Periodica Mathematica Hungarica Vol. 40 (2), (2000), pp. 85–107 A CHARACTERISATION OF A CLASS OF SEMIGROUPS WITH LOCALLY COMMUTING IDEMPOTENTS Tanveer A. Khan ∗ and Mark V. Lawson (Wales) Communicated by M´ aria B. Szendrei Abstract McAlister proved that a necessary and sufficient condition for a regular semi- group S to be locally inverse is that it can be embedded as a quasi-ideal in a semigroup T which satisfies the following two conditions: (1) T = T eT , for some idempotent e;and(2) eT e is inverse. We generalise this result to the class of semi- groups with local units in which all local submonoids have commuting idempotents. 1. Introduction This is the second of two papers (the first was [3]) in which we generalise McAlister’s theory of locally inverse regular semigroups, developed in [6] and [8], to a class of non-regular semigroups. Recall that a locally inverse regular semigroup is a regular semigroup S in which each local submonoid eSe, where e is an idempotent, is inverse or, equiva- lently, in which the idempotents in each local submonoid commute. In this paper, we replace regular semigroups by semigroups S having local units: this means that for each s ∈ S there exist idempotents e,f ∈ S such that es = s = sf . Thus our aim is to generalise McAlister’s results to semigroups with local units which have ‘locally commuting idempotents’, in the sense that the idempotents in each submonoid commute. In this paper, we concentrate on generalising the results McAlister obtained in [8], where locally inverse regular semigroups are described in the following terms. A natural class of examples of locally inverse regular semigroups are those regular semigroups T possessing an idempotent e such that T = TeT and eTe is inverse. Clearly, any regular subsemigroup of such a T is locally inverse. The converse is also Mathematics subject classification numbers. 2000. 20M10 (20M17). Key words and phrases. Semigroups with local units, semigroups in which the idem- potents in each local submonoid commute, enlargements. * Supported by the Government of Pakistan. 0031-5303/00/$5.00 Akad´ emiai Kiad´o, Budapest c  Akad´ emiai Kiad´o, Budapest Kluwer Academic Publishers, Dordrecht