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 $\alpha : X \to X$. We say 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 \linebreak class of magmas of order two, we determine all functions $\alpha$ making $X$ \linebreak Hom-associative. Furthermore, we find all such $\alpha$ that are endomorphisms of $X$. We also consider versions of these results where the binary operation \linebreak on $X$ and the function $\alpha$ only are partially defined. We use our findings to \linebreak construct numerous examples of two-dimensional Hom-associative as well as multiplicative magma algebras.