Rees matrix covers for a class of semigroups with locally commuting idempotents (Q2716987)

\textit{D. B. McAlister} [Trans. Am. Math. Soc. 277, 727-738 (1983; Zbl 0516.20039)] proved that every locally inverse semigroup can be covered by a regular Rees matrix semigroup over an inverse semigroup by means of a local isomorphism, where `\(S\) covers \(T\)' means that \(T\) is a surjective homomorphic image of a subsemigroup of \(S\), and a local isomorphism from \(S\) is a surjective homomorphism which is one-to-one on every local submonoid, i.e., subsemigroup of the form \(eSe\) with \(e\) idempotent. In the present paper the authors generalize this result to the class of semigroups with local units whose local submonoids have commuting idempotents and possessing a McAlister sandwich function. Here a semigroup \(S\) is said to have local units if, for every \(s\) in \(S\), there exist idempotents \(e,f\) such that \(es=s=sf\). The condition of possessing a McAlister sandwich function is related to regular elements and is too technical to be reproduced here. For a characterisation of the semigroups in this class see \textit{T.~A.~Khan} and \textit{M.~V.~Lawson} [Period. Math. Hung. 40, No. 2, 85-107 (2000; Zbl 0973.20053)].