Hom-associative magmas with applications to Hom-associative magma algebras

TL;DR

This paper characterizes Hom-associative magmas by identifying functions that satisfy Hom-associativity conditions, classifies these functions for magmas of order two, and constructs related algebraic structures.

Contribution

It provides a complete classification of Hom-associative structures on small magmas and introduces new Hom-associative magma algebras based on these findings.

Findings

Classified all Hom-associative functions for magmas of order two.

Identified all endomorphisms that preserve Hom-associativity.

Constructed examples of Hom-associative and multiplicative magma algebras.

Abstract

Let $X$ be a magma, that is a set equipped with a binary operation, and consider a function $\alpha : X \to X$. We that $X$ is Hom-associative if for all $x,y,z \in X$, the equality $\alpha(x)(yz) = (xy) \alpha(z)$ holds. For every isomorphism class of magmas of order two, we determine all functions $\alpha$ making $X$ Hom-associative. Furthermore, we find all such $\alpha$ that are endomorphisms of $X$. We also consider versions of these results where the binary operation on $X$ as well as the function $\alpha$ may be only partially defined. We use our findings to construct examples of Hom-associative and multiplicative magma algebras.