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

M. H. Shahzamanian

TL;DR

The paper develops explicit transformation monoid constructions with simple augmentation submodules, concentrating on $0$-minimal $\mathscr J$-classes of rank $r>2$ and graphs $\Gamma(M)$ that need not be complete. It establishes a rank-based criterion for simplicity in the rank-two case via the sandwich matrix $P'$ (identified with $I(\mathcal{E})^{\Gamma(M)}$) and extends this framework to higher ranks by carefully selecting partitions $\mathcal{B}$ and image sets $\mathcal{I}$ that satisfy a $\mathscr J$-minimal compatibility; through Rees matrix semigroup realizations, it constructs infinite families with simple augmentation and non-complete $\Gamma(M)$. The authors provide detailed $r=3$ and $r=4$ examples and a general odd/even dichotomy for $r>4$, showing that simple augmentation can persist or fail depending on the parity and choices of $\mathcal{B}$ and $\mathcal{I}$. These results expand the catalog of monoids with simple augmentation submodules beyond rank-two, highlighting intricate interactions between sandwich matrices, incidence structures, and the connectivity of $\Gamma(M)$ with potential impacts on monoid representation theory and automata-related applications.

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.

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

TL;DR

The paper develops explicit transformation monoid constructions with simple augmentation submodules, concentrating on -minimal -classes of rank and graphs that need not be complete. It establishes a rank-based criterion for simplicity in the rank-two case via the sandwich matrix (identified with ) and extends this framework to higher ranks by carefully selecting partitions and image sets that satisfy a -minimal compatibility; through Rees matrix semigroup realizations, it constructs infinite families with simple augmentation and non-complete . The authors provide detailed and examples and a general odd/even dichotomy for , showing that simple augmentation can persist or fail depending on the parity and choices of and . These results expand the catalog of monoids with simple augmentation submodules beyond rank-two, highlighting intricate interactions between sandwich matrices, incidence structures, and the connectivity of with potential impacts on monoid representation theory and automata-related applications.

Abstract

In this paper, we construct explicit families of transformation monoids whose augmentation submodules are simple and whose associated 0-minimal -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 -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.
Paper Structure (13 sections, 6 theorems, 127 equations, 2 figures)

This paper contains 13 sections, 6 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monoids
  4. $M$-sets
  5. Transformation modules and representations of monoids
  6. Simplicity of Augmentation Submodules in Monoids
  7. Simplicity of Augmentation Submodules of Rank 2
  8. Simplicity of Augmentation Submodules of Rank greater than two
  9. ($r=3$)
  10. $(r=4)$
  11. The case $r>4$
  12. ($r$ Odd)
  13. ($r$ Even)

Key Result

Lemma 3.1

Suppose that the graph $\Gamma$ is a tree on $\Omega$. Then the incidence matrix $I(\mathcal{E})$ has rank $n$ over $\mathbb{F}$ if and only if the matrix ${I(\mathcal{E})}^{\Gamma}$ has rank $n-1$ over $\mathbb{F}$.

Figures (2)

  • Figure 1: Triangular cycle of subsets of $\Omega$
  • Figure 2: Square of subsets of $\Omega$

Theorems & Definitions (10)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3