Context-Free Rewriting Systems and Word-Hyperbolic Structures With Uniqueness

Cain, Alan J.; Maltcev, Victor

doi:10.1142/s0218196712500610

Cited by 15 publications

(21 citation statements)

References 10 publications

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“…For automatic semigroups, it is possible to assume that the automatic structure has a further pleasant property, namely that every element of the semigroup is represented by a unique word in the language of representatives [KO06, Proposition 2.9(iii)]. However, there exist word-hyperbolic semigroups (indeed, word-hyperbolic monoids) that do not admit word-hyperbolic structures where the languages of representatives have this uniqueness property [CM12,Examples 10 & 11].…”

Section: 13mentioning

confidence: 99%

“…This generalization has led to a substantial amount of research on word-hyperbolic semigroups; see, for example, [CM12,FK04,HKOT02,HT03]. Some of this work has shown that word-hyperbolic semigroups do not possess such pleasant properties as word-hyperbolic groups: they may not be finitely presented, and they are not in general automatic or even asynchronously automatic [HKOT02, Example 7.7 et seq.…”

Section: Introductionmentioning

confidence: 99%

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Decision problems for word-hyperbolic semigroups

Cain

Pfeiffer

2016

Journal of Algebra

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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 exponentialtime 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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Section: 13mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Decision problems for word-hyperbolic semigroups

Cain

Pfeiffer

2016

Journal of Algebra

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“…Gray and Kambites were not the first to look for a notion of hyperbolicity for monoids but their notion is the first geometric one that takes directions into account. Previous definitions either were based on context-free rewriting systems, see e. g. [6,7,12,13,19], or looked at the underlying undirected graph of Cayley digraphs of semigroups, see e. g. [6,8,13]. Portilla et al [22] considered a notion of hyperbolicity in (a certain class of) digraphs that is also defined via hyperbolicity of their underlying undirected graphs.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolicity in semimetric spaces, digraphs and semigroups

Hamann¹

2021

Preprint

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Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We carry over some of the fundamental results of hyperbolic spaces to this new setting. In particular, we prove under some small additional geometric assumption that their notion of hyperbolicity is preserved by quasi-isometries. We also construct a boundary based on quasi-geodesic rays and anti-rays that is preserved by quasi-isometries and, in the case of locally finite digraphs, refines their ends. We show that it is possible to equip the space, if it is finitely based, with its boundary with a pseudo-semimetric and show some further results for the boundary. We also apply our results to semigroups and give a partial solution to a problem of Gray and Kambites.Funded by the German Research Foundation (DFG) -project number 448831303.Proposition 5.4. Let X be a strongly hyperbolic geodesic semimetric space that satisfies (B1) for the function f : R → R.(i) If P, Q, R are the sides of a geodesic triangle, then we havefor all ε > 0. (ii) If x, y ∈ V (D) with d(x, y) = ∞ and d(y, x) = ∞, then we have d(x, y) ≤ (d(y, x)/ε)f (δ + ε) + f (δ) for all ε > 0.Now we are able to prove that strong hyperbolicity implies hyperbolicity if (B1) is satisfied.Lemma 5.5. Every strongly hyperbolic geodesic semimetric space that satisfies (B1) or (B2) is hyperbolic.

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“…When one generalizes from groups to semigroups, there is some geometry on Cayley graphs: for instance, there are several possible definitions of hyperbolicity for semigroups [6,9,15]; one can also define ends of finitely generated semigroups and prove results about them similar to those about ends of groups [23,24]. For This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).…”

Section: Introductionmentioning

confidence: 99%

Growths of Endomorphisms of Finitely Generated Semigroups

Cain¹,

Maltcev

2016

J. Aust. Math. Soc.

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This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

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Context-Free Rewriting Systems and Word-Hyperbolic Structures With Uniqueness

Cited by 15 publications

References 10 publications

Decision problems for word-hyperbolic semigroups

Decision problems for word-hyperbolic semigroups

Hyperbolicity in semimetric spaces, digraphs and semigroups

Growths of Endomorphisms of Finitely Generated Semigroups

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