On some generalization of the bicyclic monoid

Abstract

We introduce an algebraic extension ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ of the bicyclic monoid for an arbitrary $\omega$-closed family $\mathcal{F}$ subsets of $\omega$ which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse semigroups. It is proved that ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ is a combinatorial inverse semigroup and Green's relations, the natural partial order on ${\mathbf{B}}_{\omega }^{\mathcal{F}}$, and its set of idempotents are described. We provide criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ and when ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ has the identity, is isomorphic to the bicyclic semigroup or the countable semigroup of matrix units.

13 pages, in Ukrainian