# Associativity of $\sigma$-sets for non-antielement $\sigma$-set groups

## TL;DR

This paper explores the conditions under which fusion operations on antielement free sigma-sets are associative, leading to the formation of groups that solve sigma-set equations, with theoretical proofs and new theorems.

## Contribution

It introduces a framework for associativity in sigma-sets, extending previous conditions to establish groups that solve sigma-set equations.

## Key findings

- Established conditions for associativity on sigma-set fusion
- Proved a theorem on local associativity and commutative groups
- Provided solutions to one-variable sigma-set equations

## Abstract

We study and extend the conditions for asociativity on fusion over antielement free $\sigma$-sets to introduce a group to solve $\sigma$-set equations. $\sigma$-sets as a theory of sets and antisets is sumarized and used as a framework to define the main elements of this work. A theorem on Local Asociativity, conmutative groups and solution of one variable fusion equation is presented.