nLab unital magma

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.

Related concepts

• associative magma

• commutative magma

• nonassociative group

• paraunital magma

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory