# Maximal subsemigroups of finite transformation and diagram monoids

## TL;DR

This paper provides a unified framework for describing and counting the maximal subsemigroups of various finite transformation and diagram monoids, enhancing understanding of their algebraic structure.

## Contribution

It introduces a unified approach based on a specialized algorithm to determine maximal subsemigroups across multiple classes of monoids, improving clarity and efficiency.

## Key findings

- Describes maximal subsemigroups of 12 transformation monoids
- Analyzes maximal subsemigroups of partition, Brauer, Jones, and Motzkin monoids
- Provides a unified framework for these determinations

## Abstract

We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids. Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining these maximal subsemigroups. This approach is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors. This allows us to concisely present the descriptions of the maximal subsemigroups, and to more clearly see their common features.