nLab
cancellative monoid

Contents

Context

Algebra

Monoid theory

Contents

Definition

A commutative monoid (A,)(A, \cdot) is called cancellative if

a,b,zA((az=bz)(a=b)) \underset{a,b,z \in A}{\forall} \left( \left( a \cdot z = b \cdot z \right) \Rightarrow \left( a = b \right) \right)

For a non-commutative monoid one distinguishes left and right cancellability, in the evident way

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
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

References

See also

Last revised on May 23, 2021 at 13:42:38. See the history of this page for a list of all contributions to it.