  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.
