Different similarities

Abstract

We establish the hierarchy among twelve equivalence relations (similarities) on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class Mod L (κ) of all L-structures of size κ. It turns out that, concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages (where all these similarities are different).

The poset of all copies of the random graph has the 2-localization property

“…In this section we prove Theorem 1.2. In fact, if G, ∼ is a countable graph containing a copy of the Rado graph, then these two structures are equimorphic and, by [9], the corresponding posets of copies P(G) and P(R) are forcing equivalent. So it is sufficient to prove the theorem assuming that G, ∼ is the Rado graph and we will do this in Theorem 3.3.…”

“…More generally, as a part of the investigation of the class of self-embedding monoids of first order structures, in an attempt to classify the pre-orders of the form Emb(X), R , where X is a structure and R the right Green's pre-order on the set Emb(X) of its self-embeddings (defined by f R g iff f • h = g, for some h ∈ Emb(X)), one can define two such pre-orders to be equivalent iff the antisymmetric quotients of their inverses have isomorphic Boolean completions. It is easy to see [10] that the antisymmetric quotient of the pre-order Emb(X), ( R ) −1 is isomorphic to the poset P(X), ⊂ , where P(X) = {f [X] : f ∈ Emb(X)} is the set of copies of X (that is, the set of domains of the substructures of X which are isomorphic to X) and that the isomorphism of the Boolean completions of such posets is the same as their forcing equivalence [9]. Thus, the intended classification is in fact the classification of the posets of the form P(X), ⊂ determined by their forcing-related properties.…”

Copies of the random graph

“…This investigation is related to a coarse classification of relational structures. Namely, the conditions P(X) = P(Y), P(X) ∼ = P(Y), sq P(X) ∼ = sq P(Y) and ro sq P(X) ∼ = ro sq P(Y) (where sq P denotes the separative quotient of a partial order P and ro sq P its Boolean completion) define different equivalence relations ("similarities") on the class of relational structures and their interplay with the similarities defined by the conditions X = Y, X ∼ = Y and X ⇄ Y (equimorphism) was considered in [11]. It turns out that the similarity defined by the condition ro sq P(X) ∼ = ro sq P(Y) is implied by all the similarities listed above and, thus, provides the coarsest among the mentioned classifications of relational structures.…”

“…Theorem 1.2 For each countable non-scattered graph G, ∼ and, in particular, for the Rado graph, the poset P(G) is forcing equivalent to the two-step iteration P * π, where 1 P "π is an ω-distributive forcing" and the poset P is similar to the Sacks forcing: adds a generic real, has the ℵ 0 -covering property (thus preserves ω 1 ), has the Sacks property and does not produce splitting reals. In addition, ro sq P(G) ∼ = ro(P (R)/I R ) + ∼ = ro(P * π) and these complete Boolean algebras are weakly distributive 3 .In fact, if G, ∼ is a countable graph containing a copy of the Rado graph, then these two structures are equimorphic and, by [11], forcing equivalent. So it is sufficient to prove the previous theorem assuming that G, ∼ is the Rado graph.Finally we note that the results of this paper are related to the investigation of the monoids of self-embeddings.…”

“…In fact, if G, ∼ is a countable graph containing a copy of the Rado graph, then these two structures are equimorphic and, by [11], forcing equivalent. So it is sufficient to prove the previous theorem assuming that G, ∼ is the Rado graph.…”