Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913933423542272 |
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| author | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruskuc, Nik |
| author_facet | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruskuc, Nik |
| contents | This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$ over the corresponding biordered set $E = E(P)$. In the first part of the paper we obtain a number of general presentations by generators and defining relations, in each case reflecting salient combinatorial/topological properties of the groups. In the second part we apply these to explicitly compute the groups when $P = P(P_n)$ and $E = E(P_n)$ arise from the partition monoid $P_n$. Specifically, we show that the maximal subgroup of $PG(P(P_n))$ corresponding to a projection of rank $r\leq n-2$ is (isomorphic to) the symmetric group $S_r$. In $IG(E(P_n))$, the corresponding subgroup is the direct product $Z \times S_r$. The appearance of the infinite cyclic group $Z$ is explained by a connection to a certain twisted partition monoid $P_n^Φ$, which has the same biordered set as $P_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06600 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids East, James Gray, Robert D. Muhammed, P. A. Azeef Ruskuc, Nik Group Theory Rings and Algebras This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$ over the corresponding biordered set $E = E(P)$. In the first part of the paper we obtain a number of general presentations by generators and defining relations, in each case reflecting salient combinatorial/topological properties of the groups. In the second part we apply these to explicitly compute the groups when $P = P(P_n)$ and $E = E(P_n)$ arise from the partition monoid $P_n$. Specifically, we show that the maximal subgroup of $PG(P(P_n))$ corresponding to a projection of rank $r\leq n-2$ is (isomorphic to) the symmetric group $S_r$. In $IG(E(P_n))$, the corresponding subgroup is the direct product $Z \times S_r$. The appearance of the infinite cyclic group $Z$ is explained by a connection to a certain twisted partition monoid $P_n^Φ$, which has the same biordered set as $P_n$. |
| title | Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids |
| topic | Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2507.06600 |