Amalgams of inverse semigroups and reversible two-counter machines

Abstract

We show that the word problem for an amalgam [S 1 , S 2 ; U, ω 1 , ω 2 ] of inverse semigroups may be undecidable even if we assume S 1 and S 2 (and therefore U ) to have finite R-classes and ω 1 , ω 2 to be computable functions, interrupting a series of positive decidability results on the subject. This is achieved by encoding into an appropriate amalgam of inverse semigroups 2-counter machines with sufficient universality, and relating the nature of certain Schützenberger graphs to sequences of computations in the machine.

Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups

“…In the next subsections, we recall some basic definitions of 2-counter machines, and we finally give an idea of the encoding which allow to reduce the word problem to the halting problem for such machines. Although [28] takes inspiration in the usage of Minsky machines from [20,29], there are several technical differences, starting from the necessity of considering particular subclasses of 2-counter machines more suitable to deal with inverse semigroups.…”

“…The word problem is decidable for amalgamated free products of groups and is undecidable for amalgamated free products of semigroups (even when the two semigroups are finite [29]) but it is not known under which conditions on the inverse semigroups the word problem for amalgamated free products is decidable in the category of inverse semigroups (Problem 5 of [21]). In the sequel, we will briefly illustrate some sufficient conditions on amalgams of inverse semigroups for the word problem being decidable in the amalgamated free products [7,8] and a negative recent result [28]. The paper is organized as follows.…”

Maximal subgroups of amalgams of finite inverse semigroups

“…Munn's work was greatly extended by Stephen [24] who introduced the notion of Schützenberger graphs associated with presentations of inverse semigroups. These graphs were widely used in the study algorithmic problems and the structure of several classes of inverse semigroups (see, for instance [3,4,5,6,7,8,10,12,13,20,21,22,25]). In particular Haataja, Margolis and Meakin were the first to show that Bass-Serre theory may be applied to study the structure of maximal subgroups, to obtain results for amalgams of inverse semigroups where U contained all idempotents of S 1 and S 2 .…”