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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

• Disjunctive Pseudomorphisms

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 $x$ $$\forall i,x:\left(f{U}_Z\left(i\right)\left\{x\right\}=\left(V_Z\left(i\right)\right)\left\{f\left(x\right)\right\}\vee U_Z\left(i\right)\left\{x\right\}=\varnothing \right)\text{.}$$

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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.

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

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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).

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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