Posets of copies of countable ultrahomogeneous tournaments

Abstract

The poset of copies of a relational structure X is the partial order P(X):=〈{Y⊂X:Y≅X},⊂〉 and each similarity of such posets (e.g. isomorphism, forcing equivalence = isomorphism of Boolean completions, BX:=rosqP(X)) determines a classification of structures. Here we consider the structures from Lachlan's list of countable ultrahomogeneous tournaments: Q (the rational line), S(2) (the circular tournament), and T∞ (the countable homogeneous universal tournament); as well as the ultrahomogeneous digraphs S(3), Q[In], S(2)[In] and T∞[In] from Cherlin's list. If GRado (resp. Qn) denotes the countable homogeneous universal graph (resp. n-labeled linear order), it turns out that P(T∞)≅P(GRado) and that P(Qn) densely embeds in P(S(n)), for n∈{2,3}. Consequently, BX≅ro(S⁎π), where S is the poset of perfect subsets of R and π an S-name such that 1S⊩“π is a separative, atomless and σ-closed forcing” (thus 1S⊩“π≡forc(P(ω)/Fin)+”, under CH), whenever X is a countable structure equimorphic with Q, Qn, S(2), S(3), Q[In] or S(2)[In]. Also, BX≅ro(S⁎π), where 1S⊩“π is an ω-distributive forcing”, whenever X is a countable graph containing a copy of GRado, or a countable tournament containing a copy of T∞, or X=T∞[In].