On the semigroup generating by extended bicyclic semigroup and a \(\omega\)-closed family

Abstract

Similar as in \cite{Gutik-Mykhalenych-2020}, we introduce the algebraic extension \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) of the extended bicyclic semigroup for an arbitrary \(\omega\)-closed family \(\mathscr{F}\) subsets of \(\omega\). It is proven that \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) is combinatorial inverse semigroup and Green's relations, the natural partial order on \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) and its set of idempotents are described. We gave the criteria of simplicity, \(0\)-simplicity, bisimplicity, \(0\)-bisimplicity of the semigroup \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) and when \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) is isomorphic to the extended bicyclic semigroup or the countable semigroup of matrix units. We proved that in the case when the family \(\mathscr{F}\) consists of all singletons of \(\mathbb{Z}\) and the empty set then the semigroup \(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}\) is isomorphic to the Brandt \(\lambda\)-extension of the semilattice \((\omega,\min)\).