Simplicity of Augmentation Submodules in Monoids with 0-Minimal Ideals of Rank Greater than Two

TL;DR

This paper constructs specific transformation monoids with simple augmentation submodules and explores the relationship between their structure and associated matrix properties, providing new examples and criteria.

Contribution

It introduces explicit families of monoids with simple augmentation submodules and establishes a criterion linking matrix rank to module simplicity.

Findings

Constructed monoids with simple augmentation submodules and rank > 2.

Linked matrix rank to the simplicity of augmentation modules in rank 2 cases.

Provided new examples of monoids with non-complete associated graphs.

Abstract

In this paper, we construct explicit families of transformation monoids whose augmentation submodules are simple and whose associated 0-minimal J-classes have rank greater than two. These examples provide new monoids with simple augmentation submodules and non-complete associated graphs. We also establish a connection between the sandwich matrix of a 0-minimal J-class of rank two and the simplicity of the corresponding augmentation module, yielding a criterion that determines simplicity directly from the rank of this matrix for this class of monoids.