Some Decision Problems for Inverse Monoid Presentations

This paper surveys some of the authors' recent and ongoing 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 class of algebras of type 〈2,1,0〉 and study presentations of inverse monoids from this point of view. The paper is concerned with two basic decision problems for inverse monoid presentations: the word problem and the E-unitary problem. We develop the general construction of a birooted word graph associated with an inverse monoid presentation and show how it can be used as a basic tool in the study of the word problem. We indicate several cases in which the word problem can be solved using these techniques. We study the E-unitary problem for inverse monoids of the form M=Inv〈X|w=1〉 where w is in the free inverse monoid on X. We show how the Lyndon diagrams of combinatorial group theory may be used to analyze the problem and we study several examples and special cases in detail.