$G$-set terminology When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ itself does not seem to have a special name without the "$G$" specification. 
To illustrate the point: when we have the appropriate action of a ring $R$ on a set $M$ such that $R \times M \rightarrow M$, then the set $M$ is a "module" without specification of the "$R$", and then we can refer to the $R$-module if we want to be more specific. But we cannot refer to the "set" in the same way, it has to be "$G$-set". So I am wondering is there such a name that was used in this case?
 A: If $G$ is a semigroup then one use names "polygon" [Kilp M., Knauer U., Mikhalev A.V., Monoids, acts and categories] or "operand" [A. H. Clifford, G. B. Preston, The algebraic theory of semigroups] for $G$-sets (i.e. for the sets on which $G$ acts). So you can use them also, if you would like.
A: The language of monoidal categories offers a unification of various notions spread out over mathematics. $G$-sets and $R$-modules are really the same notions. In every monoidal category, there is the notion of a monoid object, namely an object $G$ equipped with morphisms $1 \to G$ and $G \otimes G \to G$, satisfying unitality and associativity. Besides, there is the notion of a module object over that monoid object, namely an object $X$ equipped with a morphism $G \otimes X \to X$ satisfying unitality and associativity in a certain sense. For the cartesian monoidal category of sets, the monoid objects are precisely the monoids $G$ in the usual sense, and $G$-modules are usually called $G$-sets. But really, you could just stick to the general formalism and call them $G$-modules$^{(1)}$. For the monoidal category of abelian groups with the usual tensor product, the monoid objects are the rings in the usual sense, and $R$-modules in the general sense above are precisely the $R$-modules in the usual sense.
Back to the general case: For a $G$-module $(X,G \otimes X \xrightarrow{a} X)$ I would call $X$ the underlying object. It is not good to call it a module, because the action $a$ has been forgotten.
$^{(1)}$ You might object that this contradicts with the common meaning of $G$-modules as $R[G]$-modules, where $R$ is a ring which is understood. But in my opinion this only shows that this usual usage is not good. Why does one say $G$, the monoid (or often group) object in the category of sets, where one actually means the induced monoid object $R[G]$ in the category of $R$-algebras? It is important and also quite useful to distinguish objects of different categories. When $G$ is a monoid, it is understood to be a monoid object in the category of sets, and nothing else, so that every construction, in particular the definition of a module, refers to the category of sets. Therefore "$G$-modules" is not ambiguous.
