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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

• Introduction
• Semigroup with lattice structure
• Semigroup with zero degree
• Semigroup with reverses
• Semigroup with whole degrees
• Semigroup with zero degree and lattice structure
• The table
• Partially ordered monoid
• Semigroup with zero degree and lattice structure
• Monoid with lattice structure
• Monoid with full lattice structure

Introduction

WARNING: This Web page is yet very rough draft and may be with wrong or inconsistent to other articles on this Web site terminology. The terminology is however liable to changes.

This (yet partially written) article considers generalizations of the concept of dependencies and categories of dependencies as defined in my earlier articles on this topic.

Dependencies are used in algebraic theory of formulas.

We will consider several abstract algebras.

Semigroup with lattice structure

I will call a semigroup with lattice structure an algebraic system wich is both a semigroup by binary operation $\circ$ and a lattice and $\circ$ is distributive around $\cup$.

Semigroup with zero degree

I will call a semigroup with zero degree a semigroups with additional unary operation zero degree such that:

• $f^0\circ f=f\circ f^0=f$
• ${\left(g\circ f\right)}^0\subseteq f^0\cup g^0$
• ${\left(f^0\right)}^0=f^0$

Semigroup with reverses

I will call a semigroup with reverses a semigroups with additional unary operation reverse such that:

• ${\left(f^{-1}\right)}^{-1}=f$
• ${\left(g\circ f\right)}^{-1}=f^{-1}\circ g^{-1}$

Semigroup with whole degrees

I will call a semigroup with whole degrees an algebraic structure which is both a semigroup with zero degree and a semigroup with reverses.

For it any whole degree of every element can be defined by induction in the well known way.

For whole degrees hold all main equalitites except that generally $f^j\circ f^i\ne f^{i+j}$.

Semigroup with zero degree and lattice structure

The table

Identity can be completely excluded replacing it with zero degree. It can be added ??

[TODO: Normalized zero degree: $f^0\cap \left(\left(f\circ f^{-1}\right)\cup \left(f^{-1}\circ f\right)\right)$.]

Below follows a long list of axiomw. However for every particular theorem only these axioms are necessary symbols encountered in which are also encountered in theorem conditions with additional provisions that the symbols $\cup$ or $\cap$ encountered in theorem conditions mutually imply each other, $\cup$ (or $\cap$) implies $\subseteq$, and $\bigcup$ (infinite union) implies $\cup$ and $\cap$.

Semigroup

• $\circ$ is an associative binary operation;

Partial order

• $\subseteq$ is a partial order;

Lattice

• $\cup$ and $\cap$ are a lattice;
• $f\subseteq g\iff f\cup g=g$

Distributive full lattice

• $\bigcup$ (and $\bigcap$) form a full distributive lattice;

Reverse

• $f^{-1}\subseteq g^{-1}\iff f\subseteq g$
• ${\left(g\circ f\right)}^{-1}=f^{-1}\circ g^{-1}$ (needed?)

Zero degree

• $f^0\circ f=f\circ f^0=f$
• ${\left(f\cup g\right)}^0=f^0\cup g^0$
• ${\left(g\circ f\right)}^0\subseteq f^0\cup g^0$
• ${\left(f^0\right)}^0=f^0$

Zero degree and reverse

• ${\left(f^0\right)}^{-1}={\left(f^{-1}\right)}^0=f^{-1}$

Distributivity of $\circ$

• $\circ$ is distributive regarding $\cup$
• $\circ$ is infinitely distributive regarding $\bigcup$

Zero element

• $0\subseteq f$
• $0\cup f=f$
• $0\circ f=f\circ 0=f$

Let us study semigroups with some additional operations.

I will use the following original method: bring all additional operations in a triangular table. The table diagonal with express requirements for these operations. Other cells of table will express additional requirements which we will imply when there are both additional operations.

Probably tables similar to above will become wide spread with development of 21 Century Math Method, as it seems (however not yet known) that such tables may be often encountered in conjunction with 21MM. That we have encountered such a table may be a sign that we are already near to 21MM, as God, our teacher, already teaches us 21MM even before we have formalized it.

Partially ordered monoid

Consider a partially ordered (by relation $\subseteq$) monoid (with binary operation $\circ$).

Let $U$, $V$, $f$ are elements of our monoid. By definition $f\in Pseud\left(U,V\right)\iff f\circ U\subseteq V\circ f$. Such $f$ is a called a pseudomorphism from $U$ to $V$.

Proposition  If $f$ is a pseudomorphism from $U$ of to $V$ and $g$ is a pseudomorphism from $V$ to $W$ then $g\circ f$ is a pseudomorphism from $U$ to $W$.

◄  $g\circ f\circ U\subseteq g\circ V\circ f\subseteq W\circ g\circ f$.  ►

So elements of our partially ordered monoid form a category where morphisms between elements $f$ and $g$ are pseudomorphisms from $f$ to $g$. The identity morphism of every element is the identity of the monoid. (Taking in account properties of identity and reflexivity of partial order relation.)

We will call homomorphism from element $U$ of our monoid to element $V$ of our monoid such element $f$ of our monoid that $f\circ U=V\circ g$.

Obvious  Homomorphism is a pseudomorphism.

Note that homomorphism is the same as pseudomorphism of when the partial order is equality relation.

Obvious  $f$ is a homomorphism from $U$ to $V$ (in some monoid) regarding partial order $\subseteq$ if and only if it is pseudomorphism regarding $\subseteq$ and pseudomorphism regarding $\supseteq$.

The above statement allows to prove theorems only for pseudomorphisms and conclude the properties of homomorphism from properties of pseudomorphism.

I will call partially ordered monoid with inverses (or reverses) partially ordered monoid and operation (called reverse or inverse) $f\mapsto f^{-1}$ such that:

• ${\left(g\circ f\right)}^{-1}=g^{-1}\circ f^{-1}$;
• $f^{-1}\subseteq g^{-1}\iff f\subseteq g$.

By definition a monovalued element of a partially ordered monoid with inverses is such element $f$ that $f\circ f^{-1}\subseteq 1$

Semigroup with zero degree and lattice structure

I will call a semigroup with zero degree and lattice structure such a semigroup that:

• $h\circ \left(f\cup g\right)=\left(h\circ f\right)\cup \left(h\circ g\right)$;
• $\left(f\cup g\right)\circ h=\left(f\circ h\right)\cup \left(g\circ h\right)$;
• $f^0\circ f=f\circ f^0=f$
• ${\left(f\cup g\right)}^0=f^0\cup g^0$
• ${\left(g\circ f\right)}^0\subseteq f^0\cup g^0$

Remark  Additionally we could require ${\left(f^0\right)}^0=f^0$

We can turn our semigroup $G$ provided it has zero element $0$ into a monoid this way:

Elements of monoid are $G\cup \{f^{*}|f\in G,f^0\subseteq f\}$ where $f^{*}$ is a second copy of $f$ distinct of any elements of our semigroup.

Remark  $f^{*}$ is a temporary notation which will be eliminated as unnecessary below.

By definition the identity element of our monoid is $0*$.

Composition is defined as follows:

• Composition of elements of $G$ is the same as composition on our semigroup.
• $g^{*}\circ f=f\cup \left(g\circ f\right)$;
• $g\circ f^{*}=g\cup \left(g\circ f\right)$;
• $g^{*}\circ f^{*}={\left(f\cup g\cup \left(g\circ f\right)\right)}^{*}$;

Union of elements of our monoid is defined as follows:

Elements of monoid is disjoint union of elements of our semigroup and these elements of which are greater than their own zero degree.

Composition of elements of semigroup remains the same.

Monoid with lattice structure

We will call monoid with lattice structure such monoid that:

• $h\circ \left(f\cup g\right)=\left(h\circ f\right)\cup \left(h\circ g\right)$;
• $\left(f\cup g\right)\circ h=\left(f\circ h\right)\cup \left(g\circ h\right)$.

Proposition  If $\left(f\circ U\right)\cap f=\varnothing$ then $f\in Pseud\left(U,V\right)\iff f\in Pseud\left(U\cup 1,V\cup 1\right)$.

◄  $$f\in Pseud\left(U\cup 1,V\cup 1\right)\iff f\circ \left(U\cup 1\right)\subseteq \left(V\cup 1\right)\circ f\iff \left(f\circ U\right)\cup f\subseteq \left(V\circ f\right)\cup f\Leftarrow f\circ U\subseteq V\circ f\iff f\in Pseud\left(U,V\right)\text{;}$$ $$\left(f\circ U\right)\cup f\subseteq \left(V\circ f\right)\cup f\Rightarrow \left(f\circ U\right)\cap \left(\left(f\circ U\right)\cup f\right)\subseteq \left(f\circ U\right)\cap \left(\left(V\circ f\right)\cup f\right)\Rightarrow \left(f\circ U\right)\subseteq \left(f\circ U\right)\cap \left(V\circ f\right)\Rightarrow \left(f\circ U\right)\subseteq \left(V\circ f\right)\text{.}$$  ►

Theorem  If $f\in Pseud\left(U,V\right)$ then $f\in Pseud\left(U^n,V^n\right)$ for any $i=0,1,2,\dots$.

◄  We will prove it by induction. For $n=0$ the theorem is obvious. Let the statement is true for $n=i$. Then $f\circ U^i\subseteq V^i\circ f$; $$f\circ U^{i+1}=f\circ U^i\circ U\subseteq V^i\circ f\circ U\subseteq V^i\circ V\circ f=V^{i+1}\circ f\text{.}$$  ►

I will call monoid with lattice structure and reverses (or inverses) such structure which is both monoid with lattice structure and partially ordered monoid with inverses.

For monoid with lattice structure and inverses can be defined domain and image of elements of that monoid:

• $domf=\left(f^{-1}\circ f\right)\cap 1$;
• $imf=\left(f\circ f^{-1}\right)\cap 1$.

Remark  Note that domain and image could be primary concepts so eliminating inverse from consideration. However I have not yet considered this.

We can define cartesian products of a domain and an image as infinite union of all elements whose domains and image are not??

Monoid with full lattice structure

I will call monoid with full lattice structure such monoid with lattice structure, that the lattice is full that is infinite union is defined and that the monoid operation is distributive regarding the infinite union operation.

For monoid with full lattice structure we can introduce:

• $S_1\left(U\right)=U\cup U^2\cup U^3\cup \dots$;
• $S_0\left(U\right)=1\cup S_1\left(U\right)=U^0\cup U\cup U^2\cup \dots$.

Remark  Above defined, roughly speaking, may denote repeated applying operators associated with $U$ (in the third dimension of $U$ as explained in so called theory of constructs (parametrized dependencies).)

Theorem  If $\left(f\circ U\right)\cap \left(V^i\circ f\right)=\varnothing$ for any $i=2,3,4,\dots$ then $$f\in Pseud\left(U,V\right)\iff f\in Pseud\left(S_1\left(U\right),S_1\left(V\right)\right)\text{.}$$

◄

Taking in account infinite distributivity, we get $f\in Pseud\left(S_1\left(U\right),S_1\left(V\right)\right)\iff f\circ S_1\left(U\right)\subseteq S_1\left(V\right)\circ f\iff \left(f\circ U\right)\cup \left(f\circ U^2\right)\cup \left(f\circ U^3\right)\cup \dots \subseteq \left(V\circ f\right)\cup \left(V^2\circ f\right)\cup \left(V^3\circ f\right)\cup \dots$

Direct implication

Let $f\in Pseud\left(U,V\right)$ that is $f\circ U\subseteq V\circ f$.

By proved above $f\circ U^i\subseteq V^i\circ f$

Uniting left an right parts of all these formulas we get, taking in account infinite distributivity, $f\in Pseud\left(S_1\left(U\right),S_1\left(V\right)\right)$

Reverse implication

Let intersect left and right parts of the above formula with $f\circ U$. Taking in account infinite distributivity of $\cap$ regarding $\cup$ it produces: $f\circ U\cup \dots \subseteq \left(f\circ U\right)\cap \left(S_1\left(V\right)\circ f\right)$. Taking in account infinite distributivity and theorem conditions, we get $f\circ U\subseteq \left(f\circ U\right)\cap \left(V\circ f\right)$, from which follows. $f\circ U\subseteq V\circ f$.

►

Theorem  If $\left(f\circ U\right)\cap f=\varnothing$, $\left(f\circ U\right)\cap \left(V^i\circ f\right)=\varnothing$ for any $i=2,3,4,\dots$ then $$f\in Pseud\left(U,V\right)\iff f\in Pseud\left(S_0\left(U\right),S_0\left(V\right)\right)\text{.}$$

◄  By combining two previous theorems.  ►

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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Copyright © 2005 Victor Porton

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