Congruence lattices of finite diagram monoids

TL;DR

This paper provides a comprehensive description of the congruence lattices for several finite diagram monoids, revealing a unified construction underlying all congruences.

Contribution

It introduces a new construction framework that characterizes all congruences in these monoids, unifying their structure analysis.

Findings

Complete description of congruence lattices for six finite diagram monoids

All congruences are instances of a unified construction involving ideals and subgroup data

Provides a structural understanding of congruences in diagram monoids

Abstract

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.