Generators and relations for subsemigroups via boundaries in Cayley graphs

Gray, Robert D.; Ruškuc, Nik

doi:10.1016/j.jpaa.2011.03.017

Cited by 5 publications

(5 citation statements)

References 25 publications

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“…Next we claim that U is actually generated by {π M (s l ) : l ∈ L}. This is proved by a very similar argument to the one in the previous paragraph, starting from the fact that {s l : l ∈ L} is a generating set for H, expressing the generators red(r ij ) in terms of the s l , and using (7). Furthermore, the generators {π M (s l ) : l ∈ L} satisfy the relations r ′ i = 1.…”

Section: Makanin-style Presentation Theorems For Special Inverse Monoidsmentioning

confidence: 59%

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

Gray,

Ruskuc

2021

Preprint

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We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations where all the defining relations are of the form r = 1. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results to not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented. We also apply these results to establish a close connection between the question of finite presentability of the groups of units of special one-relator inverse monoids, and the open problem of whether all one-relator groups are coherent. In addition, we give a new approach via regular languages for computing invertible pieces of relators in special inverse monoids. We prove that this algorithm outperforms the classical Adjan overlap algorithm when applied to special inverse monoids, in the sense that it discovers more invertible pieces of relators.

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Section: Makanin-style Presentation Theorems For Special Inverse Monoidsmentioning

confidence: 59%

“…The fact that finite presentability is inherited by submonoids with ideal complement is well know e.g. it is a corollary of [7,Theorem B]. Therefore K Q * T W is finitely presented which implies that K Q and T W are both finitely presented being retracts of K Q * T W .…”

Section: A Constructionmentioning

confidence: 90%

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

Gray,

Ruskuc

2021

Preprint

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“…) is finitely presented, then by part (iv), it follows that V Q,W is finitely presented since its complement is an ideal. The fact that finite presentability is inherited by submonoids with ideal complement is well-known, for example, it is a corollary of [8,Theorem B]. Therefore, K Q * T W is finitely presented, which implies that K Q and T W are both finitely presented, being retracts of K Q * T W .…”

Section: The Constructionmentioning

confidence: 99%

On Groups of Units of Special and One-Relator Inverse Monoids

Gray,

Ruškuc

2023

J. Inst. Math. Jussieu

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We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$ . We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

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“…e.g. [28,44,20,21]). Even answering the question of what a "virtually free" monoid should be is hence already a non-trivial task.…”

Section: Introductionmentioning

confidence: 99%

Non-finitely Generated Maximal Subgroups of Context-free Monoids

Nyberg-Brodda¹

2021

Preprint

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A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.

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Generators and relations for subsemigroups via boundaries in Cayley graphs

Cited by 5 publications

References 25 publications

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

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

On Groups of Units of Special and One-Relator Inverse Monoids

Non-finitely Generated Maximal Subgroups of Context-free Monoids

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