nLab category of partial endofunctions

Contents

Context

Category theory

Algebra

Analysis

Contents

Idea

The essentially algebraic structure of partial endofunctions on a set SS, and specific cases for when SS is an abelian group, commutative ring, and field respectively.

Definition

In a set

Given a set SS, the category of partial endofunctions in SS, or just category of partial functions, is the concrete category Part(S)Part(S) with objects called subsets AOb(Part(S))A \in Ob(Part(S)) with the set of elements for each subset El(A)El(A), and the set of morphisms consist of functions Hom(A,(S))(AS)Hom(A, \Im(S)) \coloneqq (A \to S) for each subset AOb(Part(S))A \in Ob(Part(S)), where (S)\Im(S) is the improper subset, as well as the set of monomorphisms Hom(A,B)Hom(A, B) consisting of the subset inclusions for subsets AOb(Part(S))A \in Ob(Part(S)) and B:Ob(Part(S))B:Ob(Part(S)).

There exist a global operator representing composition of partial functions

() Part(S)(): A:Ob(Part(S)) B:Ob(Part(S))Hom(A,(S))×Hom(B,(S))Hom(AB,(S))(-)\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 fHom(A,(S))f \in Hom(A, \Im(S)), gHom(B,(S))g \in Hom(B, \Im(S)), and hHom(C,(S))h \in Hom(C, \Im(S)), given the canonical isomorphism i aHom(A(BC),(AB)C)i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C), i a(f Part(S)(g Part(S)h))=((f Part(S)g) Part(S)h)i_a \circ (f \circ_{Part(S)} (g \circ_{Part(S)} h)) = ((f \circ_{Part(S)} g) \circ_{Part(S)} h)

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)) and subset BAB \subseteq A, there is a function gHom(B,(S))g \in Hom(B, \Im(S)) such that g=f Part(S)i B,Ag = f \circ_{Part(S)} i_{B,A} for canonical injection i B,AHom(B,A)i_{B,A} \in Hom(B,A),

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)) and superset BAB \supseteq A, there is a function hHom(B,(S))h \in Hom(B, \Im(S)) such that h Part(S)i A,B=fh \circ_{Part(S)} i_{A,B} = f for canonical injection i A,BHom(A,B)i_{A,B} \in Hom(A,B),

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)), f=f Part(S)id Sf = f \circ_{Part(S)} id_S and f=id S Part(S)ff = id_S \circ_{Part(S)} f for the identity function id S:Hom((S),(S))id_S:Hom(\Im(S), \Im(S))

In an abelian group

If SS is a abelian group, then for every subset AOb(Part(S))A \in Ob(Part(S)), Hom(A,(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,

()+():{AOb(Part(S))|Hom(A,(S))}×{BOb(Part(S))|Hom(B,(S))}{(A,B)Ob(Part(S))×Ob(Part(S))|Hom(AB,(S))}(-)+(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \times \{B \in Ob(Part(S)) \vert Hom(B, \Im(S))\} \to \{(A,B) \in Ob(Part(S)) \times Ob(Part(S)) \vert Hom(A \cap B, \Im(S))\}
():{AOb(Part(S))|Hom(A,(S))}Hom(A,(S)))-(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \to Hom(A, \Im(S)))

where

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)) and gHom(B,(S))g \in Hom(B, \Im(S)) there is a partial function f+gHom(AB,(S))f + g \in Hom(A \cap B, \Im(S)) and a partial function g+fHom(BA,(S))g + f \in Hom(B \cap A, \Im(S)) such that given the canonical isomorphism i cHom(AB,BA)i_c \in Hom(A \cap B, B \cap A), i c(f+g)=(g+f)i_c \circ (f + g) = (g + f)

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)), gHom(B,(S))g \in Hom(B, \Im(S)), and hHom(C,(S))h \in Hom(C, \Im(S)), given the canonical isomorphism i aHom(A(BC),(AB)C)i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C), i a(f+(g+h))=((f+g)+h)i_a \circ (f + (g + h)) = ((f + g) + h)

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)), and supersets BAB \supseteq A for BOb(Part(S))B \in Ob(Part(S)), given the local additive unit 0 B,SHom(B,(S)0_{B,\Im{S}} \in Hom(B, \Im(S), f+0 B,S=ff + 0_{B,\Im{S}} = f and 0 B,S+f=f0_{B,\Im{S}} + f = f

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)), there is a partial function fHom(A,(S))-f \in Hom(A, \Im(S)) representing negation where the negation of FF is the local additive inverse of ff: f= A,Sf-f = -_{A,S}f

In a commutative ring

If SS is a commutative ring, then for every subset AOb(Part(S))A \in Ob(Part(S)), Hom(A,(Part(S)))Hom(A, \Im(Part(S))) is a SS-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

()():{AOb(Part(S))|Hom(A,(S))}×{BOb(Part(S))|Hom(B,(S))}{(A,B)Ob(Part(S))×Ob(Part(S))|Hom(AB,(S))}(-)\cdot(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \times \{B \in Ob(Part(S)) \vert Hom(B, \Im(S))\} \to \{(A,B) \in Ob(Part(S)) \times Ob(Part(S)) \vert Hom(A \cap B, \Im(S))\}

where

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)) and gHom(B,(S))g \in Hom(B, \Im(S)) there is a partial function fgHom(AB,(S))f \cdot g \in Hom(A \cap B, \Im(S)) and a partial function gfHom(BA,(S))g \cdot f \in Hom(B \cap A, \Im(S)) such that given the canonical isomorphism i cHom(AB,BA)i_c \in Hom(A \cap B, B \cap A), i c(fg)=(gf)i_c \circ (f \cdot g) = (g \cdot f)

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)), gHom(B,(S))g \in Hom(B, \Im(S)), and h:Hom(C,(S))h:Hom(C, \Im(S)), given the canonical isomorphism i aHom(A(BC),(AB)C)i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C), i a(f(gh))=((fg)h)i_a \circ (f \cdot (g \cdot h)) = ((f \cdot g) \cdot h)

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)), and supersets BAB \supseteq A for B:Ob(Part(S))B:Ob(Part(S)), given the local multiplicative unit 1 B,SHom(B,(S)1_{B,\Im{S}} \in Hom(B, \Im(S), f1 B,S=ff \cdot 1_{B,\Im{S}} = f and 1 B,Sf=f1_{B,\Im{S}} \cdot f = f

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)), and supersets BAB \supseteq A for B:Ob(Part(S))B:Ob(Part(S)), given the local additive unit 0 B,SHom(B,(S)0_{B,\Im{S}} \in Hom(B, \Im(S), f0 B,S=0 A,Sf \cdot 0_{B,\Im{S}} = 0_{A,\Im{S}} and 0 B,Sf=0 A,S0_{B,\Im{S}} \cdot f = 0_{A,\Im{S}}

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)), gHom(B,(S))g \in Hom(B, \Im(S)), and h:Hom(C,(S))h:Hom(C, \Im(S)), given the canonical isomorphism i lHom(A(BC),(AB)(AC)i_l \in Hom(A \cap (B \cap C), (A \cap B) \cap (A \cap C), i a(f(g+h))=(fg)+(fh)i_a \circ (f \cdot (g + h)) = (f \cdot g) + (f \cdot h)

  • for partial functions fHom(A,(S))f \in Hom(A, \Im(S)), gHom(B,(S))g \in Hom(B, \Im(S)), and h:Hom(C,(S))h:Hom(C, \Im(S)), given the canonical isomorphism i rHom((AB)C,(AC)(BC)i_r \in Hom((A \cap B) \cap C, (A \cap C) \cap (B \cap C), i a((f+g)h))=(fh)+(gh)i_a \circ ((f + g) \cdot h)) = (f \cdot h) + (g \cdot h)

In a field

If SS is a Heyting field, then for every subset AOb(Part(S))A \in Ob(Part(S)), Hom(A,(S))Hom(A, \Im(S)) is a SS-commutative algebra, with global operators corresponding to composition, addition, negation, and multiplication of partial functions. Let

Hom #0(A,im(S)){fHom(A,(S)).|xEl(A).f(x)#0}Hom_{\#0}(A, \im(S)) \coloneqq \{f \in Hom(A, \Im(S)). \vert \forall x \in 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:

1():{AOb(Part(S))|Hom(A,(S))}Hom #0(A,(S)\frac{1}{(-)}: \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \to Hom_{\#0}(A, \Im(S)

where

  • for partial function fHom(A,(S))f \in Hom(A, \Im(S)),
    f1f=id im(f) #0f \cdot \frac{1}{f} = id_{\im(f)_{\#0}}

    and

    1ff=id im(f) #0\frac{1}{f} \cdot f = id_{\im(f)_{\#0}}

and the set im(f) #0\im(f)_{\#0} is defined as

im(f) #0{xEl(A)|f(x)#0}\im(f)_{\#0} \coloneqq \{x \in El(A)\vert f(x) # 0\}

See also

References

Last revised on June 5, 2022 at 15:34:12. See the history of this page for a list of all contributions to it.