Maximal Chains of Isomorphic Suborders of Countable Ultrahomogeneous Partial Orders

Abstract

We investigate the poset P(X)∪{∅}, ⊂ , where P(X) is the set of isomorphic suborders of a countable ultrahomogeneous partial order X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in P(X) ∪ {∅}, ⊂ are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated.

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

“…A contradiction to the choice of q. q = b. Then by (8) and (7) we have {a, q}, {a + 1, q} ∈ ρ p 1 \ ρ p which implies a, a + 1 ∈ K. Since q > m H and K ⊂ H we have q > a + 1 that is b > a + 1. A contradiction again.…”

“…A contradiction to the choice of q. q = a + 1. Then by (8) we have b = q and, since a = q, by (7) we have {a, b} ∈ ρ p which implies a ∈ G p . A contradiction to the choice of q. q = b.…”

Antichains of Copies of Ultrahomogeneous Structures

“…Proof. (a) It is evident that P(B n ) = i<n {i} × C i : ∀i < n C i ∈ P(Q) (see the proof of Theorem 5.1 of [10]), which implies that P(B n ) ∼ = P(Q) n . By Lemma 5.1 there is a countable maximal antichain A = {A j : j ∈ ω} in P(Q) and, defining Āj := A j , Q, .…”

“…Third, in [7,8,9] a classification of relational structures with respect to the properties of posets P(X), ⊂ is given. Fourth, the order types of the maximal chains in the posets of copies of countable ultrahomogeneous graphs and countable ultrahomogeneous partial orders are described in [10,11]. Finally, if X is a first order structure and R right Green's pre-order on its self-embedding monoid, Emb X, the corresponding antisymmetric quotient Emb X/ ≈ R , R (right Green's order) is isomorphic to the partial order P(X), ⊃ .…”

“…Thus A is a maximal antichain in the product P(Q) n . (b) It is easy to see that the copies of B ω are of the form i∈S {i} × C i , where S ∈ [ω] ω and C i ∈ P(Q), for all i ∈ S (see the proof of Theorem 5.2 of [10]). Thus, the poset P(B ω ) is isomorphic to the countable support product cs i∈ω P i , where P i = P(Q) ∪ {∅}, ⊂ , for all i ∈ ω, and we apply Lemma 5.2.…”