If with every element of a (semi)group is associated a function, it is basically called a (semi)group action.
What if with each element of a semigroup is associated a relation? That is, formally, what if we would have a semigroup together with its homomorphism to to a semigroup of all relations on a set? Does this concept have a name? What are some of its interesting properties?
If this homomorphism is injective, this is a semigroup of relations (and a transformation semigroup in the case of functions). In the general case, if you have a homomrphism $f$ from a semigroup $S$ to the semigroup of relations on a set $Q$, then $f$ defines an action on $\mathcal{P}(Q)$ defined, for each $P \in {\cal P}( Q)$ by $$ P \cdot s = \{q \in Q \mid \text{there exists $p \in P$ such that $(p, q) \in f(s)$}\} $$