Geodesic
On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains
Algebra and discrete mathematics, Tome 15 (2013) no. 2, pp. 287-294

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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].
Keywords: $\mathrm{H}$-closed, directed complete.
Mots-clés : pospace 
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     author = {T. Yokoyama},
     title = {On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains},
     journal = {Algebra and discrete mathematics},
     pages = {287--294},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2013_15_2_a10/}
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T. Yokoyama. On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains. Algebra and discrete mathematics, Tome 15 (2013) no. 2, pp. 287-294. http://geodesic.mathdoc.fr/item/ADM_2013_15_2_a10/