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

Table of Contents

• Theory of Constructs
  • Definition and Introduction
    • Third Property $Z$. Systems of Constructs
      • Subformulas (Terminology)
    • Reindexation
  • Sequential Dependencies
    • Operation Comma and Comma-Simple Sets
    • Definition
    • Morphisms of Sequential Dependencies
  • Disjunctive Pseudomorphisms
  • Factor of Two Constructs
    • Factor of Two Constructs
    • Criterion of Pseudomorphic Image

Theory of Constructs

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.]

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.

►

Disjunctive Pseudomorphisms

I will call weakly monovalued such system of constructs $U$ that the relation $U_Z\left(i\right)$ is monovalued for any $i$.

[TODO: Theorems below can be simplified by using the dependency reverse to the dependency which maps $i$ to $U_Z\left(i\right)a$.]

Definition  A function is a disjunctive pseudomorphism from a system of constructs $U$ to a system of constructs $V$ if and only if for any $i$ exists such set $P\left(i\right)$ that $$f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f{|}_{P\left(i\right)}\text{.}$$

Remark  There exists obvious variant of this definition for the case of $U$ and $V$ being relations instead of systems of constructs. [TODO: Generalize for arbitrary dependencies?]

Obvious  A disjunctive pseudomorphism is a pseudomorphism.

Obvious  A homomorphism is disjunctive pseudomorphism.

Theorem  A function $f$ is a disjunctive pseudomorphism from $U$ to $V$ if and only if for any $i$ and any $x$ $$\forall i,x:\left(f{U}_Z\left(i\right)\left\{x\right\}=\left(V_Z\left(i\right)\right)\left\{f\left(x\right)\right\}\vee U_Z\left(i\right)\left\{x\right\}=\varnothing \right)\text{.}$$

◄

From this (taking in account that $f{U}_Z\left(i\right)\left(\left\{x\right\}\right)=\varnothing \iff U_Z\left(i\right)\left(\left\{x\right\}\right)=\varnothing$) the theorem become obvious.

►

Remark  The above theorem is the reason why disjunctive pseudomorphism is called so.

Proposition  Let $U$ and $V$ are systems of constructs, $U$ is weakly monovalued. A function is a disjunctive pseudomorphism from $U$ to $V$ if and only if it is a disjunctive pseudomorphism from $U^S$ to $V^S$ (provided that the set $indU\cup indV$ is comma-simple).

◄

Direct implication

Let $f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f{|}_{P\left(i\right)}$ for any $i$.

We will prove by induction $f\circ {U^n}_Z\left(i\right)={V^n}_Z\left(i\right)\circ f{|}_{P\left(i\right)}$. for some $P\left(i\right)$ for any $n$.

If $f\circ {U^n}_Z\left(i\right)={V^n}_Z\left(i\right)\circ f{|}_{P\left(i\right)}$. then $f\circ U_Z\left(j\right)\circ {U^n}_Z\left(i\right)=V_Z\left(j\right)\circ f{|}_{P\left(i\right)}\circ {U^n}_Z\left(i\right)$. Because $U$ is weakly monovalued $f{|}_{P\left(i\right)}\circ {U^n}_Z\left(i\right)=f\circ {U^n}_Z\left(i\right){|}_{Q\left(i\right)}$ for some set $Q\left(i\right)$. Consequently $f\circ U_Z\left(j\right)\circ {U^n}_Z\left(i\right)=V_Z\left(j\right)\circ f\circ {U^n}_Z\left(i\right){|}_{Q\left(i\right)}=V_Z\left(j\right)\circ {V^n}_Z\left(i\right)\circ f{|}_{Q\left(i\right)}$.

The statement of induction follows for the above.

From this taking in account that $indU\cup indV$ is comma-simple obviously follows $f\circ {U^S}_Z\left(i\right)={V^S}_Z\left(i\right)\circ f{|}_{P\left(i\right)}$.

Reverse implication

Let $f\circ {U^S}_Z\left(i\right)={V^S}_Z\left(i\right)\circ f{|}_{P\left(i\right)}$. From comma-simplicity obviously follows that $U^S\left(i\right)=U\left(i\right)$ and $V^S\left(i\right)=V\left(i\right)$ for $i\in indU\cup indV$. So $f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f{|}_{P\left(i\right)}$.

►

Proposition  A pseudomorphism whose destination is a weakly monovalued system of constructs is disjunctive pseudomorphism.

◄  Let $f\circ U_Z\left(i\right)\subseteq V_Z\left(i\right)\circ f$. Then because $V_Z\left(i\right)\circ f$ is monovalued (a function), $f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f{|}_{P\left(i\right)}$ for some $P\left(i\right)$.  ►

Theorem  Composition of two disjunctive pseudomorphisms is a disjunctive pseudomorphism.

◄  Let $f\circ U_Z\left(i\right)=V_Z\left(i\right)\circ f{|}_{P\left(i\right)}$ and $g\circ V_Z\left(i\right)=W_Z\left(i\right)\circ g{|}_{Q\left(i\right)}$. Then $$g\circ f\circ U_Z\left(i\right)=g\circ V_Z\left(i\right)\circ f{|}_{P\left(i\right)}=W_Z\left(i\right)\circ g{|}_{Q\left(i\right)}\circ f{|}_{P\left(i\right)}=W_Z\left(i\right)\circ g\circ f{|}_{P\left(i\right)\cap f^{-1}\left(Q\left(i\right)\right)}\text{.}$$  ►

Consequence  Disjunctive pseudomorphisms form a category (a subcategory of the category of pseudomorphisms).

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 $a$.

►

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.

►

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 $$U_Z\left(i\right){U^n}_Z\left(j\right)a\Rightarrow f{U}_Z\left(i\right)x=f{U}_Z\left(i\right){U^n}_Z\left(j\right)a=V_Z\left(i\right){V^n}_Z\left(j\right)b=V_Z\left(i\right)f{U^n}_Z\left(j\right)a=V_Z\left(i\right)fx\text{.}$$

►

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 $f=b/a$.

►

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.

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.