Twisted products of monoids

TL;DR

This work develops a unified theory of twisted products $M\times_\Phi^q S$ of monoids via a twisting $\Phi:S\times S\to \mathbb{N}$, with a focus on tight twistings that ensure robust structural behavior. It provides a general description of Green's relations, idempotents, (von Neumann) regularity, Schützenberger groups, and biordered sets for tight twisted products, in terms of the corresponding data on $M$ and $S$. The authors instantiate the framework with canonical twistings on diagram monoids (floating-component counts) and rigid, Sylvester-rank twistings, obtaining sharp results on tightness and regularity, and extending these insights to endomorphism monoids of independence algebras and to diagram-monoid variants. They also analyze idempotent-generated submonoids and stability, yielding new structural results for twisted diagram monoids and illustrating how the twisting interacts with the algebraic and combinatorial properties of the base monoids.

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.

Figures (5)

• Figure 1: Stacking Temperley--Lieb diagrams $a$ and $b$ (left). Here there are two floating loops (shown in red), so $\Phi(a,b)=2$. Removing these components leads to the product $ab$ (right).
• Figure 2: Multiplication of partitions $a,b\in\mathcal{P}_6$, with the product graph $\Pi(a,b)$ in the middle.
• Figure 3: Egg-box diagrams of the partition monoid $\mathcal{P} _2$, and two of its twisted products. See Remark \ref{['rem:eggbox']} for more details.
• Figure 4: Egg-box diagrams of the partial Brauer monoid $\mathcal{P}\mathcal{B} _2$, and two of its twisted products. See Remark \ref{['rem:eggbox']} for more details.
• Figure 5: Egg-box diagrams of the Brauer monoids $\mathcal{B}_3$ (top row) and $\mathcal{B}_4$ (bottom row), and two each of their twisted products. See Remark \ref{['rem:eggbox']} for more details.

Theorems & Definitions (78)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.7
• proof
• Lemma 4.1