On closures in semitopological inverse semigroups with continuous inversion

Oleg Gutik

Algebra and discrete mathematics, Tome 18 (2014) no. 1, pp. 59-85

Résumé

We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group $G$ is $H$-closed in the class of semitopological inverse semigroups with continuous inversion if and only if $G$ is compact, a Hausdorff linearly ordered topological semilattice $E$ is $H$-closed in the class of semitopological semilattices if and only if $E$ is $H$-closed in the class of topological semilattices, and a topological Brandt $\lambda^0$-extension of $S$ is (absolutely) $H$-closed in the class of semitopological inverse semigroups with continuous inversion if and only if so is $S$. Also, we construct an example of an $H$-closed non-absolutely $H$-closed semitopological semilattice in the class of semitopological semilattices.

Détail

Keywords: semigroup, semitopological semigroup, topological Brandt $\lambda^0$-extension, inverse semigroup, quasitopological group, topological group, semilattice, closure, $H$-closed, absolutely $H$-closed.