On completeness of H-closed pospaces

TL;DR

This paper extends the understanding of H-closed pospaces by providing a characterization involving directed completeness, down-completeness, and closure properties of chains, specifically for pospaces without infinite antichains.

Contribution

It generalizes the characterization of H-closedness for linearly ordered pospaces to a broader class without infinite antichains, using new completeness and closure conditions.

Findings

H-closed pospaces without infinite antichains are characterized by directed and down-completeness.

Supremum and infimum of chains lie in the chain's closure.

The result broadens the class of pospaces with known H-closedness criteria.

Abstract

We generalized the characterization of H-closedness for linearly ordered pospaces as follows: A pospace $X$ without an infinite antichain is an H-closed pospace if and only if $X$ is a directed complete and down-complete poset such that sup $L$ and inf $L$ are contained in the closure of $L$ for any nonempty chain $L$ in $X$.