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October 8, 2025Open Access

Green relations over finite monoids of G-equivariant functions

RRRamón H. Ruiz-MedinaUniversidad de Guadalajara VLVictor M. Lara-Gómez

Key Points

The study reveals the structure of green relations in monoids of G-equivariant functions over finite sets, crucial for understanding semigroup dynamics.
Key findings include the formulation of green relations based on elementary collapsings, providing new insights into monoid theory.
The analysis of G-action on sets helps in characterizing equivariant functions as semigroups, emphasizing their compositional nature.
Understanding these relations will enhance mathematical comprehension of semigroups, potentially influencing future research in algebraic structures.

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 f (x) =g f (x), g G, x 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.

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