nLab unital magma

Contents

Contents

Definition

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.

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.

algebraic structureoidification
magmamagmoid
pointed magma with an endofunctionsetoid/Bishop set
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
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
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on May 25, 2021 at 15:12:10. See the history of this page for a list of all contributions to it.