On some generalization of the bicyclic monoid

Abstract

We introduce the algebraic extension B ω of the bicyclic monoid for an arbitrary ω-closed family F subsets of ω which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse semigroups. It is proven that B ω is combinatorial inverse semigroup and Green’s relations, the natural partial order on B ω and its set of idempotents are described. We gave the criteria of simplicity, 0-simplicity, bisimplicity, 0-bisimplicity of the semigroup B ω and when B ω has the identity, is isomorphic to the bicyclic semigroup or the countable semigroup of matrix units.