Decision problems for word-hyperbolic semigroups

TL;DR

This paper investigates decision problems for word-hyperbolic semigroups, revealing undecidability results and providing polynomial-time algorithms for the uniform word problem, thereby advancing understanding of their computational properties.

Contribution

It refines the definition of word-hyperbolic structures, proves the undecidability of the isomorphism and automaticity problems, and offers efficient algorithms for key decision problems.

Findings

Isomorphism problem is undecidable for word-hyperbolic semigroups.

Deciding automatic, biautomatic properties is undecidable.

Uniform word problem is solvable in polynomial time.

Abstract

This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.