[[!redirects category of partial functions]] #Contents# * table of contents {:toc} ## Idea ## 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 (ring theory)|field]] respectively. ## Definition ## ### In a set ### Given a set $S$, the **category of [[partial function]]s** in $S$ is the [[concrete category]] $Part(S)$ with objects called [[subset]]s $A:Ob(Part(S))$ 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** $$(-)\circ_{Part(S)}(-): \sum_{A:Ob(Part(S))} \sum_{B:Ob(Part(S))} Hom(A, \Im(S)) \times Hom(B, \Im(S)) \to Hom(A \cap B, \Im(S))$$ 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 $id_S:Hom(\Im(S), \Im(S))$ ### In an abelian group ### 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**, $$(-)+(-): \sum_{A:Ob(Part(S))} \sum_{B:Ob(Part(S))} Hom(A, \Im(S)) \times Hom(B, \Im(S)) \to Hom(A \cap B, \Im(S))$$ $$-(-): \sum_{A:Ob(Part(S))} Hom(A, \Im(S)) \to Hom(A, \Im(S)))$$ 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 $f$: $-f = -_{A,S}f$ ### In a commutative ring ### If $S$ is a [[commutative ring]], then for every subset $A:Ob(Part(S))$, $Hom(A, \Im(Part(S)))$ is a $S$-[[commutative algebra (ring theory)|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** $$(-)\cdot(-): \sum_{A:Ob(Part(S))} \sum_{B:Ob(Part(S))} Hom(A, \Im(S)) \times Hom(B, \Im(S)) \to Hom(A \cap B, \Im(S))$$ 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 $i_r:Hom((A \cap B) \cap C, (A \cap C) \cap (B \cap C)$, $i_a \circ ((f + g) \cdot h)) = (f \cdot h) + (g \cdot h)$ ### In a field ### If $S$ is a [[Heyting field]], then for every subset $A:Ob(Part(S))$, $Hom(A, \Im(Part(S)))$ is a $S$-[[commutative algebra (ring theory)|commutative algebra]], with global operators corresponding to composition, addition, negation, and multiplication of partial functions. Let $$Hom_{\#0}(A, \im(S)) \coloneqq \sum_{f:Hom(A, \Im(S))} \prod_{x:El(A)} (f(x) # 0)$$ 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**: $$\frac{1}{(-)}: \sum_{A:Ob(Part(S))} Hom(A, \Im(S)) \to Hom_{\#0}(A, \Im(S))$$ where * for partial function $f:Hom(A, \Im(S))$, $$\frac{1}{f} \cdot \frac{1}{\frac{1}{f}} \cdot \frac{1}{f} = \frac{1}{f}$$ ## See also ## * [[partial function]] * [[2-poset of partial maps]] ## References ## * Fred Richman, [Algebraic functions, calculus style](https://web.archive.org/web/20130605213603/http://math.fau.edu/richman/Docs/Oily.pdf)