The free local semilattice on a set

Abstract

Following McAlister [3] we call a semigroup S focalfy inverse if S is regular and eSe is an inverse subsemigroup of S for each idempotent e of S. Such semigroups were introduced and studied by Nambooripad [lo] who called them ‘pseudoinverse’ semigroups. Relative to the ‘basic products’, the set of idempotents of a regular semigroup forms a partial binary algebra which has been axiomatically characterized as a ‘biordered set’ by Nambooripad [9]. The biordered set of idempotents of a locally inverse semigroup is called a local semilattice in this paper. Local semilattices were called ‘partially associative pseudo-semilattices’ by Nambooripad [lo] and ‘pseudo-semilattices’ by some subsequent authors [ 11, [5], [6], [7]. The present author used the terminology ‘local semilattice’ in [4]. A local semilattice may be viewed in several equivalent ways: (i) as a biordered set E in which iS(e,f) = 1 for all e, f E E (see Nambooripad [91v [lOI); (ii) as a biordered set E in which o(e) is a semilattice for each e E E; (iii) as a set E, together with two quasi-orders o’ and CU’ which satisfy conditions (PAl) and (PA2) of [lo] and their duals; (iv) as a binary algebra (E, A) in which A satisfies the following identities and their duals: (a) xAx=x, (b) (xAy)A(XAz) = (xAY)Az; (c) (xA_Y)A((xAz)A(xAu)) = ((~AY)A(xAz))A(xI-w). We refer the reader to the papers of Nambooripad [9], [lo] and Meakin and Pastijn [6], [7] for a discussion of the equivalence between these methods of viewing local semilattices and for all relevant notation and terminology. In particular, we regard the 0’ and o’ relations in a local semilattice (E, A) as being defined in such a way that the binary operation A extends the basic products. With this convention we have, for e, f e E,