Twisted products of monoids
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909865513844736 |
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| author | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik |
| author_facet | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik |
| contents | A twisting of a monoid $S$ is a map $Φ:S\times S\to\mathbb{N}$ satisfying the identity $Φ(a,b) + Φ(ab,c) = Φ(a,bc) + Φ(b,c)$. Together with an additive commutative monoid $M$, and a fixed $q\in M$, this gives rise a so-called twisted product $M\times_Φ^qS$, which has underlying set $M\times S$ and multiplication $(i,a)(j,b) = (i+j+Φ(a,b)q,ab)$. This construction has appeared in the special cases where $M$ is $\mathbb{N}$ or $\mathbb{Z}$ under addition, $S$ is a diagram monoid (e.g.~partition, Brauer or Temperley-Lieb), and $Φ$ counts floating components in concatenated diagrams.
In this paper we identify a special kind of `tight' twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green's relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Schützenberger groups, and more. We also consider a number of examples, including several apparently new ones, which take as their starting point certain generalisations of Sylvester's rank inequality from linear algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted products of monoids East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik Group Theory Rings and Algebras 20M10, 20M20 A twisting of a monoid $S$ is a map $Φ:S\times S\to\mathbb{N}$ satisfying the identity $Φ(a,b) + Φ(ab,c) = Φ(a,bc) + Φ(b,c)$. Together with an additive commutative monoid $M$, and a fixed $q\in M$, this gives rise a so-called twisted product $M\times_Φ^qS$, which has underlying set $M\times S$ and multiplication $(i,a)(j,b) = (i+j+Φ(a,b)q,ab)$. This construction has appeared in the special cases where $M$ is $\mathbb{N}$ or $\mathbb{Z}$ under addition, $S$ is a diagram monoid (e.g.~partition, Brauer or Temperley-Lieb), and $Φ$ counts floating components in concatenated diagrams. In this paper we identify a special kind of `tight' twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green's relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Schützenberger groups, and more. We also consider a number of examples, including several apparently new ones, which take as their starting point certain generalisations of Sylvester's rank inequality from linear algebra. |
| title | Twisted products of monoids |
| topic | Group Theory Rings and Algebras 20M10, 20M20 |
| url | https://arxiv.org/abs/2507.04486 |