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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

• Definition and Introduction
  • Third Property $Z$. Systems of Constructs
    • Subformulas (Terminology)
  • Reindexation

Definition and Introduction

Third Property $Z$. Systems of Constructs

In the previous part of this document I have introduced two special properties $X$ and $Y$. Now let introduce the third special property $Z$.

The property $Z$ is called index. (It is also commonly called parameter, but I will not use this term.)

By definition, a system of constructs™ is a class with exactly three properties $X$, $Y$, and $Z$.

That is a system of constructs is essentially a ternary relation.

Obvious  A system of constructs is a dependency.

I will call constructs elements of a system of constructs.

In practice a system of constructs is most often weakly monovalued (see below for definition of weakly monovalued) regarding $Y$, that is is a function from parent construct and index(es) to the child construct.

Remark  The theory of formulas is much more clearly expressed graphically (drawing an object as a dot with three kinds of arrows) than algebraically and a future version of this article should be illustrated, but the software for such graphics is not trivial.

Despite of being so simply defined, systems of constructs (and systems of formulas defined below) indeed produce a rather rich and important for both abstract mathematics and practice such as computer science theory.

[TODO: Example with lists of lists as a system of formulas.]

I will call the set of indices of a system of constructs $U$ the set $indU=\{f\left(Z\right)|f\in U\}$.

Theorem  For system (X, Y, Z) $f$ is a homomorphism (pseudomorphism) from $U$ to $V$ if and only if $\forall i:f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f$ ($\forall i:f\circ U_Z\left(i\right)\subseteq V_Z\left(i\right)\circ f$).

◄  It easily follows from the formulas $f\circ U_Z\left(i\right)={\left(f\circ U\right)}_Z\left(i\right)$ and $V_Z\left(i\right)\circ f={\left(V\circ f\right)}_Z\left(i\right)$.  ►

Subformulas (Terminology)

In theory of formulas the argument ($X$) is commonly called parent formula (or parent expression), the result ($Y$) is called direct child formula (or direct child expression). For the more general case of constructs it can be said parent construct to mean argument ($X$) and child construct to mean result ($Y$).

For systems of constructs the multivalued function $PrU$ is called direct parts or direct subconstructs; for systems of formulas (see below), it is also called direct subformulas or direct subexpressions.

For systems of constructs the multivalued function $Pr{U}^S$ (see below about meaning of $U^S$) is called parts or subconstructs; for systems of formulas (see below), it is also called subformulas or subexpressions.

Reindexation

In practice it may be important to bring two systems of constructs to a common system of indices.

Reindexation of a system of constructs $U$ with function $\lambda$ is $U_Z\circ \lambda$, that is composition of $U$ with $\lambda$, using $Z$ instead of $X$ as the argument of $U$ for the purposes of composition. [TODO: Write more.]

[TODO: Say about bijective, surjective, injective reindexation.]

Reindexation is an interesting operation when it is considered as a morphism in the sense of category theory. [TODO: Say more about this.]

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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