Definition:Operation

An operation is a mapping $$\circ$$ from a cartesian product of $$n$$ sets $$S_1 \times S_2 \times \ldots \times S_n$$ to a universe $$\mathbb{U}$$:

$$\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb{U}: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S_1 \times S_2 \times \ldots \times S_n: \circ \left({s_1, s_2, \ldots, s_n}\right) = t \in \mathbb{U}$$

An operation needs to be defined for all tuples in $$S_1 \times S_2 \times \ldots \times S_n$$.

Operation on a Set
An $$n$$-ary operation on a set $$S$$ is an operation where the domain is the cartesian $n$th power $$S^n$$ and the range is $$S$$:

$$\circ: S^n \to S: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \circ \left({s_1, s_2, \ldots, s_n}\right) = t \in S$$

An $$n$$-ary operation on $$S$$ needs to be defined for $$all$$ tuples in $$S^n$$.