Decision problems for word-hyperbolic semigroups

Alan J. Cain, Markus Pfeiffer

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This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan and 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. (This does not alter the class of word-hyperbolic semigroups.) 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 semigroups 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.

Original languageEnglish
Pages (from-to)287-321
Number of pages35
JournalJournal of Algebra
Publication statusPublished - 1 Nov 2016


  • Context-free languages
  • Decision problems
  • Isomorphism problem
  • Undecidability
  • Word-hyperbolic semigroups


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