Twisted products of monoids

TL;DR

This paper introduces a new class of twisted monoid products defined by special twistings, providing a detailed structural analysis and exploring various examples, including novel cases related to rank inequalities.

Contribution

It characterizes 'tight' twistings of monoids and thoroughly describes the structure of the resulting twisted products, including Green's relations and subgroup structures.

Findings

Characterization of 'tight' twistings and their properties

Structural descriptions of twisted products, including Green's relations and idempotents

Examples including new cases related to Sylvester's rank inequality

Abstract

A twisting of a monoid $S$ is a map $\Phi:S\times S\to\mathbb{N}$ satisfying the identity $\Phi(a,b) + \Phi(ab,c) = \Phi(a,bc) + \Phi(b,c)$. Together with an additive commutative monoid $M$, and a fixed $q\in M$, this gives rise a so-called twisted product $M\times_\Phi^qS$, which has underlying set $M\times S$ and multiplication $(i,a)(j,b) = (i+j+\Phi(a,b)q,ab)$. 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 $\Phi$ 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, Sch\"{u}tzenberger 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.