Amalgams of inverse semigroups and reversible two-counter machines

TL;DR

This paper demonstrates that the word problem for certain amalgams of inverse semigroups can be undecidable, even under conditions previously thought to ensure decidability, by encoding universal 2-counter machines.

Contribution

It introduces a construction showing undecidability of the word problem in amalgams of inverse semigroups with finite -classes, challenging prior positive results.

Findings

Undecidability of the word problem in specific inverse semigroup amalgams.

Encoding of universal 2-counter machines into inverse semigroup structures.

Relationship between -graph properties and machine computations.

Abstract

We show that the word problem for an amalgam $[S_1,S_2;U,\omega_1,\omega_2]$ of inverse semigroups may be undecidable even if we assume $S_1$ and $S_2$ (and therefore $U$) to have finite $\mathcal{R}$-classes and $\omega_1,\omega_2$ 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 \sch graphs to sequences of computations in the machine.