Restriction semigroups

The purpose of this paper is to investigate restriction ω -semigroups. Here a restriction ω -semigroup is a generalisation of an inverse ω -semigroup. We give a description of a class of restriction ω -semigroups, namely, restriction ω -semigroups with an inverse skeleton. We show that a restriction ω -semigroup with an inverse skeleton is an ideal extension of a (cid:2) J -simple restriction ω -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 ω -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a (cid:2) J -simple restriction ω -semigroup with an inverse skeleton, conversely, every (cid:2) J -simple restriction ω -semigroup with an inverse skeleton arises in this way.