|
Algebra and Discrete Mathematics, 2013, том 15, выпуск 2, страницы 287–294
(Mi adm426)
|
|
|
|
Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
RESEARCH ARTICLE
On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains
T. Yokoyama Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan
Аннотация:
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].
Ключевые слова:
$\mathrm{H}$-closed, pospace, directed complete.
Поступила в редакцию: 25.08.2011 Исправленный вариант: 25.01.2013
Образец цитирования:
T. Yokoyama, “On the relation between completeness and $\mathrm{H}$-closedness of pospaces without infinite antichains”, Algebra Discrete Math., 15:2 (2013), 287–294
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm426 https://www.mathnet.ru/rus/adm/v15/i2/p287
|
|