RECORD
Maximal subgroups of finitely presented special inverse monoids
Group Theory · Submitted 8 Dec 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the $E$-unitary case.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We study the maximal subgroups (also known as group $\mathcal{H}$-classes) of finitely presented special inverse monoids. We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the $E$-unitary case. We also prove that the possible groups of units are exactly the finitely generated recursively presented groups; this improves upon a result of, and answers a question of, the first author and Ru\v{s}kuc. These results give the first significant insight into the maximal subgroups of such monoids beyond the group of units, and the results together demonstrate that it is possible for the subgroup structure to have a complexity which significantly exceeds that of the group of units. We also observe that a finitely presented special inverse monoid (even an $E$-unitary one) may have infinitely many pairwise non-isomorphic maximal subgroups.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the maximal subgroups (also known as group $\mathcal{H}$-classes) of finitely presented special inverse monoids. | contribution 0.72 | 0.82 | 0.04 | asserted 0.65 |
| s1 | We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the $E$-unitary case. | result 1.00 | 0.96 | 0.77 | asserted 1.00 |
| s2 | We also prove that the possible groups of units are exactly the finitely generated recursively presented groups; this improves upon a result of, and answers a question of, the first author and Ru\v{s}kuc. | result 0.99 | 0.91 | 0.14 | asserted 1.00 |
| s3 | These results give the first significant insight into the maximal subgroups of such monoids beyond the group of units, and the results together demonstrate that it is possible for the subgroup structure to have a complexity which significantly exceeds that of the group of units. | interpretation 1.00 | 0.88 | 0.14 | asserted 0.99 |
| s4 | We also observe that a finitely presented special inverse monoid (even an $E$-unitary one) may have infinitely many pairwise non-isomorphic maximal subgroups. | result 1.00 | 0.92 | 0.07 | hedged possibility 0.95 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We study the maximal subgroups (also known as group $\mathcal{H}$-classes) of finitely presented special inverse monoids. We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the $E$-unitary case. We also prove that the possible groups of units are exactly the finitely generated recursively presented groups; this improves upon a result of, and answers a question of, the first author and Ru\v{s}kuc. These results give the first significant insight into the maximal subgroups of such monoids beyond the group of units, and the results together demonstrate that it is possible for the subgroup structure to have a complexity which significantly exceeds that of the group of units. We also observe that a finitely presented special inverse monoid (even an $E$-unitary one) may have infinitely many pairwise non-isomorphic maximal subgroups.
Human review
How the literature uses it
1 arXiv paper cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Uses its method or tool 1 paper
-
We shall also need the following lemma, which is proved by a similar technique to [ref] .
Subgroups of $E$-unitary and $R_1$-injective special inverse monoids · Special Inverse Monoids · 0.98
Cites it as background 1 paper
-
To avoid duplication we shall refrain from comprehensive historical discussion, referring instead to the survey article of Meakin [ref] for the early theory and the introduction to our own recent article [ref] for subsequent developments.
Subgroups of $E$-unitary and $R_1$-injective special inverse monoids · Introduction · 1.00
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.