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The essentially algebraic structure of partial functions on a set $S$, and specific cases for when $S$ is an abelian group, commutative ring, and field respectively.
Given a set $S$, the category of partial functions in $S$ is the concrete category $Part(S)$ with objects called subsets $A:Ob(Part(S))$ with the set of elements for each subset $El(A)$, and the set of morphisms consist of functions $Hom(A, \Im(S)) \coloneqq (A \to S)$ for each subset $A:Ob(Part(S))$, where $\Im(S)$ is the improper subset, as well as the set of monomorphisms $Hom(A, B)$ consisting of the subset inclusions for subsets $A:Ob(Part(S))$ and $B:Ob(Part(S))$.
There exist a global operator representing composition of partial functions
where
for partial functions $f:Hom(A, \Im(S))$, $g:Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_a:Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f \circ_{Part(S)} (g \circ_{Part(S)} h)) = ((f \circ_{Part(S)} g) \circ_{Part(S)} h)$
for partial function $f:Hom(A, \Im(S))$ and subset $B \subseteq A$, there is a function $g:Hom(B, \Im(S))$ such that $g = f \circ_{Part(S)} i_{B,A}$ for canonical injection $i_{B,A}:Hom(B,A)$,
for partial function $f:Hom(A, \Im(S))$ and superset $B \supseteq A$, there is a function $h:Hom(B, \Im(S))$ such that $h \circ_{Part(S)} i_{A,B} = f$ for canonical injection $i_{A,B}:Hom(A,B)$,
for partial function $f:Hom(A, \Im(S))$, $f = f \circ_{Part(S)} id_S$ and $f = id_S \circ_{Part(S)} f$ for the identity function $id_S:Hom(\Im(S), \Im(S))$
If $S$ is a abelian group, then for every subset $A:Ob(Part(S))$, $Hom(A, \Im(S))$ 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 $f:Hom(A, \Im(S))$ and $g:Hom(B, \Im(S))$ there is a partial function $f + g:Hom(A \cap B, \Im(S))$ and a partial function $g + f:Hom(B \cap A, \Im(S))$ such that given the canonical isomorphism $i_c:Hom(A \cap B, B \cap A)$, $i_c \circ (f + g) = (g + f)$
for partial functions $f:Hom(A, \Im(S))$, $g:Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_a:Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f + (g + h)) = ((f + g) + h)$
for partial function $f:Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B:Ob(Part(S))$, given the local additive unit $0_{B,\Im{S}}:Hom(B, \Im(S)$, $f + 0_{B,\Im{S}} = f$ and $0_{B,\Im{S}} + f = f$
for partial function $f:Hom(A, \Im(S))$, there is a partial function $-f:Hom(A, \Im(S))$ representing negation where the negation of $F$ is the local additive inverse of $f$: $-f = -_{A,S}f$
If $S$ is a commutative ring, then for every subset $A:Ob(Part(S))$, $Hom(A, \Im(Part(S)))$ is a $S$-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 $f:Hom(A, \Im(S))$ and $g:Hom(B, \Im(S))$ there is a partial function $f \cdot g:Hom(A \cap B, \Im(S))$ and a partial function $g \cdot f:Hom(B \cap A, \Im(S))$ such that given the canonical isomorphism $i_c:Hom(A \cap B, B \cap A)$, $i_c \circ (f \cdot g) = (g \cdot f)$
for partial functions $f:Hom(A, \Im(S))$, $g:Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_a:Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f \cdot (g \cdot h)) = ((f \cdot g) \cdot h)$
for partial function $f:Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B:Ob(Part(S))$, given the local multiplicative unit $1_{B,\Im{S}}:Hom(B, \Im(S)$, $f \cdot 1_{B,\Im{S}} = f$ and $1_{B,\Im{S}} \cdot f = f$
for partial function $f:Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B:Ob(Part(S))$, given the local additive unit $0_{B,\Im{S}}:Hom(B, \Im(S)$, $f \cdot 0_{B,\Im{S}} = 0_{A,\Im{S}}$ and $0_{B,\Im{S}} \cdot f = 0_{A,\Im{S}}$
for partial functions $f:Hom(A, \Im(S))$, $g:Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_l:Hom(A \cap (B \cap C), (A \cap B) \cap (A \cap C)$, $i_a \circ (f \cdot (g + h)) = (f \cdot g) + (f \cdot h)$
for partial functions $f:Hom(A, \Im(S))$, $g:Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_r:Hom((A \cap B) \cap C, (A \cap C) \cap (B \cap C)$, $i_a \circ ((f + g) \cdot h)) = (f \cdot h) + (g \cdot h)$
If $S$ is a Heyting field, then for every subset $A:Ob(Part(S))$, $Hom(A, \Im(Part(S)))$ is a $S$-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
and