Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank

Анотація

We study the semigroup extension ${\mathcal{I}}_{\lambda }^n\left(S\right)$ of a semigroup $S$ by symmetric inverse semigroup of a bounded finite rank $n$.  We describe  idempotents and regular elements of the semigroup ${\mathcal{I}}_{\lambda }^n\left(S\right)$ and show that the semigroup ${\mathcal{I}}_{\lambda }^n\left(S\right)$  is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup ${\mathcal{I}}_{\lambda }^n\left(S\right)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and  prove that for any infinite cardinal $\lambda$ and any positive integer $n$ the semigroup ${\mathcal{I}}_{\lambda }^n\left(S\right)$ has a  strongly tight ideal series provided so has $S$. Finally, we show that for every compact Hausdorff semitopological monoid $\left(S,{\tau }_S\right)$ there exists  its unique compact topological extension $\left({\mathcal{I}}_{\lambda }^n\left(S\right),{\tau }_{\mathcal{I}}^{\mathfrak{c}}\right)$ in the class of Hausdorff semitopological semigroups.