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Let a semigroup $S$ acts on a set $X$. Define relation $\rho=\{(x,y)\in X\times X\,|\, \exists\, a,b\in S \ \ ax=by\}$ and then take the least congruence containing $\rho$ (its classes are called sometimes orbits of the action).

If a group $G$ partially acts on $X$ (see definition in: J.Kellendonk, M.V.Lawson, Partial actions of groups, International Journal of Algebra and Computation 14 (01), 87-114) then an equvivalence arises: $x\sim y$ iff $\varnothing\ne ax=y$ for some $a\in G$.