Redirected from "category of partial functions".
Contents
Context
Category theory
Algebra
Analysis
Contents
Idea
The essentially algebraic structure of partial endofunctions on a set , and specific cases for when is an abelian group, commutative ring, and field respectively.
Definition
In a set
Given a set , the category of partial endofunctions in , or just category of partial functions, is the concrete category with objects called subsets with the set of elements for each subset , and the set of morphisms consist of functions for each subset , where is the improper subset, as well as the set of monomorphisms consisting of the subset inclusions for subsets and .
There exist a global operator representing composition of partial functions
where
-
for partial functions , , and , given the canonical isomorphism ,
-
for partial function and subset , there is a function such that for canonical injection ,
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for partial function and superset , there is a function such that for canonical injection ,
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for partial function , and for the identity function
In an abelian group
If is a abelian group, then for every subset , is a abelian group, and in addition to the global operators corresponding to composition of partial functions, there exist global operators representing addition of partial functions and negation of partial functions,
where
-
for partial functions and there is a partial function and a partial function such that given the canonical isomorphism ,
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for partial functions , , and , given the canonical isomorphism ,
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for partial function , and supersets for , given the local additive unit , and
-
for partial function , there is a partial function representing negation where the negation of is the local additive inverse of :
In a commutative ring
If is a commutative ring, then for every subset , is a -commutative algebra, and in addition to the global operators corresponding to composition, addition, and negation of partial functions, there exist a global operator representing multiplication of partial functions
where
-
for partial functions and there is a partial function and a partial function such that given the canonical isomorphism ,
-
for partial functions , , and , given the canonical isomorphism ,
-
for partial function , and supersets for , given the local multiplicative unit , and
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for partial function , and supersets for , given the local additive unit , and
-
for partial functions , , and , given the canonical isomorphism ,
-
for partial functions , , and , given the canonical isomorphism ,
In a field
If is a Heyting field, then for every subset , is a -commutative algebra, with global operators corresponding to composition, addition, negation, and multiplication of partial functions. Let
be the type of all functions whose evaluations at each element are apart from zero on the entire domain. There exists a global operator representing the reciprocal of partial functions:
where
- for partial function ,
and
and the set is defined as
See also
References