Decision problems for word-hyperbolic semigroups

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.