Definition:Iterated Binary Operation

Let $$\left({S, \circ}\right)$$ be an algebraic structure.

For each ordered $n$-tuple $$\left({a_1, a_2, \ldots, a_n}\right) \in S^n$$, the composite of $$\left({a_1, a_2, \ldots, a_n}\right)$$ for $$\circ$$ is the value at $$\left({a_1, a_2, \ldots, a_n}\right)$$ of the $n$-ary operation defined by $\circ$.

This composite is normally denoted $$\circ_n \left({a_1, a_2, \ldots, a_n}\right)$$.

If the tuple is empty, then the composite is assigned the value of the identity of the operation (if this is a structure with an identity, that is):

$$\circ_0 \left({\varnothing}\right) = e_S$$