Maximal Chains of Isomorphic Suborders of Countable Ultrahomogeneous Partial Orders

TL;DR

This paper characterizes the order types of maximal chains in the poset of isomorphic suborders of countable ultrahomogeneous partial orders, revealing their structure as certain compact sets of reals.

Contribution

It provides a complete characterization of maximal chains in the poset of isomorphic suborders for countable ultrahomogeneous partial orders, distinguishing cases based on the order's structure.

Findings

Maximal chains correspond to order types of specific compact sets of reals.

Different characterizations are given depending on whether the partial order is a countable antichain or not.

The results connect order theory with topological properties of sets of reals.

Abstract

We investigate the poset (P(X),\subset), 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)\cup \{\emptyset \},\subset) are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated.