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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 - Relations of Elements
  • Ancestry of an Element
  • Categories of Elements
  • Preimage Category
  • Ancestry Categories

Category of Dependencies - Relations of Elements

In this section we will descent from morphisms between classes to relations of individual elements.

I will call elements any objects (I do not mean objects in the sense of the section Objects and Classes here) whatsoever.

That is I will imply that any relation is a relation on the set (more precisely, set theoretic class) of elements; every member of a set is an element.

I will call elements of a dependency $U$ pairs $\left(U,a\right)$ where $a\in domU\cup imU$.

I will call the category of elements of dependencies the category $\mathrm{Elts}$ [TODO: Better notation than $\mathrm{Elts}$ to for this?]

• whose objects are elements of dependencies;
• whose morphisms from $\left(U,a\right)$ to $\left(V,b\right)$ are such functions $f$ from $U$ to $V$ that $f\left(a\right)=b$; (Recall that in my definition of categories one morphism can have several destinations and targets.)
• whose composition is composition of functions.

The identity morphisms of this category would be identity function (equality relation). To eliminate the set theoretic complexities related with domain and image of equality relation being the universal set, we can instead artificially introduce identity morphism as an abstract thing which is distinct from any function and whose left and right compositions with any function are by definition this function.

Ancestry of an Element

I will call the ancestry $Fam\left(U,a\right)$ of an element $a$ of the dependency $U$ dependency $U$ square limited to the set $\left(U^S\right)\left(\left\{a\right\}\right)$.

Remark  This can be trivially generalized for ancestries of sets of elements but I omit this for simplicity.

Remark  My notation $Fam$ is acronym of the word family.

I will call ancestry by the reverse dependency reverse ancestry.

I will call set theoretic union of ancestry and reverse ancestry double ancestry and denote it $DFam\left(U,a\right)$.

Remark  Arguments of functions $Fam$ and $DFam$ are elements of dependencies.

BTW, now we can define connected dependencies. A dependency $U$ is connected if and only if $DFamx=U$ for any element $x$ of dependency $U$.

Categories of Elements

I will call the category whose objects are elements of dependencies and whose morphisms are functions which are both morphisms of $\mathrm{Elts}$ and morphisms of some category of dependencies (e.g. category of isomorphic, homomorphic, pseudomorphic, etc. dependencies) as the category of isomorphic, homomorphic, etc. elements, correspondingly.

When elements $a$ and $b$ are related by morphisms of such categories, I will call $b$ isomorphic, homomorphic, pseudomorphic, etc. image of $a$ as appropriate.

Obvious  The relation of isomorphism of elements of dependencies is an equivalence relation.

Conjecture  Isomorphisms of elements are category theory isomorphisms for the category of homomorphisms of elements.

[TODO: Isomorphic elements can be collapsed together (similar to reduced category in category theory. Isomorphic dependencies can also be collapsed.]

Preimage Category

This subsection can be trivially generalized for any category whose morphisms are functions (i.e. any concrete category). Moreover it seems that it can be generalized for any category by replacing elements with subobjects (as defined in category theory). Haven't category theorists already done this? I'm not sure.

Let $C$ is a category, $\lambda$ is a function acting to the set of objects of $C$. I will call the inverse image of $C$ or preimage of $C$ and denote ${\lambda }^{-1}C$ the category [TODO: Standard term for this in category theory?]

• whose set of objects is $dom\lambda$,
• whose morphisms from $a\in dom\lambda$ to $b\in dom\lambda$ are all morphisms (of category $C$) from $\lambda \left(a\right)$ to $\lambda \left(b\right)$,
• whose composition of morphisms is the same as of $C$.

It is really a category because it has identity morphism $1_{\lambda a}$ for any $a\in dom\lambda$.

When category $C$ has only one object (e.g. $C$ is a category of endomorphisms or automorphisms of a dependency), we can speak about category of elements of this object. So it makes sense to speak about categories of isomorphic, homomorphic, etc. elements of a dependency.

Ancestry Categories

So we can construct the following categories:

• ${Fam}^{-1}C$ (ancestry category™ of isomorphisms, homomorphisms, pseudomorphisms, etc.);
• ${DFam}^{-1}C$ (double ancestry category of isomorphisms, homomorphisms, pseudomorphisms, etc.)

where $C$ are the categories of homomorphisms, isomorphisms, pseudomorphisms etc. of dependencies.

Objects of ancestry and double ancestry categories are elements of dependencies and morphisms are functions.

I will call intersections of these categories with the category of elements ($\mathrm{Elts}$) strict (double) ancestry categories (of isomorphisms, homomorphisms, etc.)

I will call morphisms of (strict) (double) ancestry categories induced by the categories of isomorphisms, homomorphisms, and pseudomorphisms (strict) (double) ancestry isomorphisms™, ancestry homomorphisms™, and ancestry pseudomorphism™. Accordingly this, if elements $a$ and $b$ are correspondingly source and destination of such morphism, then $b$ is called (strict) ancestry isomorphic, ancestry homomorphic, ancestry pseudomorphic, etc. image of $a$ as appropriate.

Remark  An example of ancestry category would be category whose morphisms correspond to graph isomorphisms of subgraphs produces by moving from points of some (possibly disconnected) graph. An example of strict ancestry category would be this category with only these isomorphisms which map the source point to the destination point (that is shifts by loops are not accepted in the strict ancestry category as morphisms).

Conjecture  A strict ancestry category is a full subcategory of the corresponding ancestry category.

Remark  Ancestry morphisms are important for the theory of formulas where e.g. two formulas being ancestry isomorphic means that they have the same structure of subformulas.

Conjecture  When both ancestries and reverse ancestries of some elements of dependencies are isomorphic, then these elements of dependencies are isomorphic to each other.

I deem that the above conjecture is false. Here are two weaker conjectures:

Conjecture  When double ancestries of elements of dependencies are isomorphic, then these elements of dependencies are isomorphic to each other.

Conjecture  If reverse ancestries of any two elements of a dependency are isomorphic then for any two elements of this dependency to be isomorphic is enough if their ancestries (by this dependency) are isomorphic.

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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