  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.
