Fundamental semilattices of semigroups

Abstract

We introduce the notion of a fundamental semilattice of semigroups, which is a special case of a semilattice of semigroups and a generalization of a strong semilattice of semigroups. The relations among fundamental semilattices of semigroups, refined semilattices of semigroups (originally introduced by L. Zhang, K. P. Shum and R. H. Zhang [21]) and regular semilattices of semigroups (introduced by Z. P. Wang and Y. L. Zhou [17]) are discussed. As an application, we prove that a semigroup is completely regular if and only if it is a fundamental semilattice of completely simple semigroups. The relation between this structure theorem and the theorem on completely regular semigroups by Lallement is also illustrated. The structure of some special types of completely regular semigroups is also described as fundamental semilattices of completely simple semigroups.