Semigroups admitting commutative group actions Let $(S,*)$ be a semigroup admitting a distinguished element $0$ such that $z*s = s*z = z$, for all $s \in S$. Moreover, let $(\mathbb{G},\cdot)$ be a commutative group. Consider an action
$$
\mathbb{G} \times S \to S, ~~~~~~~~~~~ (k,s) \mapsto k.s,
$$
satisfying, for all $s,t \in S$, and $k,l \in \mathbb{G}$,

*

*$~~~ k.(l.s) = (k\cdot l).s$,


*$~~~ k.(s*t) = (k.s)*t = s*(k.t)$,


*$~~~~ 1_{\mathbb{G}}.s = s$,


*$~~~~ 0.s = z.$
Does such an object have a name, or is is easily seen to be equivalent to a standard structure? If such things are studied, what can one say about them?
Such a commutative semigroup admits an equivalence relation
$$
s \simeq t, \text{ if there exists a } k \in \mathbb{G}, \text{ such that } s = k.t.
$$
Does the quotient have $S\,/\!\simeq$ have a name. For example, might one call it the projectivization of $S$?
If $S$ is a monoid does anything extra interesting happen?
EDIT: Based on the comments of M. Sapir, the definition has been refined.
 A: I'm afraid my general algebra brain has turned to mush.  I cannot give you a good reference to anything that points to items in the literature.  I cannot even suggest a good search term right now.  Let me make some other observations which might help.
Fix a similarity type which has a binary operation , and a unary operation for each field element k. Then you get a variety which extends a semigroup variety by these unary maps which are not quite semigroup morphisms. So if you have a nontrivial semigroup for a field K, you get a bunch more with more than one element in the semigroup.
In addition, you get a closure under a kind of disjoint union. For semigroups S and T and Z in the variety, where Z is the one element semigroup, take the disjoint union and extend the semigroup operation to be zero (so in Z) everywhere where two elements do not both belong to S, or to T, or to Z.  There may be another construction dual to this that involves adjoining a unit to a semigroup and making it a monoid, but I am not thinking of it.
I am sure something like this has been considered.  Other than field action on a semigroup, I don't know what search terms to use.  Embedding a field in a clone perhaps?
Gerhard "Spellcheck Still Finds Semigroups Delicious" Paseman, 2018.03.09.
