IntL (X) : σ = ρ} = {f [ρ] : f ∈ Sym(X)}; (e) Cond(ρ, σ) = {f ∈ Sym(X) : ρ ⊂ f −1 [σ]} = {f : f −1 ∈ Cond(σ c , ρc )}; (f) End(ρ) = {idX } ⇒ Cond(ρc ) = {idX }. Proof. (a) f ∈ SHom(ρ, σ) iff for each i ∈ I we have ∀x1 , . . . , xni  ∈ X ni (x1 , . . . , xni  ∈ ρi ⇔ f (x1 ), . . . , f (xni ) ∈ σi ).

(2)

But for arbitrary x1 , . . . , xni  ∈ X ni we have: f (x1 ), . . . , f (xni ) ∈ σi iff f ni (x1 , . . . , xni ) ∈ σi iff x1 , . . . , xni  ∈ (f ni )−1 [σi ] and, hence, condition (2) is equivalent to the equality ρi = (f ni )−1 [σi ]. Thus f ∈ SHom(ρ, σ) iff ρi = (f ni )−1 [σi ], for each i ∈ I, that is, iff ρ = f −1 [σ]. (b) Working like in the proof of (a) we show that f ∈ Hom(ρ, σ) iff ρ ⊂ f −1 [σ]. Clearly the conditions f [ρ] ⊂ σ and ρ ⊂ f −1 [σ] are equivalent. (c) If f ∈ Iso(ρ, σ), then f ∈ Sym(X) ∩ SHom(ρ, σ) so, by (a), ρ = f −1 [σ] and, hence, f [ρ] = f [f −1 [σ]] = σ. Conversely, if f ∈ Sym(X) and f [ρ] = σ, then, clearly, ρ = f −1 [f [ρ]] = f −1 [σ] and, by (a), f is a strong homomorphism. So f ∈ Iso(ρ, σ). (d) If σ ∈ [ρ]∼ = and f ∈ Iso(ρ, σ), then, by (c), σ = f [ρ]. Conversely, if f ∈ Sym(X), then, by (c), f ∈ Iso(ρ, f [ρ]), which implies f [ρ] ∼ = ρ, that is . f [ρ] ∈ [ρ]∼ = (e) The first equality follows from (b). If f ∈ Sym(X) then, clearly, f −1 [σ]c = f −1 [σ c ] and by (b) we have: f ∈ Cond(ρ, σ) iff ρ ⊂ f −1 [σ] iff f −1 [σ]c ⊂ ρc iff f −1 [σ c ] ⊂ ρc , that is, by (b) again, f −1 ∈ Cond(σ c , ρc ). (f) If f ∈ Cond(ρc ), then, by (e), f −1 ∈ Cond(ρ) ⊂ End(ρ) = {idX } and, hence, f = idX . 2 Condensational pre-order and equivalence The following statement contains the facts concerning the relations c and ∼c which will be used in the paper. Proposition 2.2 If L is a relational language, then for ρ, σ ∈ IntL (X) we have (a) ρ c σ ⇔ ∃ρ ∈ IntL (X) ρ ∼ = ρ ⊂ σ;

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(b) ρ ⊂ σ ⇒ ρ c σ; (c) ρ ∼ = σ ⇒ ρ ∼c σ; consequently, [ρ]∼ = ⊂ [ρ]∼c ; (d) If |ρi | < ω, for all i ∈ I, then we have [ρ]∼ = = [ρ]∼c ; c c c (e) ρ c σ ⇔ σ c ρ and ρ ∼c σ ⇔ ρ ∼c σ c . Proof. (a) If f ∈ Cond(ρ, σ), then, by Fact 2.1(d) and (b), ρ ∼ = f [ρ] ⊂ σ. Conversely, if ρ ∼ = ρ ⊂ σ, then, by Fact 2.1(d), there is f ∈ Sym(X) such that ρ = f [ρ] and by Fact 2.1(b) we have f ∈ Hom(ρ, σ). (b) If ρ ⊂ σ, then ρ ∼ = ρ ⊂ σ so, by (a), ρ c σ. ∼ ∼ (c) If ρ = σ, then ρ = σ ⊂ σ ∼ = ρ ⊂ ρ; so ρ c σ and σ c ρ, that is, ρ ∼c σ. (d) If |ρi | < ω, for all i ∈ I, and ρ ∼c σ, where σ ∈ IntL (X), then, by (a) and Fact 2.1(d), there are bijections f, g ∈ Sym(X) such that f [ρ] ⊂ σ and g[σ] ⊂ ρ. Thus for each i ∈ I we have f ni [ρi ] ⊂ σi and g ni [σi ] ⊂ ρi which implies |ρi | = |f ni [ρi ]| ≤ |σi | = |g ni [σi ]| ≤ |ρi |, so, since f ni [ρi ] ⊂ σi and these sets are finite and of the same size we have f ni [ρi ] = σi . Thus f [ρ] = σ and, by Fact 2.1(d), ρ ∼ = σ. (e) ρ c σ iff there is f ∈ Cond(ρ, σ) which is, by Fact 2.1(e) equivalent to f −1 ∈ Cond(σ c , ρc ), that is σ c c ρc . The second equivalence follows from the first one. 2 Condensations of disconnected binary structures The following concept from [8] will be used in the sequel. Let Lb = R, where ar(R) = 2, be the binary language. If X = X, ρ is an Lb -structure, then the transitive closure ρrst of the relation ρrs = ΔX ∪ ρ ∪ ρ−1 (given by x ρrst y iff there are n ∈ N and z0 = x, z1 , . . . , zn = y such that zi ρrs zi+1 , for each i < n) is the minimal equivalence relation on X containing ρ. The corresponding equivalence classes, [x], x ∈ X, are called the components of X and the structure X is called connected iff |X/ρrst | = 1. If Xi = Xi , ρi , i ∈ I, are connected Lb -structures and    Xi ∩ Xj = ∅, for different i, j ∈ I, then the structure i∈I Xi =  i∈I Xi , i∈I ρi  is the disjoint union of the structures Xi , i ∈ I, and the structures Xi , i ∈ I, are its components. Lemma 2.3 Let X, ρ and Y, τ  be binary structures and f : X → Y a homomorphism. Then for each x1 , x2 , x ∈ X (a) x1 ρrs x2 ⇒ f (x1 )τrs f (x2 ); (b) x1 ρrst x2 ⇒ f (x1 )τrst f (x2 ); (c) f [[x]] ⊂ [f (x)]; (d) f  [x] : [x] → [f (x)] is a homomorphism; (e) If f is a condensation, then f | [x] : [x] → f [[x]] is a condensation.

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Miloˇs S. Kurili´c and Nenad Moraˇca

Proof. (a) If x1 ρrs x2 , that is x1 = x2 ∨ x1 ρx2 ∨ x2 ρx1 , then, since f is a homomorphism, f (x1 ) = f (x2 ) ∨ f (x1 )τ f (x2 ) ∨ f (x2 )τ f (x1 ), that is f (x1 )τrs f (x2 ). (b) If x1 ρrst x2 , there are z0 , z1 , . . . , zn ∈ X such that x1 = z0 ρrs z1 ρrs . . . ρrs zn = x2 and, by (a), f (x1 ) = f (z0 )τrs f (z1 )τrs . . . τrs f (zn ) = f (x2 ) and, hence, f (x1 )τrst f (x2 ). (c) If x ∈ [x], then x ρrst x and, by (b), f (x )τrst f (x) so f (x ) ∈ [f (x)]. (d) If x1 , x2 ∈ [x] and x1 ρx2 then, since f is a homomorphism, f (x1 )τ f (x2 ), 2 that is, (f  [x])(x1 )τ (f  [x])(x2 ). Clearly, (e) follows from (d). Proposition 2.4 Let Xi = Xi , ρi , i ∈ I, and Yj = Yj , σj , j ∈ J, be two families of disjoint connected binary structures and X = X, ρ and Y = Y, σ their unions. Then (a) F ∈ Cond(X, Y) iff there exist asurjection f : I → J and monomorphisms gi : Xi → Yf (i) , i ∈ I, such that F = i∈I gi and (i) gi [Xi ] ∩ gi [Xi ] = ∅, for different i, i ∈ I,  (ii) i∈f −1 [{j}] gi [Xi ] = Yj , for each j ∈ J; thus, consequently, X ∼c Y implies that |I| = |J|. (b) (See [8]) F ∈ Emb(X, Y) iff there exista function f : I → J and embeddings gi : Xi → Yf (i) , i ∈ I, such that F = i∈I gi and gi (x), gi (x ) ∈ σrs , whenever i = i , x ∈ Xi and x ∈ Xi . Proof. (a) (⇒) Let F ∈ Cond(X, Y). Clearly, the sets Xi , i ∈ I, are components of X and Yj , j ∈ I, are components of Y. By Lemma 2.3(c), for i ∈ I and x ∈ Xi we have F [[x]] ⊂ [F (x)] so there is (unique) f (i) ∈ J, such that F [Xi ] ⊂ Yf (i) . By Lemma 2.3(d) and since Fis an injection, gi := F  Xi : Xi → Yf (i) is a monomorphism. Clearly F = i∈I gi and, since F is a surjection, f : I → J is a surjection as well and (ii) holds. (i) is true because F is an injection.  (⇐) Let F = i∈I gi , where the functions f : I → J and gi : Xi → Yf (i) , i ∈ I, satisfy the given conditions. By (i) and since gi ’s are injections F is an injection too. By (ii) and since f : I → J is a surjection, F is a surjection and we prove that F is a homomorphism. If u, v ∈ X and uρv, then u and v are in the same component of X, that is there is i ∈ I such that u, v ∈ Xi ; thus, since gi is a homomorphism, we have F (u) = gi (u)σf (i) gi (v) = F (v) and so F (u)σF (v). 2

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Infinite structures of non-unary languages

Figure 1 from [13] describes the hierarchy among twelve equivalence relations (similarities) on the set IntL (X), for arbitrary relational language L and set X: the equality, the isomorphism, the equimorphism, the full relation, four similarities

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of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. For example, line n denotes the statement that equimorphic structures have forcing-equivalent posets of copies. ρ ∼11 σ the full relation



r


ρ ∼10 σ
ro sq P(ρ) ∼ = ro sq P(σ) r ⇔ P(ρ) ≡f orc P(σ) o


HH H
m  n H HH 
 H 
ρ ∼8 σ H r ρ ∼9 σ  r H  ρ  σ sq P(ρ) ∼ = sq P(σ) HH

 H j l  H k  HH q
ρ ∼6 σ
r r  H ρ ∼7 σ P(ρ) ∼ = P(σ) HH
sq P(ρ) ∼ = sq P(σ) ∧ ρ  σ HH 
h H  g i H
 H  r r Hr ρ ∼c σ
ρ ∼5 σ ρ ∼4 σ  H HH p  P(ρ) ∼ P(ρ) = P(σ) HH = P(σ) ∧ ρ  σ  H  HH d HH e  f  H HH HH r H ρ ∼2 σ 3 σ r ρ ∼ ρ∼ P(ρ) = P(σ) ∧ ρ  σ =σ   b  c  r  ρ ∼1 σ  P(ρ) = P(σ) ∧ ρ ∼ =σ a

r ρ ∼0 σ ρ=σ

Figure 1: Implications between the similarities on IntL (X) By [13], concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures splits into three parts: 1. finite structures (where the diagram collapses to three points, if |X| > 1), 2. infinite structures of unary languages, and 3. infinite structures of non-unary languages, where all these similarities are different. Moreover, by Theorem 3.7 of [13] we have: If L is a non-unary relational language and κ an infinite cardinal, then in the part of the diagram from Figure 1 describing the similarities ∼k , for k ≤ 11 on the set IntL (κ) all the implications a - o are proper and there are no new implications (except the ones following from transitivity). In this section we include the relations ∼c and ≡ in the diagram and prove

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Miloˇs S. Kurili´c and Nenad Moraˇca

Theorem 3.1 If L is a non-unary relational language and κ ≥ ω a cardinal, then (a) In Figure 1 all the implications a - q are proper and there are no new implications (except the ones following from transitivity); (b) If κ = ω, then the position of the elementary equivalence, ≡, with respect to the similarities ∼k , k ≤ 11, is the same as the position of the condensation equivalence, ∼c , in Figure 1; in addition, these two similarities are incomparable. Remark 3.2 By Theorem 3.7 of [13], statement (a) is true for the part of the diagram describing the similarities ∼k , for k ≤ 11. The implication p follows from Proposition 2.2(c), q is trivial and it is clear that they can not be reversed. Thus in order to show that there are no new implications (except the ones following from transitivity) it is sufficient to show that ∼2 ⇒ ∼c (which implies ∼k ⇒ ∼c , for k ≥ 4) and that ∼c ⇒ ∼10 (which implies ∼c ⇒ ∼k , for k ≤ 10). Similarly, for a proof of (b) it is sufficient to show that ∼2 ⇒ ≡, that ≡ ⇒ ∼10 and that ≡ and ∼c are incomparable similarities. First we do the job for the class IntLb (ω), constructing structures with countable domains (which can be easily transformed into structures with domain ω). Example 3.3 ∼2 ⇒ ∼c on IntLb (ω). Let ρ, σ ∈ IntLb (ω), where ρ = {n, n + 1 : n ∈ ω} ∪ {2n, 2n : n ∈ ω} and σ = {n, n + 1 : n ∈ ω} ∪ {2n + 1, 2n + 1 : n ∈ ω}. Then P(ρ) = P(σ) = {[2n, ∞) : n ∈ ω} and ρ  σ, thus ρ ∼2 σ. Suppose that ρ ∼c σ and f ∈ Cond(ρ, σ). Then, since 0, 0 ∈ ρ we have f (0), f (0) ∈ σ and, hence, f (0) = 2n + 1, for some n ∈ ω. Since f is onto there is m ∈ ω such that f (m) = 0 and, by the previous conclusion, m > 0. Now m − 1, m ∈ ρ implies f (m − 1), f (m) ∈ σ, that is f (m − 1), 0 ∈ σ, which is false. Thus ρ ∼c σ. Example 3.4 ∼c ⇒ ∼10 on IntLb (ω). Instead of ω we take the set X = A∪B∪Q, where Q is the set of rational numbers, the sets A, B and Q are pairwise disjoint and |A| = |B| = ω. Let
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Clearly, the connectivity components of the structure X, σ are the sets A, Q and {b}, for b ∈ B. So, if f ∈ Emb(X, σ), then, by Proposition 2.4(b) and since f must preserve (ir)reflexivity, we have f [A] ⊂ A and f [Q] ⊂ Q. If b ∈ B, then b, b ∈ σ implies f (b) ∈ A. f (b) ∈ Q would imply that f (b)
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Fact 3.8 If X = X, ρ and Y = Y, σ are Lb -structures, then (a) At least one of the structures X and Xc := X, ρc  is connected; (b) P(X) = P(Xc ) and, if X is irreflexive, P(X) = P(Xre ); (c) If the structures X and Y are irreflexive, then X ≡ Y ⇔ Xre ≡ Yre . Proof. For (a) see ([8]) and for (b) see [13]. (c) By recursion, to each Lb -formula ψ(v1 , . . . , vk ) we adjoin an Lb -formula re ψ (v1 , . . . , vk ) in the following way: (vi = vj )re := vi = vj , (R(vi , vj ))re := R(vi , vj ), for i = j, (R(vi , vi ))re := ¬R(vi , vi ), (¬ψ)re := ¬ψ re , (ψ1 ∧ ψ2 )re := ψ1re ∧ ψ2re , (∃v ψ)re := ∃v ψ re . A simple induction on the complexity of formulas shows that for each irreflexive Lb -structure X, each Lb -structure Y and each Lb formula ψ(v1 , . . . , vk ) we have ∀x1 , . . . , xk ∈ X (X |= ψ[x1 , . . . , xk ] ⇔ Xre |= ψ re [x1 , . . . , xk ]),

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∀y1 , . . . , yk ∈ Y (Y |= (ψ re )re [y1 , . . . , yk ] ⇔ Y |= ψ[y1 , . . . , yk ]).

(5)

So, if X ≡ Y, then for each Lb -sentence ψ we have Xre |= ψ iff Xre |= (ψ re )re iff X |= ψ re iff Y |= ψ re iff Yre |= (ψ re )re iff Yre |= ψ and, hence, Xre ≡ Yre . Conversely, if Xre ≡ Yre , then for each Lb -sentence ψ we have X |= ψ iff Xre |= 2 ψ re iff Yre |= ψ re iff Y |= ψ and, hence, X ≡ Y. Fact 3.9 If L = Ri : i ∈ I is a relational language and X, ρ, Y, σ ∈ ModL , then we have: X, ρ ≡ Y, σ ⇔ X, ρc  ≡ Y, σ c . Proof. By recursion, to each L-formula ψ(v1 , . . . , vk ) we adjoin an L-formula ψ c (v1 , . . . , vk ) in the following way: (vi = vj )c := vi = vj , (Ri (vi1 , . . . , vini ))c := ¬Ri (vi1 , . . . , vini ), (¬ψ)c := ¬ψ c , (ψ1 ∧ ψ2 )c := ψ1c ∧ ψ2c , (∃v ψ)c := ∃v ψ c . A simple induction on the complexity of formulas shows that for each structure X, ρ ∈ ModL and each L-formula ψ(v1 , . . . , vk ) we have ∀x1 , . . . , xk ∈ X (X, ρ |= ψ[x1 , . . . , xk ] ⇔ X, ρc  |= ψ c [x1 , . . . , xk ]), (6) ∀x1 , . . . , xk ∈ X (X, ρ |= (ψ c )c [x1 , . . . , xk ] ⇔ X, ρ |= ψ[x1 , . . . , xk ]). (7) So, if X, ρ ≡ Y, σ, then for each L-sentence ψ we have X, ρc  |= ψ iff X, ρc  |= (ψ c )c iff X, ρ |= ψ c iff Y, σ |= ψ c iff Y, σ c  |= (ψ c )c iff Y, σ c  |= ψ and, hence, X, ρc  ≡ Y, σ c . Conversely, if X, ρc  ≡ Y, σ c , then for each Lsentence ψ we have X, ρ |= ψ iff X, ρc  |= ψ c iff Y, σ c  |= ψ c iff Y, σ |= ψ and, hence, X, ρ ≡ Y, σ. 2 Theorem 3.10 If κ ≥ λ ≥ ω are cardinals, L = Ri : i ∈ I a non-unary relational language and Int∗Lb (λ) = {ρ ⊂ λ2 : λ, ρ is connected ∧ ρ∩Δλ = ∅}, then there is a mapping τ : Int∗Lb (λ) → IntL (κ) such that for each ρ, σ ∈ Int∗Lb (λ)

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(a) ρ ∼k σ ⇔ τρ ∼k τσ , for each k ≤ 11; (b) ρ ∼c σ ⇔ τρ ∼c τσ ; (c) ρ ≡ σ ⇔ τρ ≡ τσ , if λ = κ. Proof. First suppose that λ < κ. Then |κ \ λ| = κ. By a well known theorem of Vopˇenka, Pultr and Hedrl´ın [17], on any set X there is an irreflexive binary relation θ such that idX is the only endomorphism of the structure X, θ. So we fix an irreflexive binary relation θ ⊂ (κ \ λ)2 such that End(κ \ λ, θ) = {idκ\λ }. Then, clearly, (8) Cond(κ \ λ, θ) = {idκ\λ } and we can assume that, in addition, the relation θ is connected and irreflexive. Namely, if θ is not connected, then by Fact 3.8, the relation θc is connected (and reflexive), by Fact 2.1(f) we have Cond(κ\λ, θc ) = {idκ\λ } and, clearly, θc \Δκ\λ is connected, irreflexive and Cond(κ \ λ, θc \ Δκ\λ ) = {idκ\λ }. The language L is not unary and we fix an i0 ∈ I such that ni0 ≥ 2. Now, for ρ ∈ Int∗Lb (λ) let the interpretation τρ = τiρ : i ∈ I ∈ IntL (κ) be defined by ⎧ ⎨ (ρ ∪ θ) × κni0 −2 if i = i0 and ni0 > 2; ρ ρ∪θ if i = i0 and ni0 = 2; τi = (9) ⎩ ∅ if i = i0 . For convenience, for ρ, σ ∈ Int∗Lb (λ), instead of Cond(κ, τρ , κ, τσ ) (respectively, Cond(λ, ρ, λ, σ)) we will write Cond(τρ , τσ ) (resp. Cond(ρ, σ)). Claim 3.11 Cond(τρ , τσ ) = {f ∪ idκ\λ : f ∈ Cond(ρ, σ)}, for ρ, σ ∈ Int∗Lb (λ). Proof. For convenience let πρ := ρ ∪ θ, for ρ ∈ Int∗Lb (λ). First we prove that Cond(κ, πρ , κ, πσ ) = {f ∪ idκ\λ : f ∈ Cond(ρ, σ)}.

(10)

By the construction, κ, πρ  = λ, ρ ∪ κ \ λ, θ and κ, πσ  = λ, σ ∪ κ \ λ, θ are partitions of the binary structures κ, πρ  and κ, πσ  into their connectivity components. If F ∈ Cond(κ, πρ , κ, πσ ), then, by Proposition 2.4(a) we have F = g1 ∪ g2 , where either g1 ∈ Mono(λ, ρ, κ \ λ, θ), which is impossible because the relation θ is irreflexive and ρ ∩ Δλ = ∅, or g1 ∈ Mono(λ, ρ, λ, σ), which, thus, must be true. Since the function f from Proposition 2.4(a) is onto we have g2 ∈ Mono(κ \ λ, θ, κ \ λ, θ) and, since F is a surjection, g1 and g2 are surjections. Hence, g1 ∈ Cond(ρ, σ) and g2 ∈ Cond(κ \ λ, θ) = {idκ\λ } so the inclusion “⊂” in (10) is proved. The inclusion “⊃” follows from Proposition 2.4(a). Now we prove Cond(κ, τiρ0 , κ, τiσ0 ) = {f ∪ idκ\λ : f ∈ Cond(ρ, σ)}.

(11)

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Miloˇs S. Kurili´c and Nenad Moraˇca

If F : κ → κ is a bijection, then F ∈ Cond(κ, τiρ0 , κ, τiσ0 ) iff for each x1 , x2 , . . . , xni0 ∈ κ we have x1 , x2 , . . . , xni0  ∈ πρ × κni0 −2 ⇒ F (x1 ), F (x2 ), . . . , F (xni0 ) ∈ πσ × κni0 −2 iff for each x1 , x2 ∈ κ we have: x1 , x2  ∈ πρ ⇒ F (x1 ), F (x2 ) ∈ πσ , iff F ∈ Cond(κ, πρ , κ, πσ ). Now (11) follows from (10). Clearly, F ∈ Cond(τρ , τσ ) iff F ∈ Cond(κ, τiρ , κ, τiσ ), for all i ∈ I. By (9) this holds iff F ∈ Cond(κ, τiρ0 , κ, τiσ0 ) and we apply (11). 2 Now we prove the theorem. Statement (a) is proved in Theorem 3.20 of [13]. (b) If ρ ∼c σ, then there are f ∈ Cond(ρ, σ) and g ∈ Cond(σ, ρ), which by Claim 3.11 implies that f ∪ idκ\λ ∈ Cond(τρ , τσ ) and g ∪ idκ\λ ∈ Cond(τσ , τρ ) and, hence, τρ c τσ and τσ c τρ , that is τρ ∼c τσ . Conversely, if τρ ∼c τσ , then there is F ∈ Cond(τρ , τσ ) and, by Claim 3.11, F  λ ∈ Cond(ρ, σ), which implies ρ c σ. Similarly we have σ c ρ and, hence, ρ ∼c σ. So, the theorem is proved for λ < κ. If λ = κ, then we define τ ρ by (9) replacing ρ ∪ θ by ρ and continue in the same way. (c) Let λ = κ, and let τρ be defined by (9), where ρ ∪ θ is replaced by ρ. First, for each Lb -formula ψ(v1 , . . . , vk ) we define an L-formula ϕψ (v1 , . . . , vk ) in the following way: ϕvi =vj := vi = vj , ϕR(vi ,vj ) := Ri0 (vi , vj , vj , . . . , vj ), ϕ¬ψ := ¬ϕψ , ϕψ1 ∧ψ2 := ϕψ1 ∧ ϕψ2 , ϕ∃v ψ := ∃v ϕψ . If ρ ∈ Int∗Lb (λ), then a simple induction on the complexity of formulas shows that for each Lb -formula ψ(v1 , . . . , vk ) we have ∀y1 , . . . , yk ∈ λ (λ, ρ |= ψ[y1 , . . . , yk ] ⇔ λ, τρ  |= ϕψ [y1 , . . . , yk ]).

(12)

So, if ρ, σ ∈ Int∗Lb (λ) and τρ ≡ τσ , then for each Lb -sentence ψ we have λ, ρ |= ψ iff λ, τρ  |= ϕψ iff λ, τσ  |= ϕψ iff λ, σ |= ψ and, hence, ρ ≡ σ. For a proof of the converse for each L-formula η(v1 , . . . , vk ) we define an Lb formula ψη (v1 , . . . , vk ) by: ψvi =vj := vi = vj , ψRi0 (vi1 ,vi2 ,...,vin ) := R(vi1 , vi2 ), i0

ψRi (vi1 ,vi2 ,...,vin ) := ¬vi1 = vi1 , for i = i0 , ψ¬η := ¬ψη , ψη1 ∧η2 := ψη1 ∧ ψη2 , i ψ∃v η := ∃v ψη . Again, for ρ ∈ Int∗Lb (λ) an induction on the complexity of formulas shows that for each L-formula η(v1 , . . . , vk ) we have ∀y1 , . . . , yk ∈ λ (λ, ρ |= ψη [y1 , . . . , yk ] ⇔ λ, τρ  |= η[y1 , . . . , yk ]).

(13)

So, if ρ, σ ∈ Int∗Lb (λ) and ρ ≡ σ, then for each L-sentence η we have λ, τρ  |= η 2 iff λ, ρ |= ψη iff λ, σ |= ψη iff λ, τσ  |= η and, hence, τρ ≡ τσ . Proof of Theorem 3.1 (a) According to Remark 3.2 we show that in the diagram for IntL (κ) we have ∼2 ⇒ ∼c and ∼c ⇒ ∼10 .

Condensational equivalence, equimorphism, elementary equivalence and ...

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By Example 3.3 there are interpretations ρ, σ ∈ Int∗Lb (ω) such that ρ ∼2 σ and ρ ∼c σ. By Theorem 3.10 we have τρ , τσ ∈ IntL (κ), τρ ∼2 τσ and τρ ∼c τσ . Thus ∼2 ⇒ ∼c in the diagram for IntL (κ). By Example 3.4 there are ρ, σ ∈ IntLb (X) such that ρ ∼c σ and ρ ∼10 σ. The structures X, ρ and X, σ are not connected but, by Fact 3.8, X, ρc  and X, σ c  are connected structures. Since for b ∈ B we have b, b ∈ ρc ∩ σ c , identifying the sets X and ω we obtain ρc , σ c ∈ Int∗Lb (ω). Now, by Proposition 2.2(e), ρ ∼c σ implies ρc ∼c σ c ; thus, by Theorem 3.10(b) we have τρc ∼c τσc . On the other hand, by Fact 3.8(b) we have P(ρc ) = P(ρ) and P(σ c ) = P(σ) thus ρ ∼10 σ, that is P(ρ) ≡f orc P(σ) implies ρc ∼10 σ c , which, by Theorem 3.10(a) gives τρc ∼10 τσc . Thus ∼c ⇒ ∼10 , in the diagram for IntL (κ). (b) Concerning the position of the elementary equivalence in the diagram for IntL (ω) first we note that, in general, ∼3 ⇒ ≡ ⇒ ∼11 . Let κ = ω. For the interpretations ρ, σ ∈ Int∗Lb (ω) considered in Example 3.5 we have ρ ∼2 σ and ρ ≡ σ and, by Theorem 3.10(a) and (c), for τρ , τσ ∈ IntL (κ) we have τρ ∼2 τσ and τρ ≡ τσ . So, in the diagram for IntL (κ) we have ∼k ⇒≡, for k ≥ 4. The interpretations ρ, σ ∈ IntLb (ω) considered in Example 3.6 are irreflexive and we have ρ ≡ σ and ρ ∼10 σ. Their reflexifications ρre and σre are in Int∗Lb (ω) and, by Fact 3.8(c) and (b), ρre ≡ σre and ρre ∼10 σre , which by Theorem 3.10(c) and (a) implies τρre ≡ τσre and τρre ∼10 τσre . So, in the diagram for IntL (κ) we have ≡ ⇒ ∼k , for k ≤ 10. By Example 3.7 the same pair of structures shows that ≡ ⇒ ∼c . Finally, the interpretations ρ0 , σ0 ∈ IntLb (ω) considered in Example 3.7 are disconnected, ρ0 ∼c σ0 and ρ0 ≡ σ0 . By Fact 3.8(a) their complements ρc0 and σ0c are in Int∗Lb (ω), by Proposition 2.2(e) we have ρc0 ∼c σ0c and, by Fact 3.9, ρc0 ≡ σ0c . By Theorem 3.10(b) and (c) we have τρc0 ∼c τσ0c and τρc0 ≡ τσ0c . So, in the diagram 2 for IntL (κ) we have ∼c ⇒ ≡. Concerning the interplay between the elementary equivalence, condensation equivalence and equimorphism, we recall that, when infinite structures of nonunary languages are in question, by Examples 3.3-3.7, these three similarities are pairwise incomparable. The following examples show that, moreover, their pairwise intersections, the similarities  ∧ ∼c , ∼c ∧ ≡ and ≡ ∧  are pairwise incomparable as well and, that, moreover again, the intersection  ∧ ∼c ∧ ≡ is different from the isomorphism (see Figure 2). Example 3.12  ∧ ∼c ⇒ ≡. For n ∈ N, let Ln denote the linear graph with n nodes and assuming that all unions are   disjoint,  let usdefine countable Lb  structures X = ω L3 ∪ ω L1 and Y = ω L3 ∪ ω L2 ∪ ω L1 . By Proposition 2.4(b) we have X  Y and, by Proposition 2.2(a), X ∼c Y. But X ≡ Y, since the graph Y has a component with exactly two elements.

Miloˇs S. Kurili´c and Nenad Moraˇca

14

≡

≡∧

r the full relation @ @ r ∼@ 10 @ @ @r ∼c r  r @ @ @ @ @ @ @ @ @ @ @r ≡ ∧ ∼c @r  ∧ ∼c r @ @ @ @ @ @r ≡ ∧  ∧ ∼c ∼ =

r

Figure 2: The similarities for infinite binary structures Example 3.13 ∼c ∧ ≡ ⇒ . It is well known that the Lb -theory Te of one equivalence relation with infinitely many equivalence classes and such that each class is infinite is complete (it is ω-categorical and has no finite models, so the Lo´sVaught Test applies, see [3], p. 113). Let {A, B} ∪ {Cα : α < ω1 } be a partition of ω1 , where |A| = |B| = ω1 and |Cα | = ω, for α < ω1 and let X = ω1 , ρ and Y = ω1 , σ, where ρ, σ ⊂ ω1 × ω1 are equivalence relations on ω1 corresponding to the partitions Πρ := {A ∪ B} ∪ {Cα : α < ω1 } and Πσ := {A, B} ∪ {Cα : α < ω1 }, respectively. Then clearly X and Y are models of Te and, hence, X ≡ Y. Since σ ⊂ ρ we have σ c ρ. Let {Bα : α < ω1 } be a partition of the set B, where |Bα | = ω, for α < ω1 , and let ρ be the equivalence relation on ω1 corresponding to the partition Πρ := {A} ∪ {Bα : α < ω1 } ∪ {Cα : α < ω1 }. Then, clearly, ρ∼ = ρ ⊂ σ and, by Proposition 2.2(a), ρ c σ, and, thus, ρ ∼c σ. By Proposition 2.4(b) we have Y  → X, so X  Y. Example 3.14 ≡ ∧  ⇒ ∼c . It is known that the Lb -theory Tl of the linear orders in which each element has an immediate predecessor and an immediate successor is complete (see [5], p. 74). So, if X and Y are the models of Tl defined

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   by X := ω (ω + Z) · Z and Y := ω (Z + ω) · Z (where ω L denotes an ω-sum of copies of a linear order L, isomorphic to the lexicographical order on ω × L), then X ≡ Y and, clearly, X  Y. Since the linear order Y does not have an initial  Y and, as in Example 3.7 we conclude that part isomorphic to ω ∗ we have X ∼ = X ∼c Y. Example 3.15  ∧ ∼c ∧ ≡ ⇒ ∼ =. Let X be a set of size ℵω and let ρ, σ ∈ IntLb (X) be the equivalence relations on X corresponding to the partitions Πρ := {X2n+1 : n ∈ ω} ∪ {Cα : α < ℵω }, Πσ := {X2n+2 : n ∈ ω} ∪ {Dα : α < ℵω }, where |Xk | = ℵk , for k ∈ N, and |Cα | = |Dα | = ω, for α < ℵω . Then ρ ≡ σ (see Example 3.13) and, using Proposition 2.4(b) we easily prove that ρ  σ and ρ ∼ = σ. Letus choose the sets Y2n+1 ∈ [X2n+2 ]ℵ2n+1 , for n ∈ ω, and let us divide the union n∈ω X2n+2 \ Y2n+1 into ℵω -many disjoint countable sets, Eα , α < ℵω . Let ρ ∈ IntLb (X) be the equivalence relations on X determined by the partition Πρ := {Y2n+1 : n ∈ ω} ∪ {Eα : α < ℵω } ∪ {Dα : α < ℵω }. Then, clearly, ρ ⊂ σ and ρ ∼ = ρ and, by Proposition 2.2(a), ρ c σ. Now, let us choosethe sets Z2n+2 ∈ [X2n+3 ]ℵ2n+2 , for n ∈ ω, and let us divide the union X1 ∪ n∈ω X2n+3 \ Z2n+2 into ℵω -many disjoint countable sets, Fα , α < ℵω . Let σ  ∈ IntLb (X) be the equivalence relations on X determined by the partition Πσ := {Z2n+2 : n ∈ ω} ∪ {Fα : α < ℵω } ∪ {Cα : α < ℵω }. Then σ  ⊂ ρ and σ  ∼ = σ and, by Proposition 2.2(a), σ c ρ. Thus ρ ∼c σ.

4

Finite and unary structures

By [13], if L is an arbitrary relational language and X a finite set, then the diagram describing the similarities ∼k , for k ≤ 11 collapses to the diagram given on Figure 3, where ∼0 = ∼1 iff |X| = 1. By Proposition 2.2(d) and since elementary equivalent finite structures are isomorphic (see [5]), we include the similarities ∼c and ≡ into the diagram putting ∼c = ≡ = the isomorphism. Now we turn to infinite structures of unary languages. By Theorem 3.5 of [13] for any unary language L and infinite cardinal κ, Figure 4 describes the hierarchy of the similarities ∼k , for k = 8, 10, on the set IntL (κ) and, in addition, ∼8 = ∼11 . Moreover, all the implications are proper and there are no new implications (except the ones following from transitivity). The position of the similarities ∼8 and ∼10 depends on the model of set theory in which we live. So, if κ is a regular cardinal and 2κ = κ+ , then ∼8 = ∼10 . In particular, by Theorem 3.6 of [13], if L is the language containing only one unary relational symbol, then on IntL (ω) we have

Miloˇs S. Kurili´c and Nenad Moraˇca

16

r ∼4 = ∼6 = ∼8 = ∼10 = ∼11 = the full relation r ∼1 = ∼2 = ∼3 = ∼5 = ∼7 = ∼9 = ∼c = ≡ = the isomorphism r ∼0 = the equality

Figure 3: The similarities on IntL (X), if 1 < |X| < ω ∼8 = ∼6 and  ∼11 ∼10 = ∼6

if the poset (P (ω)/ Fin)+ is forcing equivalent to its square, otherwise.

So, by a well known result of Shelah and Spinas [16], the equality ∼8 = ∼10 is independent of ZFC. r ∼11 = the full relation ∼6 = the isomorphism of P(X) r ∼4 r = the equality of P(X) @

@ @

@r ∼3 = ∼5 = ∼7 = ∼9 = the isomorphism = the equimorphism

@

@r ∼ 1 = ∼2 r ∼0 = the equality

Figure 4: The similarities on IntL (X) for unary L and infinite X Concerning the position of the similarities ∼c and ≡ in the diagram, we note that they are implied by the isomorphism and imply ∼11 . The following example shows that for some unary languages and infinite domains we have ∼c = ≡ = ∼ =. Example 4.1 ∼c = ≡ = ∼ =. Let L = R, where ar(R) = 1, and ρ, σ ∈ IntL (ω). If ρ ∼c σ, then, by Proposition 2.2(a), there are bijections f, g ∈ Sym(ω) such that f [ρ] ⊂ σ and g[σ] ⊂ ρ, which implies that |ρ| = |σ|. By Proposition 2.2(e) we have ρc ∼c σ c and, similarly, |ρc | = |σ c |. Now if F ∈ Sym(ω), where F [ρ] = σ, then, clearly, F ∈ Iso(ρ, σ), which implies ρ ∼ = σ. If ρ ≡ σ, then, since Th(ω, ρ) is an ω-categorical theory, we have ρ ∼ = σ.

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The next example shows that for some languages the implication ∼ = ⇒ ∼c is proper and that, moreover, ∼c do not imply ∼10 (the forcing equivalence of posets of copies) and, hence, ∼c ⇒ ∼k , for k ≤ 10. We will use the following fact. Fact 4.2 ([13]) If X = X, ρi : i ∈ I is a unary structure, ≈ the equivalence relation on the set X defined by: x ≈ y ⇔ ∀i ∈ I (x ∈ ρi ⇔ y ∈ ρi ) and X/ ≈ = {Xj : j ∈ J} the corresponding partition, then < ω} = J, then P(X) = {X}; - If J0 := {j ∈ J : |Xj |  - Otherwise, sq P(X) ∼ = j∈J\J0 (P (Xj )/[Xj ]<|Xj | )+ . Example 4.3 ∼c ⇒ ∼10 and, hence, ∼c ⇒ ∼k , for k ≤ 10. Let Q be the set of rational numbers, L = Rq : q ∈ Q a language, where ar(Rq ) = 1, for all q ∈ Q. Let ρ, σ ∈ IntL (Q) be the interpretations defined by: ρ = (−∞, q] : q ∈ Q and σ = (−∞, q−1] : q < 0∪(−∞, q] : q ≥ 0. If f ∈ Sym(Q), where f (q) = q−1, then by Fact 2.1(d) we have ρ ∼ = f [ρ] = (−∞, q − 1] : q ∈ Q ⊂ σ, and, since σ ⊂ ρ, by Proposition 2.2(a) and (b), ρ ∼c σ. Now, by Fact 4.2, since the partition corresponding to the structure Q, ρ consists of singletons, P(Q, ρ) = {Q} is the trivial forcing. On the other hand, the partition corresponding to the structure Q, σ consists of the interval [−1, 0) and singletons {q}, where q ∈ Q \ [−1, 0), so P(Q, σ) ≡f orc (P (ω)/ Fin)+ and, hence, ρ ∼10 σ. Fact 4.4 ∼4 ⇒ ∼c and, hence, ∼k ⇒ ∼c , for k ∈ {6, 8, 10, 11}, for any unary language L = Ri : i ∈ I and infinite set X. Proof. Let X = X, ρ and Y := X, σ, where ρ = ∅ : i ∈ I and σ = X : i ∈ 2 I. Then P(X) = P(Y) = [X]|X| , but, clearly, ρ ∼c σ. The following two examples show that, for some unary languages and infinite domains, the elementary equivalence and the condensational equivalence are incomparable similarities. The following statement will be used in Example 4.7. Fact 4.5 (a) If κ > ω is a regular cardinal and 2κ = κ+ , then ro(P (κ)/[κ]<κ ) ∼ = Col(ω, 2κ ) (Balcar, Vopˇenka [1]; see also [2], p. 380). ˇ =ω (b) If λ > ω is a cardinal and P a poset of size λ such that 1P  |λ| ˇ , then ∼ roP = Col(ω, λ) (see [6], p. 277). Example 4.6 ∼c ∧ ∼10 ⇒ ≡. Let Z be the set of integers, L = Rn : n ∈ Z a language, where ar(Rn ) = 1, for all n ∈ Z. Let ρ, σ ∈ IntL (Z), where ρ = (−∞, n] : n ∈ Z and σ = (−∞, n − 1] : n < 0 ∪ (−∞, n] : n ≥ 0. If f ∈ Sym(Z), where f (n) = n − 1, then by Fact 2.1(d) we have ρ ∼ = f [ρ] = (−∞, n − 1] : n ∈ Z ⊂ σ, and, since σ ⊂ ρ, by Proposition 2.2(a) and (b) we have ρ ∼c σ and, by Fact 4.2, ρ ∼10 σ. But ρ ≡ σ, because for the sentence

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Miloˇs S. Kurili´c and Nenad Moraˇca

ϕ := ∃v0 , v1 (v0 = v1 ∧ R0 (v0 ) ∧ R0 (v1 ) ∧ ¬R−1 (v0 ) ∧ ¬R−1 (v1 )) we have Z, σ |= ϕ and Z, ρ |= ¬ϕ. Example 4.7 ≡ ⇒ ∼c and, hence, ≡ ⇒ . It is consistent that ≡ ∧ ∼10 ⇒ ∼c . Let L be the language containing only one unary relational symbol, U , and let Tu be the L-theory saying that both U and its complement are infinite sets. Clearly, Tu is an ω-categorical theory having only infinite models and, hence, it is complete. Let X = ω1 , ρ and Y = ω1 , σ, where ρ, σ ⊂ ω1 and |ρ| = |ω1 \ ρ| = ω1 , but |σ| = ω. Then X ≡ Y, that is ρ ≡ σ, but, clearly, ρ c σ, which implies ρ ∼c σ. If 2ω1 = ω2 , then by Fact 4.2 we have P(X) ∼ = ((P (ω1 )/[ω1 ]<ω1 )+ )2 and the ω 1 poset P(X) is of size 2 , by Fact 4.5(a) collapses 2ω1 to ω and, by Fact 4.5(b), P(X) ≡f orc Col(ω, 2ω1 ). The same holds for the poset P(Y), because by Fact 4.2 we have P(Y) ∼ = (P (ω1 )/[ω1 ]<ω1 )+ × (P (ω)/ Fin)+ ; thus ρ ∼10 σ. Acknowledgments The authors would like to thank the referee for helpful suggestions. This research was supported by the Ministry of Education and Science of the Republic of Serbia (Project 174006).

References [1] B. Balcar, P. Vopˇenka, On systems of almost disjoint sets, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 20 (1972) 421–424. [2] B. Balcar, P. Simon, Disjoint refinement, in: J. D. Monk and R. Bonnet (Eds.), Handbook of Boolean algebras, Vol. 2, 333–388, Elsevier Science Publishers B.V., Amsterdam, 1989. [3] C. C. Chang, H. J. Keisler, Model theory, Studies in Logic and the Foundations of Mathematics, Vol. 73. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [4] P. C. Doyle, J. G. Hocking, Bijectively related spaces. I. Manifolds. Pacific J. Math. 111,1 (1984) 23–33. [5] W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993. [6] T. Jech, Set Theory, 2nd corr. edition, Springer, Berlin, 1997. [7] M. Kukiela, Reversible and bijectively related posets, Order 26,2 (2009) 119–124. [8] M. S. Kurili´c, From A1 to D5 : Towards a forcing-related classification of relational structures, J. Symbolic Logic 79,1 (2014) 279–295. [9] M. S. Kurili´c, Maximally embeddable components, Arch. Math. Logic 52,7 (2013) 793–808. [10] M. S. Kurili´c, Posets of copies of countable scattered linear orders, Ann. Pure Appl. Logic, 165 (2014) 895–912. [11] M. S. Kurili´c, Forcing with copies of countable ordinals, Proc. Amer. Math. Soc. 143,4 (2015) 1771–1784.

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