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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
  • Definition
    • Informal comments
  • Operation Comma
  • Composition of Dependencies
    • Composition by a property
    • Projection of a Dependency
    • Reverse Dependency
    • Projection, image and domain of a dependency
• Category of dependencies
  • General operations over dependencies
  • Pseudomorphisms
  • Intermorphisms

This newer version of theory of dependencies (and theory of expressions) is a very rough unfinished draft. See here for an older version.

This is a newer version of my theory of dependencies (and theory of expressions). This version is based on my category of pseudomorphisms article, which was yet not written when I was writing the old version of this article.

However the theory of dependencies can be used not only in math logic but also in many other areas 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.)

Because of being based on my newer category of pseudomorphisms work, terminology of this article may be sometimes different (incompatible) with terminology in the old version of this article.

The first section (Objects and Classes) is practically unchanged in this newer version. Changes concern mainly category theory related aspects, which are completely revised in this version.

Objects

In this section I will introduce the concepts of objects and classes (classes are essentially $n$-ary relations). I will call classes with two special properties $X$ and $Y$ (argument and result) dependencies. Dependencies are a generalization of binary relations. For dependencies like as for binary relations are defined composition and reverse.

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.

Definition

I will call an object any function. I will call the domain of an object as set of properties of the object.

I will designate two special properties $X$ and $Y$ called argument and result (or (direct) parent and (direct) child in context of theory of formulas).

I will call a directed object an object whose set of properties contains both $X$ and $Y$.

I will call a directed class or a dependency a set of directed objects.

Remark  The terms property, object, and class are from computer science. In abstract mathematics properties can be called coordinates (of a multidimensional space), objects be called points of a multidimensional space, and these classes which have the same set of properties for every member be called multidimensional relations. I will however use shorter terms properties and objects.

The above can be generalized for the cases of:

• sets of properties with argument, objects with argument, classes with argument, having only the special property $X$ instead of both $X$ and $Y$;
• sets of properties with result, objects with result, classes with result, having only the special property $Y$ instead of both $X$ and $Y$.

However for brevity I will limit our consideration only to objects having both $X$ and $Y$. (It is trivial to generalize the below for cases of having only one of $X$ or $Y$.)

Informal comments

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...

$X$ and $Y$ correspond to Father and Son in New Testament. ($Z$ will correspond to Holy Spirit.) Well, let's return to formal math.

Remark  Following the wisdom contrary to sound mind, we do not need to require dependency of the same directed class to have the same sets of properties. A reader however may assume this requirement to reduce the mess.

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.)

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

◄  ??  ►

Obvious  $dom\left(f[,]g\right)=domf\cup domg$

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

Composition of Dependencies

I will say that directed objects $f$ and $g$ (more generally, $f$ is an object with result and $g$ is an object with argument) stick when $f\left(Y\right)=g\left(X\right)$.

For any two directed objects $f$ and $g$ (I do not require them to stick to each other.) I will define their join by the formula: $$J\left(f,g\right)=\left(f{|}_{domf\setminus \left\{Y\right\}}\right)[,]\left(g{|}_{domg\setminus \left\{X\right\}}\right)\text{.}$$

So join of two directed objects is a directed object.

Lemma  $domJ\left(f,g\right)=\left(domf\setminus \left\{Y\right\}\right)\cup \left(domg\setminus \left\{X\right\}\right)$

◄  From the proposition above.  ►

Lemma  Joining directed objects is an associative binary operation.

◄

Let $f$, $g$, $h$ are directed objects.

Because the operation $[,]$ is associative,

From symmetry of this formula follows $$J\left(J\left(f,g\right),h\right)=J\left(f,J\left(g,h\right)\right)\text{.}$$.

►

Lemma

For any directed objects $f$, $g$, and $h$ (or more generally $f$ may be an object with result and $h$ may be an object with argument):

1. $J\left(f,g\right)$ is stick to $h$ if and only if $g$ is stick to $h$.
2. $f$ is stick to $J\left(f,g\right)$ if and only if $f$ is stick to $g$.

◄  It follows from the formulas $\left(J\left(f,g\right)\right)\left(Y\right)=g\left(Y\right)$ and $\left(J\left(g,h\right)\right)\left(X\right)=g\left(X\right)$.  ►

If we consider a pair as a directed object having two properties $X$ and $Y$ (argument and result), then any binary relation (set of pairs) is a dependency.

Remark  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).

Theorem  Composition of dependencies is an associative binary operation.

◄  Follows from the two lemmas above.  ►

Composition by a property

I will say that objects $f$ and $g$ stick by property $p$ when $f\left(p\right)=g\left(p\right)$.

For any two directed objects $f$ and $g$ having property $p$ (not necessarily stick by $p$) I will define their join by a property $p$ by the formula $$J_p\left(f,g\right)=\left(f{|}_{domf\setminus \left\{p\right\}}\right)[,]\left(g{|}_{domg\setminus \left\{p\right\}}\right)\text{.}$$

Composition by a property $p$ of two classes $U$ and $V$ every object of which has some property $p$ is by definition the set denoted $g{\circ }_{p}f$ of all joins by $p$ of stick by $p$ pairs of objects $f\in U$ and $g\in V$.

Projection of a Dependency

I will call the projection of a directed object $f$ a pair $\left(f\left(X\right),f\left(Y\right)\right)$. I will call the projection of a dependency the set of projections of all its members. (So projection of a dependency is a binary relation.)

Reverse Dependency

Reverse of a directed object $f$ is by definition $$f^{-1}=f{|}_{domf\setminus \left\{X,Y\right\}}\cup \left\{\left(X,f\left(Y\right)\right),\left(Y,f\left(X\right)\right)\right\}\text{.}$$

Reverse of a dependency is by definition the set of reverses of all its directed objects.

Obvious  Reverse of the reverse of a dependency is this dependency, that is ${\left(U^{-1}\right)}^{-1}=U$ for any dependency $U$.

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, image and domain of a dependency

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 $U$.

I will call slice of a class $U$ (every member of which has a property $p$) for the value $a$ of the property $p$.

I will call projection of dependency $U$ to a property $p$ the set $${Pr}_{p}U=\{f\left(p\right)|f\in U\}\text{.}$$

By definition

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

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.

Theorem  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

For any dependency $f$

• $domf=domPrf$;
• $imf=imPrf$;

Limiting and square limiting of a dependency $f$ to a set $A$ are defined correspondingly as:

• $U{|}_A=\{g\in U|g\left(X\right)\in A\}$.
• $U\square A=\{g\in U|g\left(X\right),g\left(Y\right)\in A\}$.

Category of dependencies

I will define the category of dependencies as the category whose objects are sets and whose morphisms from a set $A$ to a set $B$ are triples $\left(f,A,B\right)$ where $f$ is a dependency and

• $domf\subseteq A$;
• $imf\subseteq B$.

Composition of the morphisms is naturally induced by composition of dependencies.

The identity morphism for a set $A$ is the identity relation on the set $A$.

◄

We will prove that $I_A$ (the identity relation on a set $A$) is left identity of the category of dependencies. (For right identity the proof is symmetric.)

We need to prove that $I_A\circ U=f$ for any dependency $U$ whose image is a subset of $A$.

By definition of composition of dependencies $$I_A\circ U=\{\left(f{|}_{domU\setminus \left\{Y\right\}}\right)\left[,\right]\left\{\left(Y,y\right)\right\}|f\in U\wedge y\in A\wedge f\left(Y\right)=y\}\text{.}$$

Because image of $U$ is a subset of $A$, for any $f\in U$ exists exactly one $y\in A$ such that $f\left(Y\right)=y$, so $$I_A\circ U=\left\{\left(f{|}_{domU\setminus \left\{Y\right\}}\right)\left[,\right]\left\{\left(Y,f\left(Y\right)\right)\right\}|f\in U\right\}=\left\{f|f\in U\right\}=U\text{.}$$

►

Additionally I will define reverse of a morphism of the category of dependencies by the formula $${\left(f,A,B\right)}^{-1}=\left(f^{-1},B,A\right)\text{.}$$

$Pr$ becomes a functor from the category of dependencies to the category of sets and binary relations between sets, if it is extended so that it maps an object of the category of dependencies to itself (is identity function on the set of objects of the category of dependencies).

It is easy to prove that image, domain, image of a set (as defined in theory of endomorphisms base on the functor $Pr$) for an object of the category of dependencies is the same as of the corresponding dependency.

We can equate every dependency $U$ with the endomorphism $$\left(U,domU\cup imU,domU\cup imU\right)\text{.}$$ This allows to apply the theory of endomorphisms to dependencies and define such things as dependency series, etc.

Remark  The category $\mathbf{Set}$ of sets and functions between these is a subcategory of the category of dependencies, because functions (as well as any binary relations) are dependencies.

Theorem  The sets if isomorphisms of the category of dependencies is exactly the set of bijective functions (that is isomorphisms of the category $\mathbf{Set}$).

◄

That an isomorphism of $\mathbf{Set}$ is also an isomorphism of the category of dependencies is obvious.

Composition of two dependencies is not identity morphism if one (or both) of these dependencies is not a relation. (This follows from that this composition is obviously even not a relation.) Consequently only a relation can be an isomorphism of the category of dependencies. Any relation which is not a bijective function cannot be isomorphism because its composition with any other relation is obviously not any identity function.

►

Problem for the reader  Formulate analogous statements for retractable and sectionable morphisms instead of isomorphisms.

General operations over dependencies

It is easy to check that the properties of above defined projection, square limiting, image of a set, image and domain conform to the axioms for corresponding operators in my article about pseudomorphisms.

So the theorems and definitions from there apply to the category of dependencies.

Pseudomorphisms

The set of category of dependencies morphisms between any two given sets is partially ordered accordingly set theoretic order ($\subseteq$) of sets. (Recall that every dependency is a set.) Moreover it is infinitely distributive full lattice.

So for the category of dependencies can be applied the theory of pseudomorphisms. There exists the category of pseudomorphisms (and category of intermorphisms) between dependencies (strictly speaking, between endomorphisms of the category of dependencies).

The objects of the category of pseudomorphisms of dependencies are endomorphisms of the category of dependencies. (I will call these endo-dependencies.)

Remark  We also could limit our consideration to only such endo-dependencies $U$ that $SrcU=SrcU=domU\cup imU$ and so consider only (full) subcategories of the categories of pseudomorphism and intermorphisms of dependencies, whose objects are endo-dependencies conforming to the above formula. Limiting our consideration in this way would have the advantage that in this case every object of the category of pseudomorphism of dependencies is completely defined by a dependency (knowing dependency, its source and destination can be calculated by the above formulas).

However it is worth to note that not every pseudomorphism of the category of dependencies is interesting. For example empty $\varnothing \in Pseud\left(U,V\right)$ for any two endo-dependencies $U$ and $V$. Direct product of two sets is often also an intermorphisms of two given dependencies.

We sometimes may wish to limit our consideration to these pseudomorphisms (or intermorphisms) which correspond to monovalued functions (morphisms of the category $\mathbf{Set}$).

Excluding all other pseudomorphisms from the category of pseudomorphisms between dependencies produces a subcategory of the category of pseudomorphisms. (This follows from that composition of functions is a function and every identity morphism of the category of dependencies is a function.) I will call these categories the category of functional pseudomorphisms and the category of functional intermorphisms.

So we have noted two special classes of objects of the category of dependencies: endo-dependencies and morphisms of $\mathbf{Set}$ (functions).

Remark  These two classes intersect with each other, for example identity morphisms of the category of dependencies belong to both classes.

Intermorphisms

I recall that accordingly to the theory of pseudomorphisms isomorphisms between endomorphisms is an intermorphism which has the reverse morphism (which is also an intermorphism).

So we have the concept of isomorphisms between endo-dependencies.

In the case of dependencies the following theorem can be additionally proved:

Theorem

The following statements are equivalent:

1. $\Psi$ is an isomorphism between endo-dependencies $U$ and $V$.
2. ${\Psi }^{-1}$ is an isomorphism between endo-dependencies $V$ and $U$.
3. $\Psi$ is an isomorphism between endo-dependencies $U^{-1}$ and $V^{-1}$.
4. ${\Psi }^{-1}$ is an isomorphism between endo-dependencies $V^{-1}$ and $U^{-1}$.

◄

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

Let $\Psi$ is an isomorphism between endo-dependencies $U$ and $V$.

Then by properties of the category of pseudomorphisms $\Psi =\left(f,U,V\right)$ where $f$ is an isomorphism of the category of dependencies, that is essentially a bijective function.

$f\circ U=V\circ f$. Because $f$ is a bijection, we can multiply (compose) this equation with $f^{-1}$ at both left and right. After multiplication we get $U\circ f^{-1}=f^{-1}\circ V$. So ${\Psi }^{-1}$ is both an intermorphism and an isomorphism (because $f^{-1}$ is an isomorphism) of the category of dependencies, ${\Psi }^{-1}$ is an isomorphism of the category of pseudomorphisms.

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

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

Analogously to the previous item we get that $\Psi$ is an isomorphism of the category of pseudomorphism. [TODO: More detailed proof.]

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

►

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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