Groups and semigroups: connections and contrasts

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. Thus it is not surprising that the two subjects have developed in somewhat different directions. However, there are several areas of modern semigroup theory that are closely connected to group theory, sometimes in rather surprising ways. For example, central problems in finite semigroup theory (which is closely connected to automata theory and formal language theory) turn out to be equivalent or at least very closely related to problems about profinite groups. Linear algebraic monoids have a rich structure that is closely related to the subgroup structure of the group of units, and this has interesting connections with the well developed theory of (von Neumann) regular semigroups. The theory of inverse semigroups (i.e. semigroups of partial one-one functions) is closely tied to aspects of geometric and combinatorial group theory. In the present paper, I will discuss some of these connections between group theory and semigroup theory, and I will also discuss some rather surprising contrasts between the theories. While I will briefly mention some aspects of finite semigroup theory, regular semigroup theory, and the theory of linear algebraic monoids, I will focus primarily on the theory of inverse semigroups and its connections with geometric group theory. For most of what I will discuss, there is no loss of generality in assuming that the semigroups under consideration have an identity one can always just adjoin an identity to a semigroup if necessary so most semigroups under consideration will be monoids, and on occasions the group of units (i.e. the group of invertible elements of the semigroup) will be of considerable interest.