E-unitary inverse monoids and the Cayley graph of a group presentation

Abstract

Geometric methods have played a fundamental and crucial role in combinatorial group theory almost from the inception of that field. In this paper we initiate a study of the use of some of these methods in inverse semigroup theory. We modify a lemma of I. Simon and show how to construct E-unitary inverse monoids from the free idempotent and commutative category over the Cayley graph of the maximal group image. The construction provides an expansion from the category of X-generated groups to the category of X-generated E-unitary inverse monoids and specializes to a construction of certain relatively free E-unitary inverse monoids. We show more generally that this construction is the left adjoint of the maximal group image functor. Munn's solution to the word problem for the free inverse monoids and several of the results of McAlister and McFadden on the free inverse semigroup with two commuting generators may be obtained fairly easily from the construction. We construct the free product of E-unitary inverse monoids, thus providing an alternate construction to that of Jones.