Finite generating sets for monoids of G-equivariant functions

Given the action of a group G on a set X, the set of all G-equivariant functions, i.e., those satisfying f(g· x)=g· f(x) for all g∈ G and x∈ X, forms a monoid under composition. In this work we study their generating sets. First, we propose bounds for the cardinalities of the generating sets of their group of units, denoted by AutG(X). Subsequently, using so-called orbital infiltrations, certain transformations that turn out to be indispensable and provide relevant structural information about the monoid, we determine conditions on the group G, the set X, and the action that prevent the whole monoid EndG(X) from admitting a finite generating set.