In abstract algebra, a magma (a.k.a. a groupoid) is a basic algebraic structure consisting of a set equipped with a single, closed binary operation. (See here.) Monoids, groups, and rings are all special cases of magmas.
Is there a similar basic algebraic structure consisting of a set of scalars $S$, a set of vectors $V$, and a multiplication operation $\cdot:S\times V\rightarrow V$, which can serve as the basis for such algebraic structures as modules and vector spaces?
The homomorphisms between two such structures are precisely the homogeneous functions, i.e. those functions $f$ satisfying: for every $s \in S$ and every $v \in V$, $f(sv) = sf(v)$.
If $S$ is a magma, we can consider the special case in which for every $s,t \in S$ and every $v \in V$ we have $(st)v=s(tv)$. If $S$ is a monoid with unit $1$, we can consider the special case in which for every $v \in V$, $1v = v$. If both $S$ and $V$ are magmas, we can consider the special case in which for every $s,t \in S$ and for every $v,w \in V$ $(st)v=(sv)+(tv)$ and $s(v+w)=(sv)+(sw)$. Etc.