nLab
cancellative binary function
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
Idea
A version of cancellative magmas where the binary function’s codomain need not coincide with its domain set. This allows the concept to be generalised from magmas and monoids to other binary functions such as actions and modules .
Definition
Given sets A A , B B , and C C , a binary function f : A × B → C f:A \times B \to C is left cancellative if for all c ∈ B c \in B and d ∈ B d \in B , if a ⋅ c = a ⋅ d a \cdot c = a \cdot d for all a ∈ A a \in A , then c = d c = d , and it is right cancellative if for all a ∈ A a \in A and b ∈ A b \in A , if a ⋅ c = b ⋅ c a \cdot c = b \cdot c for all c ∈ A c \in A , then a = b a = b . It is cancellative if it is both left cancellative and right cancellative.
See also
Last revised on May 20, 2022 at 04:29:47.
See the history of this page for a list of all contributions to it.