New results on the prefix membership problem for one-relator groups

Dolinka, Igor; Gray, Robert D.

doi:10.1090/tran/8338

Cited by 5 publications

(5 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many equivalent definitions, but for our purposes, it is the best to express it in the following way: 𝑇 = Inv 𝐴 | 𝑤 𝑖 = 1 (𝑖 ∈ 𝐼) is E-unitary if in the natural homomorphism from T to its greatest group image 𝐺 = Gp 𝐴 | 𝑤 𝑖 = 1 (𝑖 ∈ 𝐼) , the only pre-images of the identity element of G are the idempotents of T. Now, yet another major result of [15] shows that when 𝑇 = Inv 𝐴 | 𝑤 𝑖 = 1 (𝑖 ∈ 𝐼) is E-unitary then, provided Gp 𝐴 | 𝑤 𝑖 = 1 (𝑖 ∈ 𝐼) has decidable word problem, the decidability of the word problem of T reduces to the membership problem for the prefix monoid of G. In the same paper, they also prove that when w is a cyclically reduced word, then Inv 𝐴 | 𝑤 = 1 is E-unitary, and thus the word problem reduces to the prefix membership problem for the corresponding one-relator group in this case. The prefix membership problem for one-relator groups has been studied by several authors (e.g., in [16,25]) and more recently by the present authors in [7]. The close connections between prefix monoids of groups and right units of special inverse monoids described above mean that it is natural to investigate both classes in parallel.…”

Section: Introductionmentioning

confidence: 80%

“…In the same paper, they also prove that when w is a cyclically reduced word, then is E -unitary, and thus the word problem reduces to the prefix membership problem for the corresponding one-relator group in this case. The prefix membership problem for one-relator groups has been studied by several authors (e.g., in [16, 25]) and more recently by the present authors in [7]. The close connections between prefix monoids of groups and right units of special inverse monoids described above mean that it is natural to investigate both classes in parallel.…”

Section: Introductionmentioning

confidence: 92%

“…Hence, The word on the right-hand side is also already in a reduced form. However, the condition and the very definition of M implies that there must be words with for all such that This is now an equality of two reduced forms in the HNN-extension , and by [7, Lemma 6.2], it is impossible unless and . However, it is clear that any element of is a unit in M , so .…”

Section: Groups Of Units and Schützenberger Groups Of Prefix Monoidsmentioning

confidence: 99%

See 2 more Smart Citations

Prefix monoids of groups and right units of special inverse monoids

Dolinka,

Gray

2023

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 92%

Section: Groups Of Units and Schützenberger Groups Of Prefix Monoidsmentioning

confidence: 99%

See 1 more Smart Citation

Prefix monoids of groups and right units of special inverse monoids

Dolinka,

Gray

2023

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

show abstract

“…As mentioned in the introduction, the prefix membership problem for one-relator groups has been shown to be decidable in a number of special cases. See also [8] for some more recent results showing that the prefix membership problem is decidable for certain classes of one-relator groups which are low down in the Magnus-Moldovanskii hierarchy.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

Gray

2019

Invent. math.

Self Cite

View full text Add to dashboard Cite

We prove the following results: (1) There is a one-relator inverse monoid Inv A | w = 1 with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg from 2008, we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid Inv A|w = 1 with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis, Meakin and Stephen in 1987.

show abstract

“…Λ = {ad, acd, abcd, abbcd}, whereas the Adian algorithm would incorrectly yield Λ = {abcdacdadabbcdacd}. We remark that the word problem for O was solved by Dolinka & Gray[15, Proposition 5.4].In view of the above example, if w is a word which is (1) positive; and (2) self-overlap free; and (3) the factorisation into minimal invertible factors of w in Inv A | w = 1 is non-trivial, then we say that Inv A | w = 1 is O'Haresque. In §4 we shall provide an infinite family of considerably simpler examples of O'Haresque monoids.…”

mentioning

confidence: 99%

On one-relator groups and units of special one-relation inverse monoids

Nyberg-Brodda¹

2021

Preprint

View full text Add to dashboard Cite

This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form Inv A | w = 1 . We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes any, red, cred, pos of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions any ⊃ cred ⊃ pos are all strict. Conditional on a natural conjecture, we prove any ⊃ red. Following this, we use the Benois algorithm recently devised by Gray & Ruškuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behaviour (which we term O'Haresque) to the O'Hare monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray & Ruškuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.

show abstract

New results on the prefix membership problem for one-relator groups

Cited by 5 publications

References 39 publications

Prefix monoids of groups and right units of special inverse monoids

Prefix monoids of groups and right units of special inverse monoids

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

On one-relator groups and units of special one-relation inverse monoids

Contact Info

Product

Resources

About