Maximal Chains in Positive Subfamilies of P(ω)

Abstract

A family P ⊂ [ω] ω is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/ Fin. For example, if I ⊂ P(ω) is an ideal containing the ideal Fin of finite subsets of ω, then P(ω) \ I is a positive family and the set Dense(Q) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(Q). We prove that, for a positive family P, the order types of maximal chains in the complete lattice P ∪ {∅}, ⊂ are exactly the order types of compact nowhere dense subsets of the real line having the minimum nonisolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0, 1) R and the poset E(Q), where E(Q) is the set of elementary submodels of the rational line.

On the structure of random hypergraphs

“…First note that a linear order L is isomorphic to the order type of a compact (nowhere dense compact) set of reals whose minimum is nonisolated if and only if it is complete (boolean), R embeddable and has a nonisolated minimum. For a proof of this fact see [5].…”

Maximal Chains of Isomorphic Suborders of Countable Ultrahomogeneous Partial Orders

“…(ii) Since ✁ p is an irreflexive relation and, by (7), q ∈ P p , by (8) the relation ✁ p 1 is irreflexive as well.…”

“…So there is If |B \A| = ω, then L is a countable and, hence, R-embeddable complete linear order. It is known that an infinite linear order is isomorphic to a maximal chain in P (ω) iff it is R-embeddable and Boolean (see, for example, [7]). By Fact 1.4 L is a Boolean order and, thus, there is a maximal chain Proof.…”

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

“…If |B \A| = ω, then L is a countable and, hence, R-embeddable complete linear order. It is known that an infinite linear order is isomorphic to a maximal chain in P (ω) iff it is R-embeddable and Boolean (see, for example, [5]). By Fact 2.2 L is a Boolean order and, thus, there is a maximal chain…”