\begin{definition}[Definition:Composite Defined by Permutation]
Let $\oplus$ be an $n$-ary operation on a set $S$.
Let $\sequence {a_k}_{k \mathop \in A}$ be a sequence of $n$ terms of $S$.
Let $\sigma: A \to A$ be a permutation of $A$.
Then the '''composite of the ordered $n$-tuple defined by the sequence $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$''' is defined as:
{{begin-eqn}}
{{eqn | l = \bigoplus_{k \mathop \in A} a_{\map \sigma k}
      | r = \begin {cases} a_{\map \sigma 1} & : n = 1 \\
\map \oplus {a_{\map \sigma 1}, a_{\map \sigma 2}, \ldots, a_{\map \sigma m} } \oplus a_{\map \sigma {n + 1} } & : n = m + 1
\end{cases}
      | c = 
}}
{{eqn | r = a_{\map \sigma 1} \oplus a_{\map \sigma 2} \oplus \cdots \oplus a_{\map \sigma n}
      | c = 
}}
{{end-eqn}}
\end{definition}