Semigroups through semilattices

Abstract

Presented in this paper is a method of constructing a compact semi- group 5 from a compact semilattice Xand a compact semigroup T having idempotents contained in X. The notions of semigroups (straight) through chains and (straight) through semilattices are introduced. It is shown that the notion of a semigroup through a chain is equivalent to that of a generalized hormos. Universal objects are obtained in several categories including the category of clans straight through a chain and the category of clans straight through a semilattice relative to a chain. An example is given of a nonabelian clan S with abelian set of idempotents E such that S is minimal (as a clan) about E. 1. Introduction. In their book Elements of compact semigroups K. H. Hofmann and P. S. Mostert present a technique for constructing a compact topological semi- group called a generalized hormos beginning with a totally ordered compact chain. Hofmann and Mostert raise the question of whether the compact chain can be replaced by a compact topological semilattice (2, p. 160). The purpose of this paper is to present one method for constructing semigroups from such semilattices. In §3 we give a definition of a semigroup through a chain and prove that the class of such semigroups coincides with the class of generalized hormi. In §4 we generalize the concept of a semigroup through a chain to the concept of a semigroup through a semilattice. Such semigroups consist of a generalized hormos through a chain in the semilattice united with homomorphic images of the generalized hormos through translates of the chain. Although we do not give a general construction for all semigroups through a semilattice, we do construct a universal one for a fixed semilattice and generalized hormos. In §5 we consider a slightly different class of semigroups and again construct a universal semigroup in this class. Commutative semigroups which are irreducible about their idempotents are examples of semigroups through semilattices. We briefly discuss this class of examples in §6.