# nLab unital magma

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A magma $(S,\cdot)$ is called unital if it has an identity element $1 \in S$, hence an element such that for all $x \in S$ it satisfies the equation

$1 \cdot x = x = x \cdot 1$

holds. The identity element is idempotent.

Some authors take a magma to be unital by default (cf. Borceux-Bourn Def. 1.2.1).

There is also a possibly empty version, where the identity element is replaced with a constant function $1:S \to S$ such that for all $x,y \in S$, $1(x)\cdot y = y$ and $x\cdot 1(y) = x$.

## Properties

The Eckmann-Hilton argument holds for unital magmas: two compatible ones on a set must be equal, associative and commutative.

## Examples

Examples include unital rings etc.

Last revised on May 25, 2021 at 11:12:10. See the history of this page for a list of all contributions to it.