The word problem for one-relation monoids: a survey

Brodda, Carl-Fredrik Nyberg

doi:10.1007/s00233-021-10216-8

Cited by 11 publications

(4 citation statements)

References 132 publications

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“…To prove this, we use an auxiliary result that is often labelled as folklore (such as in [5,26,29]). A proof of the result may be found in [4, Proposition 0.9.1], which in turn makes use of [ The previous corollary immediately raises the question: Which group-embeddable recursively presented monoids arise as prefix monoids?…”

Section: Proposition 31 a Finitely Generated Monoid Embeds Into A Fin...mentioning

confidence: 99%

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Prefix monoids of groups and right units of special inverse monoids

Dolinka,

Gray

2023

Forum of Mathematics, Sigma

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A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

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Section: Proposition 31 a Finitely Generated Monoid Embeds Into A Fin...mentioning

confidence: 99%

“…However, the question of decidability of the word problem for one-relator monoids has been a subject of a series of serious attacks, most notably by Adian [1] and his students, among others. We refer to the survey [29] for a thorough historical account of this fascinating topic.…”

Section: Introductionmentioning

confidence: 99%

Prefix monoids of groups and right units of special inverse monoids

Dolinka,

Gray

2023

Forum of Mathematics, Sigma

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“…Is it decidable whether M has context-free word problem? Note that whereas it is still an open problem whether the word problem for all onerelation monoids is decidable (see Nyberg-Brodda [62] for a survey of this problem), this does not preclude classifying those one-relation monoids with context-free word problem; cf. e.g.…”

Section: Some Decision Problemsmentioning

confidence: 99%

On the word problem for special monoids

Nyberg-Brodda

2022

Semigroup Forum

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A monoid is called special if it admits a presentation in which all defining relations are of the form $$w = 1$$ w = 1 . Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

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“…[8], whether all one-relator groups are coherent. Recently, this problem has shown to be equivalent to a problem concerning the group of units of certain one-relation inverse semigroups, and thereby also to the word problem for one-relation monoids, see [32]; see also [68,69].…”

Section: Corollary 14 Let H Be Any Finitely Generated Subgroup Of a O...mentioning

confidence: 99%

On the Diophantine problem in some one-relator groups

Nyberg-Brodda¹

2022

Preprint

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We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister-Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation a m b n = 1 is virtually a direct product of hyperbolic groups for all m, n ≥ 0, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the twogenerator, one-relator groups Gm,n with defining relation a commutator [a m , b n ] = 1, where m, n ≥ 1. In doing so, we define and study a natural class of groups (rabsags), related to right-angled Artin groups (raags). We reduce the Diophantine problem in the groups Gm,n to the Diophantine problem in groups which are virtually certain rabsags. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group G 2,2 with the single defining relation [a 2 , b 2 ] = 1. We use the recent classification by Gray & Howie of raag subgroups of one-relator groups to classify the raag subgroups of some rabsags, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups NG(p, q), which are (p + 1)-generated one-relator groups generalising the solvable Baumslag-Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.

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The word problem for one-relation monoids: a survey

Cited by 11 publications

References 132 publications

Prefix monoids of groups and right units of special inverse monoids

Prefix monoids of groups and right units of special inverse monoids

On the word problem for special monoids

On the Diophantine problem in some one-relator groups

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