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

Abstract

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

arXiv comment11 pages, in Ukrainian