Publication
Decision problems for word-hyperbolic semigroups
| Summary: | 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. |
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| Main Authors: | Cain, Alan J. |
| Other Authors: | Pfeiffer, Markus |
| Subject: | Context-free languages Decision problems Isomorphism problem Undecidability Word-hyperbolic semigroups Algebra and Number Theory |
| Year: | 2016 |
| Country: | Portugal |
| Document type: | article |
| Access type: | open access |
| Associated institution: | Universidade Nova de Lisboa |
| Language: | English |
| Origin: | Repositório Institucional da UNL |
| Summary: | 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. |
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