On some generalization of the bicyclic monoid

TL;DR

This paper introduces a new algebraic extension of the bicyclic monoid based on an arbitrary omega-closed family, generalizing several known inverse semigroups and analyzing its algebraic properties and classifications.

Contribution

It defines the algebraic extension oldsymbol{B}_{\u03c9}^{\u211f} of the bicyclic monoid for any omega-closed family, and characterizes its structure, simplicity, and isomorphism conditions.

Findings

oldsymbol{B}_{\u03c9}^{} is a combinatorial inverse semigroup.

Descriptions of Green's relations and the natural partial order are provided.

Criteria for simplicity, 0-simplicity, bisimplicity, 0-bisimplicity, and isomorphism conditions are established.

Abstract

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.