Work

Twisted products of monoids

Year: 2025

Type: article

Abstract: 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.

Source: Journal of Algebra

DOI: 10.1016/j.jalgebra.2025.10.030

Authors: James East, Robert D. Gray, P.A. Azeef Muhammed, Nik Ruškuc

Language: English

Topic: semigroups and automata theory

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,310