# Neutral element for a multiary operation

## Contents

## Definition

### All-sided neutral element

Suppose is a set and is a -ary operation for . An element is termed a **neutral element** for if the following holds: evaluated at any -tuple where of the entries are equal to and the remaining entry is , gives output . This is true regardless of where we place and is also true if .

The term neutral element, when used without qualification, is used in the context , i.e., for a binary operation, i.e., a magma.

### Left neutral element

Suppose is a set and is a -ary operation for . An element is termed a **left neutral element** for if the following holds: for all .

### Right neutral element

Suppose is a set and is a -ary operation for . An element is termed a **right neutral element** for if the following holds: for all .

### Neutral element for a given position

Suppose is a set and is a -ary operation for . Suppose . An element is termed a **neutral element** for position if if the following holds: for all where appears in the position.

## Facts

The case is special because we can deduce equality of left and right neutral element and therefore also deduce that binary operation on magma determines neutral element. For higher , there could be more than one neutral element.