$\sigma$-Sets and $\sigma$-Antisets

TL;DR

This paper explores the duality between $\sigma$-sets and $\sigma$-antisets, develops the integer space $3^{A}$ for small cardinals, and investigates algebraic properties and equations related to $\sigma$-sets.

Contribution

It introduces the $\sigma$-set-$\sigma$-antiset duality, constructs the integer space $3^{A}$ for specific cardinals, and develops properties of $\sigma$-set fusion and equations.

Findings

Established the $\sigma$-set-$\sigma$-antiset duality.

Developed the integer space $3^{A}$ for $|A|=2,3$.

Analyzed algebraic properties and equations of $\sigma$-sets.

Abstract

In this paper we present a brief study of the $\sigma$-set-$\sigma$-antiset duality that occurs in $\sigma$-set theory and we also present the development of the integer space $3^{A}=\left\langle 2^{A}, 2^{A^{-}} \right\rangle$ for the cardinals $|A|=2,3$ together with its algebraic properties. In this article, we also develop a presentation of some of the properties of fusion of $\sigma$-sets and finally we present the development and definition of a type of equations of one $\sigma$-set variable.