nLab
cancellative monoid
Contents
Context
Algebra
Monoid theory
monoid theory in algebra :
monoid , infinity-monoid
monoid object , monoid object in an (infinity,1)-category
Mon , CMon
monoid homomorphism
trivial monoid
submonoid , quotient monoid?
divisor , multiple? , quotient element?
inverse element , unit , irreducible element
ideal in a monoid
principal ideal in a monoid
commutative monoid
cancellative monoid
GCD monoid
unique factorization monoid
Bézout monoid
principal ideal monoid
group , abelian group
absorption monoid
free monoid , free commutative monoid
graphic monoid
monoid action
module over a monoid
localization of a monoid
group completion
endomorphism monoid
super commutative monoid
Contents
Definition
In set theory
A monoid ( A , ⋅ , 1 ) (A, \cdot, 1) is called left cancellative if
∀ a , b , z ∈ A ( ( z ⋅ a = z ⋅ b ) ⇒ ( 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 , z ∈ A ( ( a ⋅ z = b ⋅ z ) ⇒ ( 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 a ∈ A a \in A and b ∈ A b \in A the homotopy fiber of the functor L : A → A L:A \to A , defined as L a ( z ) ≔ a ⋅ z L_a(z) \coloneqq a \cdot z , at b b is ( − 1 ) (-1) -truncated, and is called right cancellative if for all elements a ∈ A a \in A and b ∈ B b \in B the homotopy fiber of the functor R : A → A R:A \to A , defined as R a ( z ) ≔ z ⋅ a R_a(z) \coloneqq z \cdot a at b b is ( − 1 ) (-1) -truncated. It is called cancellative if it is both left cancellative and right cancellative .
References
See also
Last revised on May 21, 2022 at 00:05:08.
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