Groups actions over sets became a powerful tool for the study of the inner structure of groups. The study of basic concepts such as orbit or stabilizer leads to the canonical bijection between the orbits of an action and the corresponding set of cosets of the group. This fundamental result becomes very useful on a big amount of problems not only in combinatorics but also in the structural study of groups.
On the other hand, the development of computer science has grown the interest of the mathematical community in monoids, which can be considered somehow the first step in the study of groups. Although the generalization of actions to monoids does not changes the original definition, they do not much resemble group actions. The goal of this talk is to present a decomposition theorem in the finite case.