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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

• Objects and Classes
  • Objects and Classes
  • Operation Comma
  • Simple Operations on Classes
  • Special Properties $X$ and $Y$
  • Operations with Classes with Argument and Result
    • Domain and Image
    • Reverse Dependency
    • Limiting a Dependency
      • Limiting a Dependency
      • Square Limiting a Dependency
    • Composition
    • Projection of a Dependency. Image of a Set. Closed Sets
    • Sequential Dependencies
• Category of Dependencies
  • Homomorphisms
  • Pseudomorphisms
• Proof of the Main Theorem about Dependencies Category
• Category of Dependencies - Relations of Elements
  • Ancestry of an Element
  • Categories of Elements
  • Preimage Category
  • Ancestry Categories
• Licensing etc.
  • About the Original Method Authorship
  • License

Objects and Classes

In this section I will introduce the concepts of objects and classes. Classes introduced in this article is a generalization of $n$-ary relations.

Notwithstanding that the idea of objects and classes has come from a particular math application namely computer science, the theory in this article is fundamental abstract mathematics, not just an application.

I will also introduce two special properties $X$ and $Y$ (argument and result). Classes with argument and result (dependencies, see below) will be a generalization of binary relations. For such classes composition (generalizing composition of binary relations) is defined.

These concepts were introduced by me in order to research properties of mathematical formulas (expressions) on a very abstract level for purposes of math logic, but they may also find other applications in other fields of mathematics.

The theory of dependencies can be considered as an intermediary between universal algebra (and model theory in general) and category theory, considered more abstract and general than universal algebra, and less general than category theory. On the other hand, dependencies can be used to model categories, in this sense the theory of dependencies may be considered as a more general theory than category theory. (The relations of the theory of classes with universal algebra and category theory are not yet thoroughly researched, however.)

Objects and Classes

Let $\mathrm{Prop}$ is a set (called properties). Anything said here is not limited to finite set of properties but you may consider only finite sets of properties, or even no more than three-elements sets. (It is enough for the theory of formulas).

I will call an object a function from a subset of $\mathrm{Prop}$.

Remark  The terms property and object are from computer science. In abstract mathematics properties can be called coordinates (of a multidimensional space), and objects be called points of a multidimensional space. I will however use shorter terms properties and objects.

A set $U$ of objects with common domain $Prop\left(U\right)$ can be called either a class (a term from computer science) or a multidimensional relation (a term from abstract mathematics). A class essentially is a $n$-ary operation, where $n=card\mathrm{Prop}$ (properties are names of the arguments of this relation).

Operation Comma

By definition comma $,$ is associative binary operation which is injective that is $a_1\ne a_2\vee b_1\ne b_2\Rightarrow \left(a_1,b_1\right)\ne \left(a_2,b_2\right)$.

The operation $\left[,\right]$ applied to two objects $f$ and $g$ is by definition $$f\left[,\right]g=f{|}_{domf\setminus domg}\cup g{|}_{domg\setminus domf}\cup \{\left(f\left(x\right),g\left(x\right)\right)|x\in domf\cap domg\}\text{.}$$ (Here comma is used in the sense of above defined operation comma.)

Obvious  The operation $\left[,\right]$ is associative.

[TODO: This article should be corrected to use semicolons instead of comma where comma is not appropriate.

Simple Operations on Classes

The value of a class $U$ on the value $a$ of the property $q$ is by definition $$U_q\left(a\right)=\{f{|}_{PropU\setminus \left\{q\right\}}|f\in U\wedge f\left(q\right)=a\}\text{.}$$

The projection of a class $U$ to the property $q$ is by definition $${Pr}_{q}U=\left\{f\left(q\right)|f\in U\right\}\text{.}$$

Special Properties $X$ and $Y$

I will designate two special properties $X$ and $Y$ (argument and result).

Binary relations can be considered as classes with exactly two properties $X$ and $Y$. Strictly speaking, the indices of elements in a pair are $1$ and $2$ (or $0$ and $1$ if we number not accordingly tradition but accordingly contemporary mathematics), but writing instead $X$ and $Y$ is less confusing (at least today; in the future mathematical notation shall be more unified, and hopefully we will eventually agree whether to start numbering from zero or one).

Oh, well. Now the concepts of argument and result are formalized. Haven't I discovered $X$ and $Y$, yeah? Anyway, in former times these were called unknown...

BTW, $X$ here denotes the Father and and $Y$ denotes the Son. (Index as defined below will be Holy Spirit.) Well, let's return to formal math.

I will call classes with properties $X$ and $Y$ classes with argument and result or dependencies (of $Y$ from $X$).

Remark  Alternatively we could call a dependency a system of a class and a pair of its properties, but that variant is much less fortunate than carefully crafted variant that two special common properties $X$ and $Y$ are designated to be special for all classes instead of having different special properties for different classes.

Operations with Classes with Argument and Result

Having special properties $X$ and $Y$ we can define some operations on classes, which are generalizations of the corresponding operations on binary relations.

Domain and Image

Domain and image of dependencies are generalization of domain and image of relations.

By definition

• $domU={Pr}_{X}U$;
• $imU={Pr}_{Y}U$.

Reverse Dependency

Reverse dependency is a generalization of reverse relation.

By definition $U^{-1}=\{f{|}_{Prop\left(U\right)\setminus \left\{X,Y\right\}}\cup \left\{\left(X,f\left(Y\right)\right),\left(Y,f\left(X\right)\right)\right\}|f\in U\}$.

Obvious  ${\left(U^{-1}\right)}^{-1}=U$ for any dependency $U$.

Limiting a Dependency

There are two kinds of limiting a dependency to a set: limiting and square limiting.

Limiting a Dependency

A dependency $U$ limited to a set $A$ is by definition $U{|}_A=\{g\in U|g\left(X\right)\in A\}$.

Square Limiting a Dependency

A dependency $U$ square limited to a set $A$ is by definition $U{\square }_A=\{g\in U|g\left(X\right),g\left(Y\right)\in A\}$.

Composition

By definition composition of classes $U$ and $V$ is $$V\circ U=\{\left(f{|}_{Prop\left(U\right)\setminus \left\{Y\right\}}\right)\left[,\right]\left(g{|}_{Prop\left(V\right)\setminus \left\{X\right\}}\right)|f\in U\wedge g\in V\wedge f\left(Y\right)=g\left(X\right)\}\text{.}$$

Recall that we consider binary relations as a particular case of classes (with only $X$ and $Y$ properties). So left and right composition of a dependency with a binary relation (and specifically with a unary function) is defined. (We will use this below to define morphisms between dependencies.)

Theorem  Composition of dependencies is an associative operation.

◄

(I suspect that there should be a shorter proof...)

Let $U$, $V$, $W$ are classes (with special properties).

$$W\circ \left(V\circ U\right)=\{\left(c{|}_{Prop\left(V\circ U\right)\setminus \left\{Y\right\}}\right)\left[,\right]\left(h{|}_{Prop\left(W\right)\setminus \left\{X\right\}}\right)|c\in V\circ U\wedge h\in W\wedge c\left(Y\right)=h\left(X\right)\}\text{.}$$ $c=\left(f{|}_{Prop\left(U\right)\setminus \left\{Y\right\}}\right)\left[,\right]\left(g{|}_{Prop\left(V\right)\setminus \left\{X\right\}}\right)$ where $f\in U\wedge g\in V\wedge f\left(Y\right)=g\left(X\right)$; $c\left(Y\right)=g\left(Y\right)$

So $$c{|}_{Prop\left(V\circ U\right)\setminus \left\{Y\right\}}=\left(f{|}_{Prop\left(U\right)\setminus \left\{Y\right\}}\right)\left[,\right]\left(g{|}_{Prop\left(V\right)\setminus \left\{X,Y\right\}}\right)$$

$W\circ \left(V\circ U\right)=\left(W\circ V\right)\circ U$ follows from symmetry of the above formula.

►

Proposition  ${\left(V\circ U\right)}^{-1}=U^{-1}\circ V^{-1}$ for any dependencies (and even any classes) $U$ and $V$.

◄  Follows from symmetry.  ►

Projection of a Dependency. Image of a Set. Closed Sets

[TODO: Split this section into several smaller sections.]

Projection of a dependency (onto the direction $X\to Y$) is defined as $$PrU=\left\{f{|}_{\left\{X,Y\right\}}|f\in U\right\}=\left\{\left(f\left(X\right),f\left(Y\right)\right)|f\in U\right\}\text{.}$$

Such projection is a binary relation.

I will call $U\left(A\right)=\{f\left(Y\right)|f\in U\wedge f\left(X\right)\in A\}$ the image of the set $A$ by a dependency $U$.

Definition  A set $A$ is closed regarding a dependency $U$ if and only if $U\left(A\right)\subseteq A$

Obvious  Image of a set by a dependency is image of the set by the projection of the dependency that is $U\left(A\right)=\left(PrU\right)\left(A\right)$ for any dependency $U$ and set $A$.

Consequence  A set is closed regarding a dependency if and only if it is closed regarding its projection.

Remark  The predicate of being closed is asymmetric (it changes, if $X$ and $Y$ are interchanged with each other). This suggest the idea that some kind of an asymmetric category of dependencies should be introduced, in addition to the symmetric $\mathrm{HomDep}$ (see below) category. (Maybe, it is related with pseudomorphisms, see below.)

Obvious  Composition of projection of dependencies is projection of compositions, that is $Pr\left(V\circ U\right)=\left(PrV\right)\circ \left(PrU\right)$

Obvious  Image of a set by composition of dependencies is the image of the image of this set, that is $\left(V\circ U\right)A=V\left(U\left(A\right)\right)$.

Obvious  Image of a dependency is image of its domain by this dependency, that is $imU=U\left(domU\right)$.

Sequential Dependencies

Sequential dependency $U^S$ corresponding to dependency $U$ is defined by the formula: $$U^S=\left(=\right){|}_{domU\cup imU}\cup U\cup \left(U\circ U\right)\cup \left(U\circ U\circ U\right)\cup \dots \text{.}$$

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 $\Psi \circ U\supseteq V\circ \Psi$.]

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 $V$.

◄

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.

►

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.

►

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 $V$.

◄

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 $V$.

►

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).]

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.

Licensing etc.

About the Original Method Authorship

21st Century Math Method, Operator Theory of Formulas, and Algebraic General Topology and Math Synthesis were developed by Victor Porton (http://ex-code.com/~porton/). See www.mathematics21.org

Please donate to the method author to help in further research.

License

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• About the Original Method Authorship,

with the Front-Cover Texts being

• More on the site www.mathematics21.org,

and with the Back-Cover Texts being

• This method was developed by Victor Porton during his revolutionary research Algebraic General Topology and Math Synthesis. See the Web site www.mathematics21.org

For using this document in Wiki like collaboration areas this license requirement is weakened by allowing to apply to front and back covers invariant sections rules instead of front and back cover rules in Wiki collaboration areas.

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.

License

See here about licensing of this text etc.

Related Web Categories

See also my suggestions for new mathematics classification categories.

• /Computers/Computer_Science/Theoretical/
• /Science/Math/Algebra/
• /Science/Math/Logic_and_Foundations/
• /Science/Math/Logic_and_Foundations/Foundations/
• /Science/Math/Logic_and_Foundations/Model_Theory/
• /Science/Math/Algebra/Category_Theory/
• /Science/Math/Publications/Online_Texts/
• /Science/Math/Logic_and_Foundations/Education/
• /Science/Math/Education/

Copyright © 2005 Victor Porton

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