On the semigroup which is generated by extended bicyclic semigroup and ω-closed family
Олег Гутік, Inna Pozdniakova · Математичні методи та фізико-механічні поля · 2021
The algebraic extension $\mathbf{B}_\mathbb{Z}^{\it F}$ of the extended bicyclic semigroup is introduced for an arbitrary ω-closed family ${\it F}$ of subsets of ω. It is proved that $\mathbf{B}_\mathbb{Z}^{\it F}$ is a combinatorial inverse semigroup. Green’s relations and the natural partial order on the semigroup $\mathbf{B}_\mathbb{Z}^{\it F}$ and its set of idempotents are described. The criteria of simplicity, 0-simplicity, bisimplicity, 0-bisimplicity of the semigroup $\mathbf{B}_\mathbb{Z}^{\it F}$, and the criterion for $\mathbf{B}_\mathbb{Z}^{\it F}$ 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 F consists of all singletons of ω and the empty set, the semigroup $\mathbf{B}_\mathbb{Z}^{\it F}$ is isomorphic to the Brandt λ-extension of the semilattice (ω,min). Cite as: O. V. Gutik, I. V. Pozdniakova, “On the semigroup which is generated by extended bicyclic semigroup and ω-closed family,” Mat. Met. Fiz. Mekh. Polya , 64 , No. 1, 21–34 (2021), https://doi.org/10.15407/mmpmf2021.64.1.21-34