nLab unital magma




A magma (S,)(S,\cdot) is called unital if it has an identity element 1S1 \in S, hence an element such that for all xSx \in S it satisfies the equation

1x=x=x1 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:SS1:S \to S such that for all x,ySx,y \in S, 1(x)y=y1(x)\cdot y = y and x1(y)=xx\cdot 1(y) = x.


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


Examples include unital rings etc.


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,)(C, I, \otimes) is an object ACA \in C with morphisms ι:IA\iota:I \to A and π:AAA\pi:A \otimes A \to A such that the following diagrams commute:

IA ιid A AA λ A π A id A A, \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 } \,,
AI id Aι AA ρ A π A id A 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 λ A:IAA\lambda_A:I \otimes A \to A and ρ A:AIA\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.

algebraic structureoidification
pointed magma with an endofunctionsetoid/Bishop set
unital magmaunital magmoid
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
monoidal category2-category/bicategory

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