nLab cancellative monoid

Contents

Context

Algebra

Monoid theory

Contents

Definition

In set theory

A monoid (A,,1)(A, \cdot, 1) is called left cancellative if

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

and called right 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)

It is called cancellative if it is both left cancellative and right cancellative.

In infinity-groupoid theory

A monoid ( 0 0 -truncated A 3 A_3 -space) (A,,1)(A, \cdot, 1) is called left cancellative if for all objects aAa \in A and bAb \in A the homotopy fiber of the functor L:AAL:A \to A, defined as L a(z)azL_a(z) \coloneqq a \cdot z, at bb is (1)(-1)-truncated, and is called right cancellative if for all elements aAa \in A and bBb \in B the homotopy fiber of the functor R:AAR:A \to A, defined as R a(z)zaR_a(z) \coloneqq z \cdot a at bb is (1)(-1)-truncated. It is called cancellative if it is both left cancellative and right cancellative.

algebraic structureoidification
truth valuepreorder
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

References

See also

Last revised on May 20, 2022 at 20:05:08. See the history of this page for a list of all contributions to it.