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Decision problems for word-hyperbolic semigroups

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Pfeiffer_2016_JA_Word_hyberbolicSemigroups_AM.pdf (447.6Kb)
Date
01/11/2016
Author
Cain, Alan James
Pfeiffer, Markus Johannes
Keywords
Word-hyperbolic semigroups
Decision problems
Undecidability
Isomorphism problem
Context-free languages
QA Mathematics
NDAS
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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
DOI
https://doi.org/10.1016/j.jalgebra.2016.07.007
ISSN
0021-8693
Type
Journal article
Rights
© 2016, Elsevier Ltd. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.jalgebra.2016.07.007
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/11263

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