Twisted products of monoids

Abstract

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