Definition:N-Ary Operation Induced by Binary Operation

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Let $(G, \oplus)$ be a magma.

Let $n\geq 1$ be a natural number.

Let $G^n$ be the $n$th cartesian power of $G$.

The $n$-ary operation induced by $\oplus$ is the $n$-ary operation $\oplus_n : G^n \to G$ defined as:

$\oplus_n (f) = \displaystyle \bigoplus_{i \mathop = 1}^n f(i)$

where $\bigoplus$ denotes indexed iteration of $f$ from $1$ to $n$.

Induced nullary operation

Let $(G, \oplus)$ be a unital magma with identity $e$.

The $0$-ary operation induced by $\oplus$ is the nullary operation equal to the element $e$.

Also see