# On signed generators of groups and algebras

## TL;DR

This paper explores signed groups and algebras, focusing on generators, their classifications, and the structure of groups with mixed commutativity, extending concepts like spin matrices and quaternions.

## Contribution

It provides a complete classification of anticommutative generators and finite groups with mixed commutativity, advancing the understanding of signed group structures.

## Key findings

- Classified anticommutative generators
- Classified finite groups with mixed commutativity
- Extended notions of spin matrices and quaternions

## Abstract

Operators acting on the discrete random chaos yield signed multiplicative systems, extending the notion of spin matrices and quaternions. We investigate signed groups through the associated sign matrices, focusing on generators and their replacements. Of particular interest are anticommutative generators leading to a complete classification of the generated groups. The classification of finite groups of mixed commutativity is also obtained.