### E-unitary inverse semigroups over semilattices

D. B. McAlister
1978 Glasgow Mathematical Journal
1. Introduction. An inverse semigroup is called E-unitary if the equations ea = e = e 2 together imply a 2 = a. In a previous paper  , the author showed that any E-unitary inverse semigroup is isomorphic to a semigroup constructed from a triple (G, 26, "3/) consisting of a down-directed partially ordered set 26, an ideal and subsemilattice % of 2£ and a group G acting on 26, on the left, by order automorphisms in such a way that 36 = G<&. This semigroup is denoted by P{G, 26, %; it consists
more » ... 26, %; it consists of all pairs (a,g)e< ! S/xG such that g^a e % under the multiplication (a, g)(b, h) = (aAgb, gh). The aim of this paper is to give necessary and sufficient conditions on an inverse semigroup in order that it should be isomorphic to some P(G, 26, %) with 26 a semilattice. As well, we consider those congruences p on an inverse semigroup P(G, 26, %) for which the quotient has the form P(H, % V) for some triple H, % Y as above, with °U a semilattice. We shall assume familiarity with the construction and properties of P(G, 26, <30 from , . Undefined notation and terminology is that of Clifford and Preston . In particular, when we are considering a partial order on an inverse semigroup, the partial order being referred to is the natural partial order; it is defined by a ^ b if and only if a = eb for some e 2 = eeS. Throughout the paper, when the terminology "triple {G,26,^/)" is used, it means that 26 is a down-directed partially ordered set with "2/ an ideal and subsemilattice of 26, and that G is a group acting on 26 by order automorphisms in such a way that 26 = Gty. DEFINITION 1.1. Let S be an inverse semigroup. Then we say that S is an E-unitary inverse semigroup over a semilattice if S = P(G, 26, "3/) for some triple (G, 26, "30 with SC a semilattice. In terms of Definition 1.1, the aim of this paper is therefore to characterize fi-unitary inverse semigroups over a semilattice. The general case DEFINITION 2.1. Let S be a partially ordered set and let 0:S-> T be a mapping of S into a set T. Then 6 is an m-map if, for each teT, the set {seS:s6 = t} has a maximum member.