Proper restriction semigroups – semidirect products and W-products

Abstract

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product \({T\ltimes {\mathcal {Y}}}\) of a semilattice \({\mathcal {Y}}\) by a monoid T, where the action of T on \({\mathcal {Y}}\) is injective with images of the action being order ideals of \({\mathcal {Y}}\). Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form \({{\mathcal {Y}}\rtimes T}\).