The essentially algebraic structure of partial functions on a set , and specific cases for when is an abelian group and commutative ring respectively.
Definition
Given a set , the category of partial functions in is the concrete category with objects called subsets with the set of elements for each subset , and the set of morphisms only consist of functions , where is the improper subset, as well as injections representing the subset inclusions.
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 ,
for partial function and superset , there is a function such that for canonical injection ,
for partial function , and for the identity function
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 ,
for partial functions , , and , given the canonical isomorphism ,
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 :
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.
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
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 ,