3 Inverse Monoids

Abstract

An important class of regular monoids is the class of inverse monoids. In fact, many semigroup theorists would assert that inverse monoids form the most important class of monoids outside of groups. They abstract the notion of partial symmetry in much the same way that groups abstract the notion of symmetry. For a detailed discussion of this viewpoint, see Lawson [Law98]. From the perspective of this book they provide a natural class of monoids whose representation theory we can understand as well as that of groups. Namely, we shall see in Chapter 9 that the algebra of an inverse monoid can be explicitly decomposed as a direct product of matrix algebras over the group algebras of its maximal subgroups (one per \(\mathcal{J}\)-class).