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This document is rough unfinished draft. See also an older version of formulas theory article.

Table of Contents

• Factor of Two Constructs
  • Factor of Two Constructs
  • Criterion of Pseudomorphic Image

Factor of Two Constructs

Lemma  If $U$ and $V$ are binary relations, $U$ is monovalued (a function), and $a$ and $b$ are elements of $U$ and $V$ correspondingly, then there exist no more than one strict ancestry disjunctive pseudomorphism from $a$ to $b$ (regarding $U$ and $V$).

◄

We will prove that for this strict ancestry disjunctive pseudomorphism $f$ (if it exists) $f{U}^i\left(a\right)=V^i\left(b\right)$ for $i=0,1,2,\dots$ (until either $U^i\left(a\right)=\varnothing$ or all natural numbers are exhausted), and so $f$ can take only one certain value on every element of the ancestry of $a$.

For $i=0$ the formula to prove is $f\left(a\right)=b$ which is true because $f$ is strict.

For $i>0$ it can be proved by induction (in assumption $U^i\left(a\right)\ne \varnothing$): We have $f\left(U^i\left(a\right)\right)=f\left(U\left(U^{i-1}\left(a\right)\right)\right)=V\left(f\left(U^{i-1}\left(a\right)\right)\right)=V\left(V^{i-1}\left(b\right)\right)=V^i\left(b\right)$. I have taken into account that $f\left(U\left(x\right)\right)\ne \varnothing \Rightarrow f\left(U\left(x\right)\right)=V\left(f\left(x\right)\right)$ (as $card{U}^i\left(x\right)=1$) together with the fact that $f$ is defined on all ancestry of $a$.

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Consequence  If $U$ and $V$ are monovalued binary relations (functions) and $a$ and $b$ are elements of $U$ and $V$ correspondingly, then there exist no more than one strict ancestry pseudomorphism from $a$ to $b$ (regarding $U$ and $V$).

◄  It follows from that any pseudomorphism from a part of $U$ to a part of $V$ is a disjunctive pseudomorphism.  ►

Theorem  If $U$ and $V$ are systems of constructs, $U$ is weakly monovalued, and $a$ and $b$ are elements of $U$ and $V$ correspondingly, then there exist no more than one strict ancestry disjunctive pseudomorphism from $a$ to $b$ (regarding $U$ and $V$).

◄

Let $f$ is such a morphism that is disjunctive pseudomorphism from $U_a=U{\square }_{U^S\left(a\right)}$ to $V_b=V{\square }_{V^S\left(b\right)}$. Note that

• ${\left(U_a\right)}_Z\left(i\right)=U_Z\left(i\right){\square }_{U^S\left(a\right)}$;
• ${\left(V_b\right)}_Z\left(i\right)=V_Z\left(i\right){\square }_{V^S\left(b\right)}$.

We have:

• $domf=U^S\left(a\right)$;
• $imf=V^S\left(b\right)$;

• $dom\left({\left(U_a\right)}_Z\left(i\right)\right)=dom{U}_Z\left(i\right)\cap dom{U}^S\left(a\right)$;
• $im\left({\left(U_a\right)}_Z\left(i\right)\right)=im{U}_Z\left(i\right)\cap im{U}^S\left(a\right)$;

• $dom\left({\left(V_a\right)}_Z\left(i\right)\right)=dom{V}_Z\left(i\right)\cap dom{V}^S\left(a\right)$;
• $im\left({\left(V_a\right)}_Z\left(i\right)\right)=im{V}_Z\left(i\right)\cap im{V}^S\left(a\right)$.

For any $i$ exists such set $P\left(i\right)$ that $f\circ {\left(U_a\right)}_Z\left(i\right)={\left(V_b\right)}_Z\left(i\right)\circ f{|}_{P\left(i\right)}$.

Taking in account that $im\left({\left(U_a\right)}_Z\left(i\right)\right)\subseteq dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)$ and $im\left({\left(U_a\right)}_Z\left(i\right)\right)\subseteq dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)$, we get $f{|}_{dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)}\circ {\left(U_a\right)}_Z\left(i\right)={\left(V_b\right)}_Z\left(i\right)\circ f{|}_{P\left(i\right)\cap \left(dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)\right)}$

So $f{|}_{dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)}$ is a disjunctive pseudomorphism from ${\left(U_a\right)}_Z\left(i\right)$ to ${\left(V_b\right)}_Z\left(i\right)$, by the lemma $f{|}_{dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)}$ is defined in a certain single possible way. (I have taken in account that $f\left(a\right)=b$.) $$domU\cup imU={\bigcup }_{i}dom{U}_Z\left(i\right)\cup {\bigcup }_{i}im{U}_Z\left(i\right)={\bigcup }_i\left(dom{U}_Z\left(i\right)\cup im{U}_Z\left(i\right)\right)\text{;}$$ so $f$ is defined in one certain way on the entire set $domf\cap \left(domU\cup imU\right)=domf$, that is $f$ is defined in one certain way.

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Consequence  If $U$ and $V$ are weakly monovalued systems of constructs and $a$ and $b$ are elements of $U$ and $V$ correspondingly, then there exist no more than one strict ancestry pseudomorphism from $a$ to $b$ (regarding $U$ and $V$).

◄  It follows from that any pseudomorphism from a part of $U$ to a part of $V$ is a disjunctive pseudomorphism.  ►

It seems that two above theorems can be somehow generalized for the case of arbitrary dependencies.

Factor of Two Constructs

Let $U$ and $V$ are systems of constructs, $U$ is weakly monovalued. Let $a$ and $b$ are elements of $U$ and $V$ correspondingly.

As by a theorem above there exists no more than one strict ancestry disjunctive pseudomorphism from an element $a$ to an element $b$ we can introduce factor $b/a$ of $b$ by $a$ which is the strict ancestry disjunctive pseudomorphism from $a$ to $b$.

In the case when both $U$ and $V$ are weakly monovalued there exists even no more than one strict ancestry pseudomorphism from $a$ to $b$.

Theorem  $\forall i:\left({U^S}_Z\left(i\right)a\ne \varnothing \Rightarrow \left(b/a\right){U^S}_Z\left(i\right)a={V^S}_Z\left(i\right)b\right)$ if $U$ and $V$ are systems of constructs, $U$ is weakly monovalued and $b/a$ exists.

◄  It follows from that $b/a$ is a disjunctive pseudomorphism from $U^S$ to $V^S$.  ►

Consequence  $\forall i:\left(b/a\right){U^S}_Z\left(i\right)a\subseteq {V^S}_Z\left(i\right)b$ under the theorem conditions.

◄  Obvious. (Note that this statement can alternatively be proved independently of the theorem, analogously to the theorem itself.)  ►

The reverse statement of this theorem is:

Theorem  If $\forall i:\left({U^S}_Z\left(i\right)a\ne \varnothing \Rightarrow f{U^S}_Z\left(i\right)a={V^S}_Z\left(i\right)b\right)$ for some function from ancestry of element $a$ of a weakly monovalued system of constructs $U$ to ancestry of element $b$ of a system of constructs $V$ then $f=b/a$, provided that the set $indV$ is comma-simple.

◄

As under theorem conditions exists no more than one strict ancestry disjunctive pseudomorphism, it is enough to prove that $f$ is a disjunctive ancestry pseudomorphism.

It cannot be $fa=\varnothing$ by definition of ancestry. So from theorem conditions $fa=\left\{b\right\}$.

So we need to prove only that $f$ is a pseudomorphism.

It is enough to prove that $U_Z\left(i\right)x\ne \varnothing \Rightarrow f{U}_Z\left(i\right)x=V_Z\left(i\right)fx$ for $x={U^n}_Z\left(j\right)a$. We have $$U_Z\left(i\right){U^n}_Z\left(j\right)a\Rightarrow f{U}_Z\left(i\right)x=f{U}_Z\left(i\right){U^n}_Z\left(j\right)a=V_Z\left(i\right){V^n}_Z\left(j\right)b=V_Z\left(i\right)f{U^n}_Z\left(j\right)a=V_Z\left(i\right)fx\text{.}$$

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Analogous theorems also take place for ancestry homomorphism:

Theorem  $\forall i:\left(b/a\right){U^S}_Z\left(i\right)a={V^S}_Z\left(i\right)b$ if $b/a$ is a homomorphism.

◄  It follows from the formula $\left(b/a\right)\circ {U^S}_Z\left(i\right)={V^S}_Z\left(i\right)\circ \left(b/a\right)$, which in turn follows from that $b/a$ is a homomorphism from $U^S$ to $V^S$.  ►

Theorem  If $\forall i:f{U^S}_Z\left(i\right)a={V^S}_Z\left(i\right)b$ for some function from ancestry of element $a$ of a system of constructs $U$ to ancestry of element $b$ of a weakly monovalued system of constructs $V$ then $f$ is an strict ancestry homomorphism from $a$ to $b$, provided that the set $indV$ is comma-simple.

◄  The proof is a trivial simplification of the proof of the analogous theorem above about pseudomorphism.  ►

Proposition  $b/a=\{\left(U^S\left(i\right)a,V^S\left(i\right)b\right)|U^S\left(i\right)a\ne \varnothing \}$ if $U$ and $V$ are weakly monovalued systems of constructs and $b/a$ exists.

◄  It follows from a theorem above.  ►

Remark  The above statement allows to extend the definition of factor for the case when there are no pseudomorphism, but then the factor would be a relation rather than a function.

Proposition  If $b$ is a strict ancestry disjunctive pseudomorphic image of $a$ and $f$ is a strict ancestry disjunctive pseudomorphism then $f\circ \left(b/a\right)=\left(f\left(b\right)\right)/a$.

◄  As $f\circ \left(b/a\right)$ is a strict ancestry disjunctive pseudomorphism, it is enough to prove that $f\circ \left(b/a\right)$ maps $a$ to $f\left(b\right)$, what is obvious.  ►

Criterion of Pseudomorphic Image

Theorem  Let $U$ and $V$ are weakly monovalued systems of constructs. An element $b$ of $V$ is a strict ancestry pseudomorphic image of an element $a$ of $U$ if and only if $$\forall i,j:\left(\left({U^S}_Z\left(i\right)\right)\left(a\right)=\left({U^S}_Z\left(j\right)\right)\left(a\right)\ne \varnothing \Rightarrow \left({V^S}_Z\left(i\right)\right)\left(b\right)=\left({V^S}_Z\left(j\right)\right)\left(b\right)\right)\text{,}$$ provided that domains and images of these systems of constructs are subsets of some comma-simple sets.

◄

Direct implication

It follows from the formula ${U^S}_Z\left(i\right)a\ne \varnothing \Rightarrow \left(b/a\right){U^S}_Z\left(i\right)a={V^S}_Z\left(i\right)b$.

Reverse implication

From the formula in the condition $\left({U^S}_Z\left(i\right)\right)\left(a\right)\ne \varnothing \Rightarrow \left({V^S}_Z\left(i\right)\right)\left(b\right)=f\left(\left({U^S}_Z\left(i\right)\right)\left(a\right)\right)$ for some function $f$. (I have taken into account that ${U^S}_Z\left(i\right)$ and ${V^S}_Z\left(i\right)$ and are monovalued and so values of $f$ can be taken arbitrarily independently of each other.) By a theorem above $f=b/a$.

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This formula is a formalization of vague informal statement $V$ is a pseudomorphic image of $U$ if and only if equivalent paths in $U$ are equivalent in $V$.

[TODO: Simplify this formula using reindexation.]

[TODO: Consider system of sequences, the system of constructs corresponding to binary operation comma. (Probably morphisms with this system of constructs may also be used to simplify the above formula.)]

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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