Green relations over finite monoids of $G$-equivariant functions

TL;DR

This work investigates the Green's relations in the monoid $End_G(X)$ of $G$-equivariant transformations on a finite $G$-set $X$, introducing and exploiting elementary collapsings as a central generating class. The authors leverage stabilizer data $G_x$, orbit structure, and conjugacy/normalizer information via $N_G(H)$ to characterize the principal ideals and Green's classes, establishing criteria for when a map factors or is equivalent to another in terms of kernels, images, and stabilizers. A key result is that every elementary collapsing is $\mathcal{R}$-related to a fixing elementary collapsing, and that the Green's relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{D}$, $\mathcal{H}$) reflect the orbit-stabilizer geometry of $X$; some elementary collapsings are not $\mathcal{H}$-related to any fixing collapsing, highlighting intricate interactions between orbit structure and Green's relations. The paper provides a framework, with explicit examples, to compute and reason about Green's classes in this class of monoids, offering foundations for further study in equivariant semigroup theory and related areas of algebraic combinatorics.

Abstract

For a group $G$ acting over a set $X$, the set of all the $G$-equivariant functions, i.e., the set of functions which conmute with the action, ($g\cdot f(x)=g\cdot f(x), \forall g\in G, \forall x\in X$), is a monoid with the composition. The Green Relations are powerful tools to comprehend the structure of a semigroup. We study the case where $X$ is a finite set and compute the green relations for its monoid of $G$-equivariant functions, attempting to describe them based on some particular elements in the monoid called elementary collapsings.

Figures (4)

• Figure 1: Green's relations of a given $\mathbb{Z}_{2}$-set.
• Figure 2: Functions with the same image.
• Figure 3: Elementary collapsings of the same type not $\mathcal{L}$-related.
• Figure 4: Elementary collapsings not $\mathcal{H}$-related to a fixing elementary collapsing.

Theorems & Definitions (63)

• Lemma 1
• Proposition 2
• proof
• Example 3
• Proposition 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof