Green relations over finite monoids of G-equivariant functions

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.