# Definition:N-Ary Operation Induced by Binary Operation

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## Definition

Let $\struct {G, \oplus}$ be a magma.

Let $n \ge 1$ 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:

- $\map {\oplus_n} f = \ds \bigoplus_{i \mathop = 1}^n \map f i$

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

### Induced nullary operation

Let $\struct {G, \oplus}$ be a unital magma with identity $e$.

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