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The algebraic extension $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ of theextended 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 inversesemigroup. Green's relations, the natural partial order on the semigroup$\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and its set of idempotents aredescribed. 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 beisomorphic to the extended bicyclic semigroup or the countable semigroup ofmatrix 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, thesemigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is isomorphic to theBrandt $\lambda$-extension of the semilattice $(\omega,\min)$.

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