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.
Citation
Cain , A J & Pfeiffer , M J 2016 , ' Decision problems for word-hyperbolic semigroups ' , Journal of Algebra , vol. 465 , pp. 287-321 . https://doi.org/10.1016/j.jalgebra.2016.07.007
Publication
Journal of Algebra
Status
Peer reviewed
ISSN
0021-8693Type
Journal article
Collections
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