Partitioning of an infinite set with a non surjective injective map and the study of a remarkable semi-group

In this article, we will present a particularly remarkable partitioning method of any infinite set with the help of a non-surjective injective map to itself. All these maps constitute a semi-group for the law of composition bundled with certain properties allowing us to prove the existence of remarkable elements. Not to mention a compatible equivalence relation that allows transferring the said law to the quotient set which can be provided with a lattice structure. Finally, we will present the concept of Co-injectivity and some of its properties.