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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

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: $$U^S=\left(=\right){|}_{domU\cup imU}\cup U\cup \left(U\circ U\right)\cup \left(U\circ U\circ U\right)\cup \dots \text{.}$$

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

Copyright © 2005 Victor Porton

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