Finite generating sets for monoids of G-equivariant functions

Let G be a group acting on a set X. The set of all G-equivariant functions on X forms a monoid under composition, whose group of units consists precisely of the G-equivariant bijections. In this paper we investigate the problem of finite generation for both the group of units and the whole monoid. When X is a finite G-set, we establish bounds for the rank of $$\textrm{Aut}_{G}(X)$$ by constructing generating sets from the orbit structure of the action. We further refine the upper bound by introducing a grouping procedure that exploits generators of pairwise coprime orders, leading naturally to a possible new invariant associated with the action. We then study the monoid $$\textrm{End}_{G}(X)$$ . To this end we introduce orbital infiltrations, a family of indispensable equivariant transformations that reveal structural obstructions to finite generation. As a consequence, we obtain necessary conditions on the group, the G-set, and the action for $$\textrm{End}_{G}(X)$$ to admit a finite generating set.