On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains

We study the relation between completeness and $\mathrm{H}$-closedness for topological partially ordered spaces. In general, a topological partially ordered space with an infinite antichain which is even directed complete and down-directed complete, is not $\mathrm{H}$-closed. On the other hand, for a topological partially ordered space without infinite antichains, we give necessary and sufficient condition to be $\mathrm{H}$-closed, using directed completeness and down-directed completeness. Indeed, we prove that {a pospace} $X$ is $\mathrm{H}$-closed if and only if each up-directed (resp. down-directed) subset has a supremum (resp. infimum) and, for each nonempty chain $L \subseteq X$, $ \bigvee L \in \mathrm{cl} {\mathop{\downarrow} } L$ and $ \bigwedge L \in \mathrm{cl} {\mathop{\uparrow} } L$. This extends a result of Gutik, Pagon, and Repovš [GPR].