On one-relator inverse monoids and one-relator groups

Abstract

It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon A w = 1 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M = Mon A u = v where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation of the form MWe show that a positive solution to the word problem for such monoids for all reduced words w would imply a positive solution to the word problem for all one-relation monoids. We prove a conjecture of Margolis, Meakin and Stephen by showing that every inverse monoid of the form M = Inv A w = 1 , where w is cyclically reduced, must be E-unitary. As a consequence the word problem for such an inverse monoid is reduced to the membership problem for the submonoid of the corresponding one-relator group G = Gp A w = 1 generated by the prefixes of the cyclically reduced word w. This enables us to solve the word problem for inverse monoids of this type in certain cases.

Groups and semigroups: connections and contrasts

“…In this case, Ivanov, Margolis and Meakin [64] proved the following structural result about such monoids.…”

“…Consider the associated one relator inverse monoid M = Inv X : aubc −1 v −1 a −1 = 1 . Note that aubc −1 v −1 a −1 is a reduced word (but not a cyclically reduced word) since b = c. In [64], Ivanov, Margolis and Meakin show, by using some results of Adian and Oganessian [3] and by using the inverse hull of the semigroup S, that S embeds in M . It follows that if the word problem is decidable for M , then it must also be decidable for S.…”

“…In [64], Ivanov, Margolis and Meakin made heavy use of Schützenberger graphs as well as van Kampen diagram arguments to study one relator inverse monoids. In view of the difficulty of solving the word problem for one-relation semigroups of the form Sgp X : u = v where u, v ∈ X + , I will restrict attention to one-relator inverse monoids of the form Inv X : w = 1 .…”

“…However, in general this problem remains unsolved. In fact, the following result in [64] shows that this problem is at least as difficult as the one-relation semigroup problem, even if we restrict to the case where w is a reduced word.…”

“…Consequently, we need to show that every word labelling the boundary of every van Kampen diagram over G must be an idempotent in M . A lemma in [64] shows that this happens if, for every van Kampen diagram ∆ over G, there is some cell Π of ∆ and some vertex α ∈ ∂∆ ∩ ∂Π such that the cyclic conjugate of w ±1 obtained by reading around the boundary of Π, starting at α, is a unit cyclic conjugate of w ±1 . In this situation, we say that a unit cyclic conjugate of w starts on the boundary of ∆.…”