Group with operators

Definition
This notion of group with operators is sometimes called $$\Omega$$-group (pronounced omega group) because the operator set is usually denoted $$\Omega$$. An $$\Omega$$-group is a group $$G$$ along with a set $$\Omega$$ with a function $$n:\Omega \to \mathbb{N}_0$$ such that for every $$\omega \in \Omega$$, $$\omega$$ is a $$n(\omega)$$-ary operation from $$G$$ to $$G$$, i.e., a function $$G^{n(\omega)} \to G$$, and further, $$\omega(e,e,\dots,e) = e$$ where $$e$$ is the identity element of $$G$$.