On Representations of Semigroups

Abstract

Semigroup representations are one of the oldest areas in semigroup theory. In 1933, Suschkewitch published the first paper on the topic. Since then, the area has been approached largely by Clifford, Munn, Ponizovskii, and then by Hewitt and Zuckerman, Lallement and Petrich, Preston, McAlister, and Rhodes. Following an intense period of development during the 1950’s and 1960’s, the theory witnessed a dormant era during the 1970’s and 1980’s. There was a resurgence of interest in the subject in the late 1990’s in the work of Putcha, Brown and others. The lack of continuity of research in the theory is intriguing. This thesis addresses the discontinuous development of the theory and the reasons behind it. The Clifford-Munn-Ponizovskii correspondence states that the irreducible representations of a semigroup are in one-to-one correspondence with the irreducible representations of its maximal subgroups. Since the principal approach to identify representations of semigroups is this correspondence, we start with the observation that the lack of interest in semigroup representation theory could have been because the Clifford-Munn-Ponizovskii theory reduces the whole problem of finding irreducible representations to group representation theory. It turns out that this is a wrong assumption. It is not clear that during the dormant period the correspondence was widely known. By the time Munn and others stopped working on the theory, the correspondence was not stated in a fully-fledged form. The Clifford-Munn-Ponizovskii correspondence was subsequently formulated by others and emerged much later than in the work of Clifford and Munn. In the thesis we first discuss the subject using modern mathematical language, starting with groups and then with semigroups. With this hindsight, we then turn to the subject as it was developed for groups and semigroups.