Nomenclature for distributive action between semigroups

Question

Let $(G,*)$ and $(H,+)$ be semigroups. Let $\cdot$ be an action of $G$ on $H$, such that $\cdot$ distributes over $*$.

[I.e., $(g_1 * g_2) \cdot h = g_1*(g_2\cdot h)$, and $g\cdot(h_1+h_2) = (g\cdot h_1) + (g\cdot h_2)$.]

Is there a canonical name for this kind of structure? (In the same sense that 'module' describes a structure involving rings and groups.)

Answer 1

I think, it isn't. If $H$ is commutative, you can call it a semimodule over $G$ (i.e over the semigroup ring $\mathbb{Z}G$).