On Special Inverse Monoids with the Strong $F$-Inverse Property
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Dolinka, Igor Kudryavtseva, Ganna |
| author_facet | Dolinka, Igor Kudryavtseva, Ganna |
| contents | An inverse monoid $S$ is called $F$-inverse if each $σ$-class of $S$, where $σ$ is the minimum group congruence of $S$, has a maximum element with respect to the natural order of $S$. Since the property of an inverse monoid being $F$-inverse immediately implies that it must be $E$-unitary, it follows that every $X$-generated $F$-inverse monoid with canonical maximum group image $G$ must be isomorphic to a quotient of the Margolis-Meakin expansion $M(G,X)$. If this is realised in such a way that all the maximal elements of each $σ$-class of $M(G,X)$ get identified, thus producing the top element of the corresponding $σ$-class of $S$, we say that $S$ is strongly $F$-inverse. Consequently, there is a universal $X$-generated inverse monoid $M_{sF}(G,X)$ with maximum group image $G$ and the strongly $F$-inverse property. We provide a presentation for this inverse monoid and show it can be further simplified upon introducing additional assumptions on the group $G$ (which will include all one-relator groups). We use this to provide a full description of all one-relator special inverse monoids with a cyclically reduced relator word that are strongly $F$-inverse. We also discuss some further examples and non-examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Special Inverse Monoids with the Strong $F$-Inverse Property Dolinka, Igor Kudryavtseva, Ganna Group Theory Primary 20M05, Secondary 20M18, 20F05 An inverse monoid $S$ is called $F$-inverse if each $σ$-class of $S$, where $σ$ is the minimum group congruence of $S$, has a maximum element with respect to the natural order of $S$. Since the property of an inverse monoid being $F$-inverse immediately implies that it must be $E$-unitary, it follows that every $X$-generated $F$-inverse monoid with canonical maximum group image $G$ must be isomorphic to a quotient of the Margolis-Meakin expansion $M(G,X)$. If this is realised in such a way that all the maximal elements of each $σ$-class of $M(G,X)$ get identified, thus producing the top element of the corresponding $σ$-class of $S$, we say that $S$ is strongly $F$-inverse. Consequently, there is a universal $X$-generated inverse monoid $M_{sF}(G,X)$ with maximum group image $G$ and the strongly $F$-inverse property. We provide a presentation for this inverse monoid and show it can be further simplified upon introducing additional assumptions on the group $G$ (which will include all one-relator groups). We use this to provide a full description of all one-relator special inverse monoids with a cyclically reduced relator word that are strongly $F$-inverse. We also discuss some further examples and non-examples. |
| title | On Special Inverse Monoids with the Strong $F$-Inverse Property |
| topic | Group Theory Primary 20M05, Secondary 20M18, 20F05 |
| url | https://arxiv.org/abs/2506.14047 |