The maximal semilattice decomposition of a semigroup

Abstract

A semigroup is a non-empty set on which an associative multiplication is defined. A semilattice is a commutative semigroup each of whose elements is idempotent. The maximal semilattice homomorphic image of an arbitrary semigroup S is the semilattice Y such that every semilattice homomorphic image of S is also a homomorphic image of Y. The maximal semilattice decomposition of S is the decomposition of S into equivalence classes which are complete inverse images of members of Y. We identify these classes with members of Y. YAMADA [20] and CLIFFORD [3] have characterized such a decomposition for an arbitrary semigroup, TnlERRIN [18] for a groupoid, TAMURA and KIMURA [15] for a commutative semigroup, and MCLEAN [11] for a band. We give several characterizations of the maximal semilattice decomposition for arbitrary semigroups and others for some special classes of semigroups. Further, we consider some properties of members of such a decomposition. In section 2 we give a characterization of all semilattice decompositions of S in terms of prime ideals of S (2.3). In section 3 we show that a member N~ (containing x~S) of the maximal semilattice decomposition Y of S is the set of all elements y of S such that the smallest face of S containing x is also the smallest face of S containing y (3.2). Moreover, we show that no ideal of any N~ contains a proper prime ideal (3.4). These results are of fundamental importance in the remainder of the paper. We prove, among other things, that N x is the largest subsemigroup of S containing x and containing no proper prime ideals (3.10). In section 4 we establish several properties of each N, in terms of properties of elements of S and in terms of (several kinds of) ideals of S. Here we partly make use of the decomposition theory of CROISOT [6] and CLIFFORD [1], [2]. In section 5 we perform a similar analysis as in section 4 under the additional hypothesis that Y be linearly ordered. Finally, in section 6 we give explicit expressions for the smallest face of S containing an element x of S for some classes of semigroups. In particular we do this for a class of semigroups considered by THIERRIN [19] (6.1), a class of semigroups which we call weakly commutative (6.5), and a class of periodic semigroups (6.7 and 6.9).