The word problem for some classes of Adian inverse semigroups

Abstract

Abstract. We show that all of the Schützenberger complexes of an Adian inverse semigroup are finite if the Schützenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).

New results on the prefix membership problem for one-relator groups

“…Motivated by the above discussion, both the word problem for one-relator inverse monoids and the prefix membership problem for one-relator groups, with cyclically reduced defining relator, have already received a great deal of attention in the literature; see e.g. [14,16,19,20,32,34] and [5,Question 13.10]. In this paper we will make several new contributions towards resolving these open problems.…”

“…We finish this subsection by presenting yet another application of Theorem 7.2 which concerns the prefix membership problem for one-relator groups defined by Adjan presentations [16,32] over a two-letter alphabet. Recall that a one-relator group, inverse monoid, or monoid presentation is an Adjan presentation if it is of the form X | u = v , where u, v ∈ X * are positive words such that the first letters of u, v are different, and also the last letters of u, v are different.…”