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.

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