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Axiomatic Theory of Formulas pages on this site.

This document is rough unfinished draft. See also an older version of formulas theory article.

Table of Contents

• Category of Dependencies
  • Homomorphisms
  • Pseudomorphisms

Category of Dependencies

I will use a variation of the standard definition of category with one morphism having multiple sources and destination:

Definition

A category is a system of:

• a set $\mathrm{Ob}$ of objects; (Do not confuse category theory objects with objects defined above.)
• a set of morphisms (also called arrows);
• set of source objects $Srcf$ and set of destination objects $Dstf$ for every morphism $f$;
• composition $gf$ of every two morphisms $f$ and $g$ such that $Dstf\cap Srcg\ne \varnothing$ with properties:
  • $gf$ is a morphism;
  • $Src\left(gf\right)=Srcf$ and $Dst\left(gf\right)=Dstg$;
  • for any object $U$ exists identity morphism $1_U$ with source and destination containing $U$ such that $1_{U}f=f{1}_U=f$ for any such morphism $f$ whose source and destination contain $U$.

This definition has the advantage that I can to not include destination and source into the value of a morphism when I don't need to.

Let it be a problem for the reader to show how this definition relates with standard definition of a category where every morphism has exactly one source and destination object.

I will call a dependency (dependency from $X$ to $Y$ is implied) a class having both special properties $X$ and $Y$.

Obvious  Composition of two dependencies is a dependency.

So a dependency is a generalization of both multidimensional relation and of binary relation. It has advantages of both, on one hand it is multidimensional and on the other hand composition of dependencies like as for binary relations is defined (unlike multidimensional relations for which composition is not defined).

Traditionally a dependency $f$ is often denotes like $Y=f\left(X\right)$. I however will not use this notation because it is vague and indeterminate.

Dependencies are often encountered in multiple variable analysis, e.g. in theory of partial differential equations and in physics. But I introduced dependencies with an other purpose, to research mathematical expressions (formulas), that is in the field of discrete mathematics. The concept of dependency (and category of dependencies) is however interesting by itself and will grow into a separate math discipline not limited to any one of the above mentioned applications.

Dependencies form a category with morphisms being pseudomorphisms (defined below) and subcategory of homomorphisms which are a special case of pseudomorphisms. Why I introduced the new concept of dependency at all? Because these have interesting morphisms and form an important category.

A function $\Psi$ is called a extended function from a dependency $U$ to a dependency $V$ if and only if both:

1. $dom\Psi \supseteq domU\cup imU$;
2. $im\Psi \subseteq domV\cup imV$.

When the first requirement of the above definition is strengthened to be instead $dom\Psi =domU\cup imU$, I will call it function from a dependency $U$ to a dependency $V$.

Remark  I consider functions a more important concept than extended functions, because extended functions don't well agree with category theory morphisms (see below). Extended functions is a supplementary topic.

Lemma  $domV\supseteq imU\Rightarrow dom\left(V\circ U\right)=domU$ for any dependencies $U$ and $V$ (provided that $V\circ U$ is defined).

◄

That $dom\left(V\circ U\right)\subseteq domU$ is obvious. Let's prove $dom\left(V\circ U\right)\supseteq domU$.

$domV\supseteq imU$ means $\forall f\in U:\exists g\in V:g\left(X\right)=f\left(Y\right)$. So for any $x\in domU$ exists $f\in U$ such that $f\left(X\right)=x$ and $g\in V$ such that $g\left(X\right)=f\left(Y\right)$. Consequently $h=\left(f{|}_{Prop\left(U\right)\setminus \left\{Y\right\}}\right)\left[,\right]\left(g{|}_{Prop\left(V\right)\setminus \left\{X\right\}}\right)\in V\circ U$; $h\left(X\right)=f\left(X\right)=x$; $x\in domV\circ U$.

►

Theorem  If $f$ is an extended function from a dependency $U$ to a dependency $V$ and $g$ is an extended function from a dependency $V$ to a dependency $W$, then $g\circ f$ is an extended function from the dependency $U$ to the dependency $W$.

◄

We have $imf\subseteq domV\cup imV\subseteq domg$. So by the lemma $dom\left(g\circ f\right)=domf\supseteq domU\cup imU$. Obviously $im\left(g\circ f\right)\subseteq img\subseteq domW\cup imW$.

►

Consequence  If $f$ is a function from dependency $U$ to a dependency $V$ and $g$ is a function from a dependency $V$ to dependency $W$, then $g\circ f$ is a function from the dependency $U$ to the dependency $W$.

So, dependencies with (extended) functions form a category. (Dependencies are category theory objects, functions are the arrows, identity function on $domU\cup imU$ is the identity morphism for a dependency $U$.)

Below there will be introduced some more specific categories.

[TODO: The above (and below about morphisms) can be generalized for relations (rather than only mono-valued functions). Moreover it can be generalized for morphisms themselves to be classes (so introducing concept of multidimensional morphisms, in a n other way than in the multidimensional category theory).]

Homomorphisms

An (extended) function $\Psi$ from a dependency $U$ to a dependency $V$ is called an (extended) dependency homomorphism if and only if $\Psi \circ U=V\circ \Psi$.

Homomorphism as defined above is a particular case of morphisms in category theory. It (seems) also to be a generalization of homomorphism in universal algebra ($U$ and $V$ can be algebras, with properties being finite sequence of elements and indices (see below) being algebraic operations). So algebraization of mathematics has reached a worth-note point, algebraization of algebra itself started, the concept of homomorphism is defined in algebraic terms.

Theorem  If $f$ is an (extended) homomorphism from a dependency $U$ to a dependency $V$ and $g$ is an (extended) homomorphism from a dependency $V$ to a dependency $W$, then $g\circ f$ is an (extended) homomorphism from the dependency $U$ to the dependency $W$.

◄

That $g\circ f$ is a function from the dependency $U$ to the dependency $W$ is already proved.

$g\circ f\circ U=g\circ V\circ f=W\circ g\circ f$. (Associativity of composition was taken into account.)

►

Below for short I will skip anything about extended morphisms as these are not important and confusing.

Injective dependency homomorphism is called dependency monomorphism.

Dependency homomorphism such that $im\Psi =domV\cup imV$ is called dependency epimorphism (with the destination $V$).

Dependency homomorphism is called dependency isomorphism if and only if it is both dependency monomorphism and dependency epimorphism.

I will call the homo-category of dependencies ($\mathrm{HomDep}$) the category whose objects are dependencies, and whose arrows are dependency homomorphisms. (Identity morphisms for a dependency $U$ is the identity function on $domU\cup imU$.)

Remark  I suggest to denote the supplementary category whose arrows are instead extended dependency homomorphisms as $\mathrm{ExtHomDep}$.

Theorem  Dependency homomorphisms, dependency monomorphisms, and dependency isomorphisms are the same as morphisms, monomorphisms, and isomorphisms of $\mathrm{HomDep}$, correspondingly.

I will prove this theorem below in a separate section.

Theorem

The following statements are equivalent:

1. $\Psi$ is a dependency isomorphism from a dependency $U$ to a dependency $V$;
2. ${\Psi }^{-1}$ is a dependency isomorphism from a dependency $V$ to a dependency $U$;
3. $\Psi$ is a dependency isomorphism from a dependency $U^{-1}$ to a dependency $V^{-1}$.
4. ${\Psi }^{-1}$ is a dependency isomorphism from a dependency $V^{-1}$ to a dependency $U^{-1}$.

◄

$\text{(1)}\Rightarrow \text{(2)}$

Let $\Psi$ is a dependency isomorphism from a dependency $U$ to a dependency $V$.

Because $\Psi$ is a proper homomorphism, $im{\Psi }^{-1}=dom\Psi =domU\cup imU$, that is ${\Psi }^{-1}$ is an epimorphism. ${\Psi }^{-1}$ is injective as any inverse of a function. Because $\Psi$ is a epimorphism, $dom{\Psi }^{-1}=im\Psi =domV\cup imV$. So ${\Psi }^{-1}$ is a monomorphism.

$\Psi \circ U=V\circ \Psi$; we can multiply (compose) this equation with ${\Psi }^{-1}$ at both left and right because ${\Psi }^{-1}$ is bijective and $im\Psi \supseteq domV\wedge dom\Psi \supseteq imU$. After multiplication we get $U\circ {\Psi }^{-1}={\Psi }^{-1}\circ V$. So ${\Psi }^{-1}$ is a homomorphism from $V$ to $U$. [TODO: More detailed proof.]

$\text{(2)}\Rightarrow \text{(3)}$

From ${\Psi }^{-1}\circ V=U\circ {\Psi }^{-1}$ we get $\Psi \circ U^{-1}=V^{-1}\circ \Psi$.

Taking in account that $domU\cup imU=dom{U}^{-1}\cup im{U}^{-1}$ and likewise for $V$, we get that $\Psi$ is a dependency isomorphism from a dependency $U^{-1}$ to a dependency $V^{-1}$.

The rest implications to prove equivalence of all four items easily follow from the above two implications:

$\text{(3)}\Rightarrow \text{(4)}$

From $\text{(1)}\Rightarrow \text{(2)}$.

$\text{(4)}\Rightarrow \text{(1)}$

From $\text{(2)}\Rightarrow \text{(3)}$.

►

Theorem  If $\Psi$ is a homomorphism from a dependency $U$ to a dependency $V$, then $\Psi$ is an epimorphism from $U$ to $V{\square }_{im\Psi }$.

◄

That $im\Psi =domV{\square }_{im\Psi }\cup imV{\square }_{im\Psi }$ is obvious.

$\Psi \circ U=V\circ \Psi$; from this follows (??) that $\Psi \circ U=V{\square }_{im\Psi }\circ \Psi$.

►

Consequence  If $\Psi$ is a monomorphism from a dependency $U$ to a dependency $V$, then $\Psi$ is an isomorphism from $U$ to $V{\square }_{im\Psi }$.

Pseudomorphisms

Pseudomorphisms defined below are important for the theory of formulas (expressions) which is developed by me for the purposes of math logic and computer science. Pseudomorphisms may also find other usages for themselves.

A function $\Psi$ from a dependency $U$ to a dependency $V$ is called a dependency pseudomorphism™ if and only if $\Psi \circ U\subseteq V\circ \Psi$.

Theorem  If $f$ is a pseudomorphism from a dependency $U$ to a dependency $V$ and $g$ is a pseudomorphism from a dependency $V$ to a dependency $W$, then $g\circ f$ is a pseudomorphism from the dependency $U$ to the dependency $W$.

◄

That $g\circ f$ is a function from the dependency $U$ to the dependency $W$ is already proved.

►

So, dependencies with pseudomorphisms form a category. I will call this category pseudo-category™ of dependencies and denote it $\mathrm{PseudDep}$.

Obvious  A dependency homomorphism is a dependency pseudomorphism.

Consequence  The category of dependencies with homomorphisms as morphisms is a subcategory of the category of dependencies with pseudomorphisms as morphisms.

I will call injective dependency pseudomorphism pseudo-monomorphism.

I will call dependency pseudomorphism $\Psi$ from $U$ to $V$ a pseudo-epimorphism if and only if $im\Psi =domV\cup imV$.

I will call a function $\Psi$ pseudo-isomorphism from $U$ to $V$ when it is both pseudo-monomorphism and pseudo-epimorphism from $U$ to $V$.

Theorem  If $\Psi$ is an injective function and $\Psi$ and ${\Psi }^{-1}$ are pseudomorphisms from $U$ to $V$ and from $V$ to $U$ correspondingly, then $\Psi$ is dependency isomorphism.

◄

$\Psi \circ U\subseteq V\circ \Psi$; ${\Psi }^{-1}\circ V\subseteq U\circ {\Psi }^{-1}$; ${\Psi }^{-1}\circ V\circ \Psi \subseteq U$; Because ${\Psi }^{-1}$ is a pseudomorphism from $V$ then $im\Psi =domV\cup imV\supseteq imV$; this (taking in account that $\Psi$ is a bijection) allows to multiply (compose) both sides of the previous formula with $\Psi$ at the left side: $V\circ \Psi \subseteq \Psi \circ U$. So $\Psi \circ U=V\circ \Psi$ that is $\Psi$ is a homomorphism. Taking in account that $\Psi$ is injective and that $im\Psi =domV\cup imV$; we prove that $\Psi$ is an isomorphism.

►

[TODO: Generalize the main theorem about category of dependencies to pseudomorphisms.]

Theorem  If $\Psi$ is a pseudomorphism from a dependency $U$ to a dependency $V$, then $\Psi$ is an pseudo-epimorphism from $U$ to $V{\square }_{im\Psi }$.

◄

That $im\Psi =domV{\square }_{im\Psi }\cup imV{\square }_{im\Psi }$ is obvious.

$\Psi \circ U\subseteq V\circ \Psi =V{|}_{im\Psi }\circ \Psi$; from this follows that $\Psi \circ U\subseteq V{\square }_{im\Psi }\circ \Psi$.

►

Consequence  If $\Psi$ is a pseudo-monomorphism from a dependency $U$ to a dependency $V$, then $\Psi$ is a pseudo-isomorphism from $U$ to $V{\square }_{im\Psi }$.

[TODO: We could also introduce reverse pseudomorphisms by the formula $\Psi \circ U\supseteq V\circ \Psi$.]

Related Links

• Axiomatic Theory of Formulas pages on this site.
• This article as one XHTML+MathML file.
• Axiomatic Theory of Formulas (old version of this theory, essentially based on universal algebra; but a more ready document than this). Victor Porton. 2005.
  • first part;
  • second part (Representation Related Properties of Formulas).
• Definition of category with multiple domains and ranges.
• 21 Century Math Method (rough draft), a logical method which will use this theory of formulas.
• 21 Century Math Method in Programming (rough draft).
• Algebra of transformations of formulas (a weblog message by Victor Porton).
• Software for 21 Century Math Method.
• 21 Century Math Method - Further Research Plans.
• www.mathematics21.org
• Journal of post-Axiomatic Mathematics and Logic.

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