A generalization of anti-homomorphisms

TL;DR

This paper explores properties of anti-homomorphisms, introduces a new composition operation, and develops factorization categories to generalize and study anti-homomorphisms in a broad framework.

Contribution

It introduces a new $*$-composition for anti-homomorphisms and develops factorization categories to generalize their properties and behaviors.

Findings

$*$-composition preserves anti-homomorphism structure

Factorization categories generalize anti-homomorphisms

Several properties of anti-homomorphisms are established in generality

Abstract

We prove some nice properties of anti-homomorphisms, some of which are analogic to that of homomorphisms. Meanwhile, we develop a new kind of composition called $*$-composition such that the $*$-composition of two anti-homomorphisms is still an anti-homomorphism. Moreover, we develop a certain kind of categories called factorization categories, which generalize anti-homomorphisms and provide a general framework to study "anti" phenomenon. We deduce several results of anti-homomorphisms in great generality.