Restriction $$\omega $$ ω -semigroups

Abstract

The purpose of this paper is to investigate restriction $$\omega $$ -semigroups. Here a restriction $$\omega $$ -semigroup is a generalisation of an inverse $$\omega $$ -semigroup. We give a description of a class of restriction $$\omega $$ -semigroups, namely, restriction $$\omega $$ -semigroups with an inverse skeleton. We show that a restriction $$\omega $$ -semigroup with an inverse skeleton is an ideal extension of a $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse $$\omega $$ -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup with an inverse skeleton, conversely, every $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup with an inverse skeleton arises in this way.