# 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.

## Generalizations

This concept could be generalized from the category of sets to any monoidal category:

A unital magma object or unital algebra object in a monoidal category $(C, I, \otimes)$ is an object $A \in C$ with morphisms $\iota:I \to A$ and $\pi:A \otimes A \to A$ such that the following diagrams commute:

$\array{ I \otimes A &\overset{\iota \otimes \mathrm{id}_A}{\longrightarrow}& A \otimes A \\ \downarrow^{\lambda_A} & & \downarrow^{\pi} \\ A &\overset{\mathrm{id}_A}{\longrightarrow}& A } \,,$
$\array{ A \otimes I &\overset{\mathrm{id}_A \otimes \iota}{\longrightarrow}& A \otimes A \\ \downarrow^{\rho_A} & & \downarrow^{\pi} \\ A &\overset{\mathrm{id}_A}{\longrightarrow}& A } \,,$

where $\lambda_A:I \otimes A \to A$ and $\rho_A:A \otimes I \to A$ are the left and right unitors of the monoidal category.

In the category of modules, unital magma objects are called nonassociative unital algebras, and in the category of abelian groups, unital magma objects are called nonassociative rings.

Last revised on October 4, 2023 at 17:15:43. See the history of this page for a list of all contributions to it.