Partition de n'importe quel ensemble infini avec des applications injectives non-surjectives, et étude d'un demi-groupe remarquable

In this article, we will present a particularly remarkable partitioningmethod of any infinite set with the aid of non-surjective injective maps. Thenon-surjective injective maps from an infinite set to itself constitute a semigroup for the law of composition bundled with certain properties allowing usto prove the existence of remarkable elements. Not to mention a compatibleequivalence relation that allows transferring the said law to the quotientset which can be provided with a lattice structure. Finally, we will presentthe concept of Co-injectivity and some of its properties.