Groups and semigroups: connections and contrasts
John C. Meakin · Cambridge University Press eBooks · 2007
Introduction Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley's theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set to itself. Of course both group theory and semigroup theory have developed significantly beyond these early viewpoints, and both subjects are by now integrally woven into the fabric of modern mathematics, with connections and applications across a broad spectrum of areas. Nevertheless, the early viewpoints of groups as groups of permutations, and semigroups as semigroups of functions, do permeate the modern literature: for example, when groups act on a set or a space, they act by permutations (or isometries, or automorphisms, etc.), whereas semigroup actions are by functions (or endomorphisms, or partial isometries, etc.). Finite dimensional linear representations of groups are representations by invertible matrices, while finite dimensional linear representations of semigroups are representations by arbitrary (not necessarily invertible) matrices. The basic structure theories for groups and semigroups are quite different — one uses the ideal structure of a semigroup to give information about the semigroup for example — and the study of homomorphisms between semigroups is complicated by the fact that a congruence on a semigroup is not in general determined by one congruence class, as is the case for groups.