Abstract
In this article, we will present a particularly remarkable partitioning method of any infinite set with the aid of non-surjective injective maps. The non-surjective injective maps from an infinite set to itself constitute a semigroup 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.
Highlights
The concept of map in mathematics has a primordial role in understanding the links that exist between the different mathematical fields and structures
A map is binary relation over two sets that associates to every element of the first set exactly one element of the second set, sometimes with a specific property
In this article we will show how non-surjective injective maps allow to partitioning an infinite set in several ways
Summary
The concept of map in mathematics has a primordial role in understanding the links that exist between the different mathematical fields and structures. A map is binary relation over two sets that associates to every element of the first set exactly one element of the second set, sometimes with a specific property. A “map” is a “linear transformation” in linear algebra, a “continuous function” in topology, operators in analysis and representations in group theory, etc. In this article we will show how non-surjective injective maps allow to partitioning an infinite set in several ways
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