El monoide de endomorfismos de G-conjuntos: estructuras y otras propiedades algebraicas

2024/07/12 by Ramón H. Ruiz-Medina, Ruiz-Medina, Ramón H.
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2408.10209

openalex publication_date 2024/07/12 · openalex created_date 2024/10/01 · openalex updated_date 2026/07/28

Abstract

Given the action of a group G on a set X, an endomorphism of X is a function f:X → X which is G-equivariant, that is, it commutes with the action, i.e., f(g⋅ x)= g⋅ f(x), for all x∈ X. The set of endomorphisms of a G-set X is a monoid, with the composition of functions , which we will denote End_G(X). Given subsets U,N⊆ M, we say that U generates M modulo N if it is satisfied that M= ⟨ U ∪ N ⟩. The relative rank of M modulo N is the minimum cardinality of a set U to generate M modulo N. In this work we address the particular case in which G and X are finite to calculate the relative rank of the endomorphism monoid End_G(X) modulo its group of units, denoted by Aut_G(X). We also address structure situations, such as isomorphisms of Aut_G(X) and End_G(X) with other known structures.

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