# Definition:N-Ary Operation Induced by Binary Operation

**Navigation:** Iterated Operation → Indexed → Induced $n$-ary Operation

## Definition

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 $e$.