nLab cancellative monoid

Definition

In set theory

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

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

and called right cancellative if

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

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

In infinity-groupoid theory

A monoid ($0$-truncated $A_3$-space) $(A, \cdot, 1)$ is called left cancellative if for all objects $a \in A$ and $b \in A$ the homotopy fiber of the functor $L:A \to A$, defined as $L_a(z) \coloneqq a \cdot z$, at $b$ is $(-1)$-truncated, and is called right cancellative if for all elements $a \in A$ and $b \in B$ the homotopy fiber of the functor $R:A \to A$, defined as $R_a(z) \coloneqq z \cdot a$ at $b$ is $(-1)$-truncated. It is called cancellative if it is both left cancellative and right cancellative.

Related concepts

• Grothendieck group
• cancellative category
• integral monoid
• GCD ring
• multiplicative submonoid of cancellative elements
• cancellative binary function

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

References

See also

• Wikipedia, Cancellative semigroup