nLab
absorption monoid

Contents

Context

Algebra

Monoid theory

Contents

Definition

An absorption monoid or annihilation monoid is a monoid (M,1,)(M,1,\cdot) that is also an absorption magma (M,0)(M,0).

Properties

Zero divisors

A non-zero element aMa \in M is a zero divisor if there exists a non-zero element bMb \in M such that ab=0a \cdot b = 0.

Localization and group completion

The localization of an absorption monoid at 00 is the trivial group. Because of this, the group completion of any absorption monoid is the trivial group. This is why one speaks of division monoids instead of groups in the context of absorption monoids, and in particular, why the additive identity element in any nontrivial field has no multiplicative inverse.

Examples

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 October 20, 2021 at 05:13:12. See the history of this page for a list of all contributions to it.