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
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
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
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

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