On some generalization of the bicyclic monoid

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