Home News Donate || Math Web browser

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

• Sequential Dependencies
  • Operation Comma and Comma-Simple Sets
  • Definition
  • Morphisms of Sequential Dependencies

Sequential Dependencies

Operation Comma and Comma-Simple Sets

I recall that by definition operation comma is an associative binary operation $,$ such that $$\forall a_1,a_2,b_1,b_2:\left(a_1,a_2\right)=\left(b_1,b_2\right)\Rightarrow a_1=b_1\wedge a_2=b_2\text{.}$$

I will call a comma-simple set such a set $A$ that $\forall n>1,a_1,\dots ,a_n\in A:\left(a_1,\dots ,a_n\right)\notin A$. The term comma-simple set corresponds to the informal notation of one dimensional (not multi-dimensional) set.

Proposition  Let $A$ is a comma-simple set. Let $a_1,\dots ,a_n,b_1,\dots ,b_m\in A$. Then $\left(a_1,\dots ,a_n\right)=\left(b_1,\dots ,b_m\right)\Rightarrow n=m\wedge a_i=b_i$.

◄  It is enough to prove that $n=m$, the rest is obvious. Suppose $n<m$. Then, taking in account associativity of comma, $a_n=\left(b_n,\dots ,b_m\right)$, what is impossible because $A$ is a comma-simple set.  ►

Definition

A dependency $U$ (raised) into degree $n$ where $n$ is a whole number is defined by recursive formulas:

• $U^0=\left(=\right){|}_{domU\cup imU}$ (identity relation on the set $domU\cup imU$);
• $U^n=U\circ U^{n-1}=U^{n-1}\circ U$ for positive $n$;
• $U^n=U^{-1}\circ U^{n+1}=U^{n+1}\circ U^{-1}$ for negative $n$.

It is easy to show that $U$ in the degree $-1$ is the reverse dependency of $U$ and that $U^1=U$.

I recall that sequential dependency $U^S$ corresponding to dependency $U$ is defined by the formula: $$U^S=U^0\cup U^1\cup U^2\cup \dots \text{.}$$

Sequential dependencies are important for the theory of formulas.

Remark  Sequential systems of constructs normally are not with simple indices.

Theorem  ${(V\circ U)}_Z\left(a,b\right)=V_Z\left(b\right)\circ U_Z\left(a\right)$ for two systems of constructs $U$ and $V$. [TODO: Can be generalized for any dependencies?]

◄  By definition $$V\circ U=\{f{|}_{\left\{X,Z\right\}}\left[,\right]g{|}_{\left\{Y,Z\right\}}|f\in U\wedge g\in V\wedge f\left(Y\right)=g\left(X\right)\}=\{\left(\left(X,f\left(X\right)\right);\left(Y,g\left(Y\right)\right);\left(Z,\left(f\left(Z\right),g\left(Z\right)\right)\right)\right)|f\in U\wedge g\in V\wedge f\left(Y\right)=g\left(X\right)\}\text{;}$$ $${\left(V\circ U\right)}_Z\left(a,b\right)=\{\left(\left(X,f\left(X\right)\right);\left(Y,g\left(Y\right)\right)\right)|f\in U\wedge g\in V\wedge f\left(Y\right)=g\left(X\right)\wedge \left(f\left(Z\right),g\left(Z\right)\right)=\left(a,b\right)\}=\{f{|}_{\left\{X\right\}}\left[,\right]g{|}_{\left\{Y\right\}}|f\in U_Z\left(a\right)\wedge g\in V_Z\left(b\right)\wedge f\left(Y\right)=g\left(X\right)\}=V_Z\left(b\right)\circ U_Z\left(a\right)\text{.}$$  ►

Consequence  ${\left(U_n\circ \dots \circ U_1\right)}_Z\left(a_1,\dots ,a_n\right)={U_n}_Z\left(a_n\right)\circ \dots \circ {U_1}_Z\left(a_1\right)$ for systems of constructs.

◄  It can be proved by induction: $${\left(U_n\circ \dots \circ U_1\right)}_Z\left(a_1,\dots ,a_n\right)={\left(U_n\circ \dots \circ U_2\right)}_Z\left(a_n,\dots ,a_2\right)\circ {U_1}_Z\left(a_1\right)={U_n}_Z\left(a_n\right)\circ \dots \circ {U_1}_Z\left(a_1\right)\text{.}$$  ►

Consequence  ${U^n}_Z\left(a_1,\dots ,a_n\right)=U_Z\left(a_n\right)\circ \dots \circ U_Z\left(a_1\right)$ for a system of constructs $U$.

Consequence  ${U^S}_Z\left(a_1,\dots ,a_n\right)=U_Z\left(a_n\right)\circ \dots \circ U_Z\left(a_1\right)$ for a system of constructs $U$ whose set of indices is comma-simple.

◄  It is enough to prove that $\left(a_1,\dots ,a_n\right)\notin ind{U}^k$ for $k\ne n$. $ind{U}^k$ consist exactly of sequences of $n$ comma separated elements of $indU$. The statements to prove follows from the proposition that for comma-simple sets only sequences of equal length can be equal.  ►

Obvious  $U^S\left(A\right)={\left(PrU\right)}^S\left(A\right)=U^0\left(A\right)\cup U^1\left(A\right)\cup U^2\left(A\right)\cup \dots$.

Theorem  A set is closed regarding a dependency if and only if it is closed regarding corresponding sequential dependency.

◄  It follows from the statement about image of composition of dependencies.  ►

Morphisms of Sequential Dependencies

Obvious  $dom{U}^S\cup im{U}^S=domU\cup imU$

Obvious  $\left(U\circ V\right)\cup \left(U\circ W\right)=U\circ \left(V\cup W\right)$

Theorem  A function is homomorphism (pseudomorphism) from $U$ to $V$ if and only if it is homomorphism (pseudomorphism) from $U^S$ to $V^S$. (For the reverse implication to be true is additionally required that the sets of indices of $U$ and $V$ to be subsets of some comma-simple set.)

◄

Direct implication

Homomorphism

$f\circ U=V\circ f\Rightarrow f\circ U\circ U=V\circ V\circ f\Rightarrow \dots$ Uniting all these equalities we get (taking in the account the statement above) that [TODO: More detailed proof.] $f\circ U^S=V^S\circ f$.

Pseudomorphism

Analogous.

Reverse implication

Homomorphism

Let $f\circ U^S=V^S\circ f$. Then $\forall a:f\circ {U^S}_Z\left(a\right)={V^S}_Z\left(a\right)\circ f$. Let $indU\cup indV=K$. By theorem conditions $K$ is a comma-simple set. So taking $i\in K$ we get $f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f$ and consequently $f\circ U=V\circ f$.

Pseudomorphism

Analogous.

►

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

Produced with XSLT Stylesheets for Math.