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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

• Proof of the Main Theorem about Dependencies Category

Proof of the Main Theorem about Dependencies Category

Let's reprise the theorem again:

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 as several separate lemmas.

Lemma  A function is a dependency isomorphism from a dependency $U$ to a dependency $V$ if and only if it is a $\mathrm{HomDep}$ isomorphism from $U$ to $V$.

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Direct implication

Let $\Psi$ is a dependency isomorphism from a dependency $U$ to a dependency $V$. Because $\Psi$ is injective, the reverse function ${\Psi }^{-1}$ is defined and

• $dom\left({\Psi }^{-1}\circ \Psi \right)=im\left({\Psi }^{-1}\circ \Psi \right)=dom\Psi =domU\cup imU$;
• $dom\left(\Psi \circ {\Psi }^{-1}\right)=im\left(\Psi \circ {\Psi }^{-1}\right)=im\Psi =domV\cup imV$.

So $\Psi \circ {\Psi }^{-1}$ and ${\Psi }^{-1}\circ \Psi$ are identity morphisms of $U$ and $V$ correspondingly.

Reverse implication

Let $\Psi$ is a category theory isomorphism from a dependency $U$ to a dependency $V$ in category of dependencies with dependency homomorphisms as morphisms. Then there exists some morphism $g$ from $V$ to $U$ such that $\Psi \circ g$ and $g\circ \Psi$ are identity morphisms on $domV\cup imV$ and $domU\cup imU$ correspondingly. But because $\Psi$ and ${\Psi }^{-1}$ are dependency homomorphisms from $U$ to $V$ and from $V$ to $U$ correspondingly, then $dom\Psi =domU\cup imU$ and $im\Psi =domV\cup imV$. Consequently $g={\Psi }^{-1}$ is the reverse function of $\Psi$. $\Psi$ is injective, it is proper dependency homomorphism and dependency epimorphism. So $\Psi$ is a dependency isomorphism.

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Lemma  A function is a dependency monomorphism from a dependency $U$ to a dependency $V$ if and only if it is a $\mathrm{HomDep}$ monomorphism from $U$ to $V$.

◄

Direct implication

Let $\Psi$ is a dependency monomorphism from a dependency $U$ to a dependency $V$. Then $\Psi$ is a dependency isomorphism from $U$ to $V{\square }_{im\Psi }$. Consequently $\Psi$ is a category theory isomorphism from $U$ to $V{\square }_{im\Psi }$. Consequently $\Psi$ is a category theory monomorphism from $U$ to $V{\square }_{im\Psi }$. It is the same that $\Psi$ is a category theory monomorphism from $U$ to $V$.

Reverse implication

If a dependency homomorphism $\Psi$ is not a dependency monomorphism, then $\Psi$ is not injective that is $\Psi \left(a\right)=\Psi \left(b\right)$ for some $a\ne b$. From this easily follows that $\Psi \circ \left\{\left(a,a\right)\right\}=\Psi \circ \left\{\left(b,b\right)\right\}$. Because $\left\{\left(a,a\right)\right\}$ and $\left\{\left(b,b\right)\right\}$ are morphisms acting to $U$, $\Psi$ is not a $\mathrm{HomDep}$ monomorphism.

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Lemma  A function is a dependency epimorphism from a dependency $U$ to a dependency $V$ if and only if it is a $\mathrm{HomDep}$ epimorphism from $U$ to $V$.

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Direct implication

Let $\Psi$ is a dependency epimorphism from $U$ to $V$, that is $im\Psi =domV\cup imV$. We need to prove that for any morphisms $f$ and $g$ from $V$ holds $f\ne g\Rightarrow f\circ \Psi \ne g\circ \Psi$ that is $$\forall b\in domV\cup imV:\left(f\left(b\right)\ne g\left(b\right)\Rightarrow \exists a\in dom\Psi :f\left(\Psi \left(a\right)\right)\ne g\left(\Psi \left(a\right)\right)\right)\text{.}$$ This is true because for any $b\in im\Psi =domV\cup imV$ exists $a\in dom\Psi$ such that $b=\Psi \left(a\right)$.

Reverse implication

If $\Psi$ is not a dependency epimorphism from $U$ to $V$ then exists some $c\in domV\cup imV$ such that $c\notin im\Psi$. Then replacing in any morphism $f$ from $V$ the value at the point $c$ with any other value, we get some morphism $g$ such that $f\circ \Psi =g\circ \Psi$. So $\Psi$ is not a $\mathrm{HomDep}$ epimorphism from $U$ to $V$.

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So the theorem is proved.

[TODO: The above can be easily generalized for certain subcategories of $\mathrm{HomDep}$ such as the categories of dependencies with certain number of properties and/or the category of dependencies on a set (of at least two elements, because we need $a\ne b$ in the proof).]

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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Copyright © 2005 Victor Porton

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