$\sigma$-Relations, $\sigma$-functions and $\sigma$-antifunctions

TL;DR

This paper introduces and develops the concepts of $\sigma$-relations, $\sigma$-functions, and their antifunctions within $\sigma$-Set Theory, extending classical set theory notions with new $\sigma$-antifunctions and related structures.

Contribution

It defines $\sigma$-relations, $\sigma$-functions, and introduces $\sigma$-antifunctions, expanding the theoretical framework of $\sigma$-Set Theory with new concepts and relationships.

Findings

Defined $\sigma$-relations and $\sigma$-functions.

Constructed $\sigma$-antifunctions, antidentity, and antinverse.

Identified 16 related $\sigma$-functions in bijective cases.

Abstract

In this article we develop the concepts of $\sigma$-relation and $\sigma$-function, following the same steps as in Set Theory. First we define the concept of ordered pair and then we build the Cartesian Product of $\sigma$-sets so that we can define the concepts of $\sigma$-relation and $\sigma$-function. Now, as in $\sigma$-Set Theory there exist the concepts of $\sigma$-antielement and $\sigma$-antiset, we can build the new concepts of $\sigma$-antifunction, antidentity and antinverse. Finally, in the case that a $\sigma$-function $f:A\to B$ is bijective and there exist $A^{\st}$ and $B^{\st}$ $\sigma$-antiset of $A$ and $B$, we get 16 different $\sigma$-functions which are related in a diagram of $\sigma$-functions.