Definition:Composite Defined by Permutation

Definition

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:

\(\ds \bigoplus_{k \mathop \in A} a_{\map \sigma k}\)

\(=\)

\(\ds \begin {cases} a_{\map \sigma 1} & : n = 1 \\ \map \oplus {a_{\map \sigma 1}, a_{\map \sigma 2}, \ldots, a_{\map \sigma m} } \oplus a_{\map \sigma {n + 1} } & : n = m + 1 \end{cases}\)

\(\ds \)

\(=\)

\(\ds a_{\map \sigma 1} \oplus a_{\map \sigma 2} \oplus \cdots \oplus a_{\map \sigma n}\)

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations