Presentations of inverse monoids

Abstract

The theory of generators and relations for groups is closely related to the geometry of certain digraphs with edges labeled from the generating set. Examples of this connection are provided by the Cayley graph of a group and the coset enumeration scheme of Coxeter. This paper introduces the author's recent work aimed at developing a theory of presentations of inverse monoids analogous to the theory of generators and relations for groups. We regard inverse monoids as a variety of type «2, 1, 0å and study presentations from this point of view. This paper is primarily concerned with the word problem for inverse monoid presentations. We develop the general construction of birooted, labeled digraphs associated with an inverse monoid presentation and show how they can be used to approach the word problem. We indicate several cases in which the word problem can be solved using these techniques.