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