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

Abstract

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