Negative Answer of Hilbert’s Sixth Mathematical Conjecture

—Mathematical Reasoning of Philosophical Intervening Principle Based on Yin Yang Wu Xing Theory in Traditional Chinese Philosophy (I) ()

—Mathematical Reasoning of Philosophical Intervening Principle Based on Yin Yang Wu Xing Theory in Traditional Chinese Philosophy (I) ()

Ziqing Zhang^{1}, Yingshan Zhang^{2}

^{1}School of Chinese Language and Literature, Faculty of Humanities and Social Sciences, East China Normal University, Shanghai, China.

^{2}School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai, China.

**DOI: **10.4236/ojpp.2021.114037
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Hilbert (David Hilbert, 1862-1943 a.d) is a well-known German mathematician. He introduced 23 of the most important problem for twentieth century mathematicians to study, this is the famous “Hilbert 23 problems”. Hilbert’s sixth mathematical conjecture has attracted the attention of many mathematicians over the past century. Traditional Chinese Philosophy (TCP) differs from Western philosophy mainly in that TCP contains biological energy called qi or Chi or energy of the system. This biological energy is everywhere. Logically, therefore, no single axiom system is recognized. It is considered that the world is composed of several incompatible axiom systems with the killing relationship and the loving relationship, which form five kinds of states of qi or Chi or energy of the system. Therefore, human understanding of the world cannot proceed from observation and hypothesis, but from the five kinds of the whole, using non-generative logic, using obtaining image from classification or classification taking image technology, through the global to local to consider the killing relationship and the loving relationship, reproducibility results can be obtained. This philosophical thought is the opposite of the philosophical implications of Hilbert’s sixth mathematical conjecture. Therefore, on the basis of explaining eastern philosophy, this paper gives a negative answer to Hilbert’s sixth mathematical conjecture from the perspective of philosophy and mathematics. The main frame in a more direct way of TCP includes the non-authigenic logic as the no Axiom system, the loving analysis as the preconception principle, the killing analysis as the integration coordination combination principle, the killing reproducibility as the logic layering principle and the loving reproducibility as the automation principle.

Keywords

Hilbert’s Sixth Mathematical Conjecture, Traditional Chinese Philosophy (TCP), Traditional Chinese Mathematics (TCMath), Yin Yang Wu Xing Theory, Steady Multilateral Systems, Incompatibility Relations, Non-Authigenic Logic

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Zhang, Z. and Zhang, Y. (2021) Negative Answer of Hilbert’s Sixth Mathematical Conjecture

—Mathematical Reasoning of Philosophical Intervening Principle Based on Yin Yang Wu Xing Theory in Traditional Chinese Philosophy (I).*Open Journal of Philosophy*, **11**, 552-619. doi: 10.4236/ojpp.2021.114037.

—Mathematical Reasoning of Philosophical Intervening Principle Based on Yin Yang Wu Xing Theory in Traditional Chinese Philosophy (I).

1. Introduction

Hilbert (David Hilbert, 希尔伯特, 1862-1943 a.d) is a well-known German mathematician. He was born in East Prussia Konigsberg. Since 1895, in University of Goettingen, is the life tenure of professor, becoming a member of the royal society in 1928. In general, he in geometry and mathematics on the basis of the profound study is the most famous; Hilbert’s Programme prompted Computability Theory.

He collected 23 problems, now known as Hilbert’s Problems, the process of twentieth century mathematics development had a profound impact; among them, there are still many problems unsolved.

His achievements include the Hilbert basal Theorem of ring theory (Hilbert’s Basis Theorem), the geometry Basis (Grundlagen der Geometrie), and Hilbert Space and other theory research.

The Hilbert, a German mathematician, into the basic algebraic invariant, algebraic number theory, geometry, variational method, the Hilbert space, etc., have a great contribution, was the greatest mathematician of his time.

He advocated mathematical axiomatization, and put forward “Hilbert 23 problems”, promoting the 20th century development of mathematics.

1.1. Mathematical Contribution

In 1862, Hilbert was born in Konigsberg (at that time, as the capital of east Prussia, drawing into the Russian territory) world war ii, in 1880 into the local university, a doctorate in 1884, 1886 teaching in the university, become a professor and married in 1892. In 1895 became a professor at the University of Goettingen, until death.

Each period Hilbert math characteristic is only focused on one area, the main problem to solve, and turned to another area.

From 1884 to 1892, which focuses on the Hilbert algebraic invariants, proves that any transformation group of algebraic expression invariant, has a limited set of basement, and can actually be constructed.

From 1892 to 1898, focusing on algebraic number theory, laid the foundation of theory of the class body. In 1898 began to focus on the problem of surface geometry axiomatic, results in the next year to complete, author, the basis of geometry for plane geometry axiomatic system is established.

From 1899 to 1901 is the period of variational method, the Hilbert with strict proof, and established the Dirichlet principle: under the boundary curve and boundary value slightly limitation, for several consecutive partial derivatives of the given boundary value and all the possible function, there will be a function of double integral value reaches the minimum.

In 1902, the Hilbert turned to integral equations, the corresponding infinite dimensional linear space (Hilbert space), for the subsequent quantum physics reserves the sharp mathematical tools.

Besides in various fields have outstanding achievements, the Hilbert geometry axiomatic strict ideas soon spread to mathematics in various fields, and Hilbert himself also seriously studied physics, to all branches of physics axiomatic; But his achievements in physics axiomatization are limited.

In 1922, the Hilbert axiomatic itself to research, hope to prove that the axiomatic system of the general in independence, consistency and completeness is not a problem. But in the 1930s, GöDel’s few papers make this hope fail to be fully implemented.

In addition, the Hilbert Paris in 1900, the 2nd international conference on mathematical speech also deeply affect the development of 20th century mathematics. He says the problem is a fountain of mathematics activities, and some problems from the experience and natural phenomena, because others will do a study of logic integration, generalization and specialization. The theory and experience of interaction make mathematics is very useful. He in the name of “math” and a half after the speech, cites the 23 to the twentieth century mathematicians to solve problems, to illustrate the background one by one. These are the famous Hilbert 23 problems, they did in the 20th century an important role in the development of mathematics.

Hilbert made it for knowledge has always been optimistic, corresponding to the philosopher du Bois-Reymond pessimistic view: we are ignorant, and I will have been ignorant, Hilbert presents a lifelong belief is: we have to know, we will know.

After the 70-year-old Hilbert’s body was not so good, memory loss. One day, Hasse and Hilbert about class body theory, the basic concept of the Hilbert has demanded the class body theory and results explain to him, then Hilbert’s reaction was: this is really beautiful, is who invented?

Hitler came to power in 1933, began to persecute jews, the old disciple of Hilbert left Germany. The Hilbert became alone, with no energy, to drag the rest.

1.2. Hilbert 23 Problems

Hilbert (Hilbert d., 1862.1.23-1943.2.14) is the 20th century one of the greatest mathematicians in Germany and the world. He runs across two centuries of 60 years of research career, through nearly all of the modern mathematics forefront, and his thoughts are deeply penetrating into the whole of modern mathematics. Hilbert is the core of the Gottingen school, with his hard work and sincere personal qualities attracted youth researchers scholars from around the world, make Gottingen traditional influence in the world. Hilbert died, published in the journal nature, Germany the view that is rare in the world have a mathematician work, originated from Hilbert’s work is not in some way. He likes maths world Alexander, in the whole map of mathematics, left his glorious name. Conference in Paris in 1900, Hilbert mathematicians introduced 23 of the most important problem for the twentieth century mathematicians to study, this is the famous “Hilbert 23 problems”.

In 1975, the University of Illinois in the United States at a meeting of the international mathematics, mathematicians three-quarters of a century are reviewed the research progress of Hilbert 23 issues. At that time, about half of the problem has been solved, the remaining half of most also have significant progress.

Mathematicians in the United States, in 1976, the selection of the United States since 1940 in ten big achievements of mathematics, three is the Hilbert 1, 5, 10 the solution of the problem. Therefore, to solve the problem of Hilbert, contemporary mathematician is considered a great honour.

The following excerpt is published in 1987, “the mathematician small dictionary” and some other literature collection of Hilbert 23 issues:

1) The continuum hypothesis

Cantor (康托, 1874) speculation in a countable set between the base and the base of real Numbers is no other base, this is the famous continuum hypothesis.

Godel (哥德尔, 1938) proved the continuum hypothesis and the universally recognized Merlot-Frenkel (策梅洛·弗伦克尔) axiomatic set theory system of no contradiction.

American mathematician Cohen (科亨, 1963) proved that continuous assumption and Merlot-Frenkel (策梅洛·弗伦克尔) axiomatic set theory were independent of each other.

Continuum hypothesis, therefore, cannot be inside of Merlot-Frenkel (策梅洛·弗伦克尔) axiom system to prove its correctness or not. Hilbert problem 1 in this sense has been resolved.

2) The compatibility of arithmetic axiom

The compatibility of Euclidean (欧几里得) geometry boils down to the compatibility of axiom at arithmetic.

Hilbert had formalism plan proof theory method is put forward to prove it. Godel (哥德尔, 1931) publishing the incompleteness theorem denied this view.

German mathematician Genc (根茨, 1936), under the condition of using transfinite induction, proved arithmetic axiom of compatibility.

Published in 1988, the “China encyclopedia” mathematics, points out that mathematical compatibility problems are unresolved.

3) The two bottom contour, such as the volume of a tetrahedron is equal

Question means, there are two equilateral tetrahedron contours, they cannot be decomposed into a finite number of small tetrahedron, make the two groups of tetrahedron congruent to each other.

Dern (德恩, 1990) was to give a positive answer to this question.

4) The distance in a straight line between two points is the most short term problem

This question is too general. Meet the geometry of the nature of many, and therefore need to increase some constraints.

Soviet union mathematician Pogorelov (波格列洛夫, 1973) announced that in the case of symmetric distance, problem solved.

“China encyclopedia”, said after Hilbert, in terms of structure and discusses various special metric geometry has a lot of progress, but the problem is not solved.

5) The concept of Lie (李) group, the definition of a continuous transformation of the functions of this group of don’t assume that is differentiable

The problem referred to as the analyticity of continuous group, namely: whether every local Euclidean (欧几里得) group there must be a Lie (李) group?

Soviet union mathematicians Neumann (冯·诺伊曼, 1933) for a compact group, Bonderyagin (邦德里雅金, 1951) for a group of exchange, and Xie (谢瓦荚, 1941) for the solvable group obtained the positive answer of this problem.

Gleason (格利森, 1952), Montgomery (蒙哥马利, 1952) and Zippin (齐宾, 1852) solving together, completely positive results were obtained.

6) The axiomatization of physics

Hilbert suggests using mathematical method of axiomatic push all physical performance, the first is the probability and mechanics.

Soviet union mathematician Kolmogorov (柯尔莫哥洛夫, 1933) implemented the axiomatization of probability theory.

Later, in quantum mechanics, quantum field theory has achieved great success.

But whether physics can come axiomatization, many people doubt it.

7) The number of some irrational and transcendence

For any algebraic number of alpha indicates, it is to prove the irrational and transcendence. For any algebraic number of alpha indicates and for any irrational algebraic number of beta indicates, it is to prove the alpha beta of transcendence.

Gelfand (A.O. 盖尔方德, 1934) and Schneider (T. 施奈德, 1934) independently solved the second part of the problem, namely, for any algebraic number of alpha indicates in which alpha is not equal to 0 or 1, and for any irrational algebra of beta indicates, it can be to prove the alpha beta of transcendence.

8) A prime number

There are a lot of problems, including the Riemann (黎曼) hypothesis, Goldbach (哥德巴赫) conjecture, and problem of twin prime Numbers.

In general the Riemann (黎曼) hypothesis is still to be solved.

For Goldbach (哥德巴赫) conjecture, the best result belongs to the trained Chen (陈景润, 1966), but is distance from the most to solve themselves.

The best results of the problem of the prime twins are trained by Chen (陈景润), Tao (陶哲轩), Zhang (张益唐) and others. Zhang (张益唐) proved that there is an infinite number difference of the prime Numbers less than 70 million.

9) In any number of domains has proved the most general reciprocity law

The issue was given a positive answer by Japanese mathematician Takagi (高木贞治, 1921) and German mathematician Atin (阿廷1927).

10) The solvability of Diophantine (丢番图) equations

The solvability of Diophantine (丢番图) equation is able to calculate the integer root of an integer coefficient equation.

Hilbert asked, can use a general algorithm composed of limited steps to judge a solvability of diophantine equation?

Soviet union mathematician Matiyasevich (IO.B. 马季亚谢维奇, 1933) proved that the Hilbert’s desired algorithm does not exist.

11) A coefficient for arbitrary quadratic algebraic number

Hasse (H. 哈塞, 1929) and Siegel (C.L. 西格尔, 1936, 1951) were to obtain important results on this issue.

12) Crocker (克罗克) theorem to Abel (阿贝尔) domains to any rational domain algebra

This problem only some sporadic results, is far from solved also differ.

13) With only two variables function solution of the general equation of seven times

With only two variables function solution of the general equation of seven times seven equation root depends on three parameters *a*, *b*, *c*, namely *x* = *x* (*a*, *b*, *c*). This function can be expressed in binary function?

Soviet union mathematician Arnold (阿诺尔德, 1957) solved the continuous function.

Soviet union mathematician Weissjin (维士斯金, 1964) put it to the situation continuously differentiable function.

But if the requirement is analytic function, the problem hasn’t been solved yet.

14) Prove that certain types of complete function are limited

Prove that certain types of complete function of the finiteness of the algebraic invariant and problems.

An counter example was given by Japanese mathematician Nagata (永田雅宜, 1958).

15) Schubert (舒伯特) count calculus based strictly

Schubert (舒伯特) count strict basic problem is a typical calculus: there are four lines in three dimensional space, can ask a few lines and the four lines intersect?

An intuitive method was presented by Schubert (舒伯特).

Hilbert called for the problem of generalization and give strict basis.

Having some computable method, it and algebraic geometry are not closely related.

But has yet to establish the basis of strict.

16) The algebraic curve and algebraic curve surface topology

The problem is classified into two parts. First part involves an algebra curve containing the maximum number of branch of the closed curve. Later asked to discuss the maximum number of limit cycles and the relative position of the *X*, *Y* is *X*,*Y*, *n* polynomial.

Soviet union mathematician Piotrowski (彼得罗夫斯基), declared the limit cycle number of when *n* = 2 is not more than 3, but this is wrong, have been elaborated by the Chinese mathematicians to cite the counter example in 1979.

17) The sum of the squares of the other forms of positive semidefinite expression

The sum of the squares of the other forms of positive semidefinite said that if a real coefficient polynomial n yuan for all arrays (*x*_{1}, *x*_{2}, ..., *x _{n}*) are constant is greater than or equal to 0, whether can be written in the form of sum of squares?

German mathematician Artin (阿廷, 1927) testified that this is right.

18) Use congruent polyhedral structure space

Using congruent polyhedral structure space, German mathematicians BobMach (比勃马赫, 1910) and Podbyhart (荚因哈特, 1928) made a part of the solution.

19) The regular solution of variational problem whether certain parsing

The regular solution of variational problem whether certain analytic very few studies on this issue.

Some results were drawn by Soviet union mathematicians Bernstein (C.H. 伯恩斯坦) and Piotrowski (彼得罗夫斯基), etc.

20) General boundary value problem

Advances in general boundary value problem for this issue very quickly, has become a big branch of mathematics. Continue to study.

21) Have to order value of the existence of the solution of the linear differential equation

Having to order value of the existence of the solution of the linear differential equation has been given by German mathematicians Hilbert (希尔伯特, 1905), and Oral (H. 罗尔, 1957) solution work.

22) The composed of automorphic functions uniformization of analytic function

Composed of self-controlled function analytical function of uniformization involves the hardships of the Riemann (黎曼) surface.

Krupp (P. 克伯, 1907) won the important breakthrough, other aspects haven’t been solved yet.

23) The further development of the variational method

The variational method to further develop is not a math problem, just talked about general attitudes towards the variational method.

Since the 20th century variational method had the very big development.

The 23 question involves the most important areas of modern mathematics, to promote the development of mathematics in the twentieth century.

For Hilbert’s sixth mathematical conjecture, in addition, many scholars tried to use the Chinese theory of Yin and Yang to establish an axiomatic system in west. These scholars are including Zhang (2009), Aucher (2015), Zhang & Zhang (2004), Zhang et al. (2016), Zhang (2002), Zhang (1998), Chen et al. (2014), Zhang & Kandel (2012), Fang (2012), Kim et al. (2014), Fang & Faure (2011), and so on. Also some scholars, for example, Zhang & Chen (2009), tried to use the concept of Yin Yang Wu Xing in Traditional Chinese Philosophy (TCP) or Traditional Chinese Mathematics (TCMath) to improve the axiomatic system in west. However, in this paper, to explain is: mathematics of trying to cannot be completely successful, could only partial success. The kernel of this paper is the existence and reasoning of the incompatibility relations, and it accords with the oriental or eastern thinking model.

In fact, at present a lot of many concepts in TCP or TCMath have been accepted by many scholars. In TCP or TCMath, for example, there is a very important concept—*Chi *or* Qi* as energy of the system, which is partially accepted by Hilbert as Hilbert manifolds to be studied by many mathematical researchers. The Burghelea & Kuiper (1969) and Ruelle (1982) are the most prominent. It is because they are all to identify the change problem of any input variables for any output variables but don’t worry about the contents of the variable.

The philosophical meaning of five aspects of non-authigenic logic is an important concept of TCP or TCMath. And it is necessary to discuss the five basic principles and their rules of non-authigenic logic. According to the Yin Yang Wu Xing philosophical meaning, it is known that the world of logic should have the meaning of the five aspects. That is to say: taking the research object is classified into five categories, *i.e.*, obtaining image from classification or classification taking image (比类取象), to achieve real purpose of learning about the world.

Zhang (1993, 2006, 2007a, 2007b, 2011a, 2011b, 2012, 2018a, 2018b 2020) has started a great interest and admired works for TCP or TCMath, where, through mathematical reasoning, they demonstrate the presence of incompatibility relations, which are predominant in daily life, yet absent in traditional Aristotelian Western logic.

The main issue to be researched shall be led in a more direct way. The problem of logic is the key to all issues. The non-authigenic logic or the authigenic logic is the main logic of eastern culture, or western culture, respectively. The philosophical meaning of five aspects of non-authigenic logic can be used to TCP itself. The main frame in a more direct way of TCP includes the non-authigenic logic (不自生逻辑) as the no Axiom system, the loving analysis (相生分析) as the preconception principle, the killing analysis (相克分析) as the integration coordination combination principle, the killing reproducibility (相克再现性) as the logic layering principle and the loving reproducibility (相生再现性) as the automation principle. The paper mainly demonstrates the non-authigenic logic. Other questions will be dealt with separately.

Information of the theoretical basis shall be given in advance to lay foundation for following research. The non-authigenic thinking rule or the no Axiom system is the key to the non-authigenic logic, liking wood. The loving analysis is the preconception principle of the non-authigenic logic, liking fire. The killing analysis is the integration coordination combination principle of the non-authigenic logic, liking earth. The killing reproducibility is the logic layering principle of the non-authigenic logic, liking metal, And the loving reproducibility is the automation principle of the non-authigenic logic liking water. There are five philosophical bases of TCP.

Many people as Western persons are beyond all doubt that the Yin Yang Wu Xing theory is superior to the traditional true-false logic, which does not contemplate incompatibility relations, which Zhang & Shao (2012) have expertly explained from a mathematical standpoint.

The work Zhang (1993, 2007a) has started, allows many people like Western person to think of a true re-foundation of mathematical language, to make it a better suited tool for the needs of mankind mathematical complex system and the environment. Even so, Zhang & Shao (2012) also bring to light the difficulty of establishing the values of both the intervention reaction coefficients ${\rho}_{1},{\rho}_{2}$ and the self-protection coefficient ${\rho}_{3}$ as parameters with due accuracy.

Latest works discovered the academic relationships between Leibnitz and Jesuit priest J. Bouvet, linking the deduction of the Trigrams to German binary system situation that coincidentally appears in quoted paper. This will clearly realize the scientific-mathematical fundamentals of Traditional Chinese Medicine (TCM, see Contributions).

There are some publications Inchauspe, (2014, 2015a, 2015b, 2017), Inchauspe & Inchauspe (2019) and Inchauspe & Tachini (2020), devoted to demonstrate in Occident the exact basis of Chinese Medicine, *i.e.* as Chinese Five Elements find precise consistency with Euclid’s Five Regular Polyhedra, as well as the trigrammatic order of the I-Ching hexagrams.

The article proceeds as follows. Section 2 contains the basic generalized relations, generalized reasoning and steady multilateral systems of non-authigenic logic. And a negative philosophical answer of Hilbert’s sixth mathematical conjecture is demonstrated in Section 3. And a negative mathematical answer of Hilbert’s sixth mathematical conjecture is demonstrated in Section 4. Referential significance of this study could be further enhanced in Section 5. And conclusions are drawn in Section 6.

2. Generalized Relations, Generalized Reasoning and Steady Multilateral Systems of Non-Authigenic Logic

2.1. Generalized Relations and Reasoning of Non-Authigenic Logic

Suppose all of the research objects are a collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$. The authigenic logic is first to look at the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ by observation, and discovered some research objects which constitute of a sub-collection ${V}_{0}^{m}=\left\{{y}_{1},\cdots ,{y}_{m}\right\}\subseteq {V}_{}^{n}$, then according to the nature of the sub-set ${V}_{0}^{m}$ of axioms and assumptions, and according to the assumption to judge a certain proposition true or false. If a logical reasoning is based on the observations as ${V}_{0}^{m}$, then the corresponding logic is called authigenic logic.

But the non-authigenic logic is not to look at the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects, just according to themselves needs, classification taking images corresponding to some group, put forward and want to know what is the relation between different classes of the research objects? If a logical reasoning has nothing to do with the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects, then the corresponding logic is called non-authigenic logic in Zhang (2018a).

From oriental or eastern philosophical thought, if a person will go to solve an objective problem of a mathematical complex system, and if this person can get some analysis conclusion only by using non-authigenic logical reasoning or analysis, then the analysis conclusion can ensure to conform to the objective facts and to be reproduced by all others because the logical reasoning has nothing to do with the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects. This property is known as the reproducibility.

Reproducibility is the fundamental meaning of balanced measurement if the real conclusion of objects cannot be known. In general, the reproducibility was defined in a stable environment. The same object is measured by a lot of operators and not only by using the similar measuring instruments and methods. But the measured values of average variation should be similar to each other, and should be similar to the objective facts. Reproducibility also refers to the average variation is smaller between measuring systems and measuring conditions.

Reproducibility is the most basic requirements for mathematical scholars. If there is no reproducibility, so there would be no classical in TCP. Because mathematical scholars are to use the data to study the science law behind the data, the law has nothing to do with mathematical scholars, and has nothing to do with how much data observed by mathematical scholars, and has nothing to do with how what reasonable methods used by mathematical scholars. The mathematical method is reasonable if the method can at least ensure the reproducibility of the mathematical analysis conclusion. As long as mathematical scholars using mathematical method have reproducibility, so the analysis of the conclusions should be unique based on similar data.

In mathematics, let ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ be a non-empty set (the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects). Then there is a non-empty Descartesian set

${V}_{}^{n}\times {V}_{}^{n}=\left\{\left(x,y\right):x\in {V}_{}^{n},y\in {V}_{}^{n}\right\}$.

The non-empty subset $R\subseteq {V}_{}^{n}\times {V}_{}^{n}$ is called a relation of ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$. TCP mainly researches general relation rule for a general ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ rather than for a special subset ${V}_{0}^{m}=\left\{{y}_{1},\cdots ,{y}_{m}\right\}\subseteq {V}_{}^{n}$. So the general set ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ cannot be supposed. The thing can only be done to research the structure of the set of relations ${\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$ because the structure has nothing to do with the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects and without considering the specific content of the research object.

For a relation set
${\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$, define both an inverse relationship
${*}^{-1}$ of
${R}_{i}\in {\Re}_{}^{m}$ and a relation multiplication * between
${R}_{i}\in {\Re}_{}^{m}$ and
${R}_{j}\in {\Re}_{}^{m}$ as follows:*
${R}_{i}^{-1}=\left\{\left(x,y\right):\left(y,x\right)\in {R}_{i}\right\}$ *and
${R}_{i}*{R}_{j}=\left\{\left(x,y\right):\text{Thereisa}u\in {V}_{}^{n}\text{suththat}\left(x,u\right)\in {R}_{i}\text{and}\left(u,y\right)\in {R}_{j}\right\}$ *.*

The relation ${R}_{i}$ is called a reasonable relation if ${R}_{i}^{-1}\in {\Re}_{}^{m}$.

A generalized reasoning of a general set ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ is defined as for any a non empty relation set ${R}_{i}*{R}_{j}\ne \varnothing $ there is a relation ${R}_{k}\in {\Re}_{}^{m}$ such that

${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i}^{},{R}_{j},{R}_{k}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$.

Lemma 2.1. *For any a relation set
${\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$ *,* the following statements are true*.

(1) *The associative law is formed naturally*,* i.e.*,

$\left({R}_{i}*{R}_{j}\right)*{R}_{l}={R}_{i}*\left({R}_{j}*{R}_{l}\right),\forall {R}_{i}^{},{R}_{j},{R}_{l}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}.$

(2) *Inverse operation is similar to that of matrix*,* i.e.*,

${\left({R}_{i}*{R}_{j}\right)}^{-1}={R}_{j}^{-1}*{R}_{i}^{-1},\forall {R}_{i}^{},{R}_{j}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}.$

(3) *Maintain collections subordinate relations of the relation multiplication **,* i.e.*,

${R}_{l}*{R}_{i}\subseteq {R}_{l}*{R}_{j},{R}_{i}*{R}_{l}\subseteq {R}_{j}*{R}_{l},$

if ${R}_{i}\subseteq {R}_{j},\forall {R}_{i}^{},{R}_{j},{R}_{l}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}.$

(4) *Maintain collections subordinate relations of the *inverse relationship
${*}^{-1}$,* i.e.*,

${\left({R}_{i}\right)}_{}^{-1}\subseteq {\left({R}_{j}\right)}_{}^{-1},$

if ${R}_{i}\subseteq {R}_{j},\forall {R}_{i}^{},{R}_{j}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}.$

Proof. (1). For any one $\left(x,y\right)\in \left({R}_{i}*{R}_{j}\right)*{R}_{l}\ne \varnothing $, by the definition of the relation multiplication *, there is a $u\in {V}_{}^{n}$ such that $\left(x,u\right)\in {R}_{i}*{R}_{j}$, $\left(u,y\right)\in {R}_{l}$.

Thus, there is a $v\in {V}_{}^{n}$ such that $\left(x,v\right)\in {R}_{i},\left(v,u\right)\in {R}_{j},\left(u,y\right)\in {R}_{l}$.

Hence, there is a $v\in {V}_{}^{n}$ such that $\left(x,v\right)\in {R}_{i},\left(v,y\right)\in {R}_{j}*{R}_{l}$.

Therefore, $\left(x,y\right)\in {R}_{i}*\left({R}_{j}*{R}_{l}\right)$. It means that $\varnothing \ne \left({R}_{i}*{R}_{j}\right)*{R}_{l}\subseteq {R}_{i}*\left({R}_{j}*{R}_{l}\right)\ne \varnothing $.

Similarly to prove:

$\varnothing \ne {R}_{i}*\left({R}_{j}*{R}_{l}\right)\subseteq \left({R}_{i}*{R}_{j}\right)*{R}_{l}\ne \varnothing .$

If $\left({R}_{i}*{R}_{j}\right)*{R}_{l}=\varnothing $, then must be ${R}_{i}*\left({R}_{j}*{R}_{l}\right)=\varnothing $. Otherwise, if ${R}_{i}*\left({R}_{j}*{R}_{l}\right)\ne \varnothing $, then there is

$\varnothing \ne {R}_{i}*\left({R}_{j}*{R}_{l}\right)\subseteq \left({R}_{i}*{R}_{j}\right)*{R}_{l},$

such that*
$\left({R}_{i}*{R}_{j}\right)*{R}_{l}\ne \varnothing $ *. This is a contradiction. Thus, the following result can be obtained
${R}_{i}*\left({R}_{j}*{R}_{l}\right)=\left({R}_{i}*{R}_{j}\right)*{R}_{l}$.

(2) For any one $\left(x,y\right)\in {\left({R}_{i}*{R}_{j}\right)}^{-1}$, by the definition of the inverse relationship ${*}^{-1}$, there is $\left(y,x\right)\in {R}_{i}*{R}_{j}$.

By the definition of the relation multiplication *, there is a $u\in {V}_{}^{n}$ such that $\left(y,u\right)\in {R}_{i},\left(u,x\right)\in {R}_{j}$.

Thus, by the definition of the inverse relationship ${*}^{-1}$ again, there are

$\left(u,y\right)\in {R}_{i}^{-1},\left(x,u\right)\in {R}_{j}^{-1}\text{or,}\left(x,u\right)\in {R}_{j}^{-1},\left(u,y\right)\in {R}_{i}^{-1},$

Hence, by the definition of the relation multiplication *, the following result can be obtained $\left(x,y\right)\in {R}_{j}^{-1}*{R}_{i}^{-1}$.

Therefore, ${\left({R}_{i}*{R}_{j}\right)}^{-1}\subseteq {R}_{j}^{-1}*{R}_{i}^{-1},\forall {R}_{i}^{},{R}_{j}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$.

Similarly, for any one $\left(x,y\right)\in {R}_{j}^{-1}*{R}_{i}^{-1}$, by the definition of the relation multiplication *, there is a $u\in {V}_{}^{n}$ such that $\left(x,u\right)\in {R}_{j}^{-1},\left(u,x\right)\in {R}_{i}^{-1}$.

Thus, by the definition of the inverse relationship ${*}^{-1}$ again, there are

$\left(u,x\right)\in {R}_{j}^{},\left(y,u\right)\in {R}_{i}^{}\text{or,}\left(y,u\right)\in {R}_{i}^{},\left(u,x\right)\in {R}_{j}^{}$.

Hence, by the definition of the relation multiplication *, the following result can be obtained*
$\left(y,x\right)\in {R}_{i}^{}*{R}_{j}^{}$ *.

Then, by the definition of the inverse relationship ${*}^{-1}$ again, the following result can be gotten $\left(x,y\right)\in {\left({R}_{i}*{R}_{j}\right)}_{}^{-1}$.

Therefore,*
${\left({R}_{i}*{R}_{j}\right)}^{-1}\supseteq {R}_{j}^{-1}*{R}_{i}^{-1},\forall {R}_{i}^{},{R}_{j}\in {\Re}_{}^{m}$ *.

Comprehensive above, the desired conclusion can be achieved: ${\left({R}_{i}*{R}_{j}\right)}^{-1}={R}_{j}^{-1}*{R}_{i}^{-1},\forall {R}_{i}^{},{R}_{j}\in {\Re}_{}^{m}$.

(3) Assume that ${R}_{i}\subseteq {R}_{j},\forall {R}_{i},{R}_{j},{R}_{l}^{}\in {\Re}_{}^{m}$.

For any one $\left(x,y\right)\in {R}_{l}^{}*{R}_{i}^{}$, by the definition of the relation multiplication *, there is a $u\in {V}_{}^{n}$ such that $\left(x,u\right)\in {R}_{l}^{},\left(u,y\right)\in {R}_{i}^{}$.

Thus, by the definition of collections subordinate relations ${R}_{i}\subseteq {R}_{j}$, $\forall {R}_{i},{R}_{j}\in {\Re}_{}^{m}$, there are

$\left(x,u\right)\in {R}_{l}^{},\left(u,y\right)\in {R}_{j}^{}.$

Thereby, by the definition of the relation multiplication * again, there is $\left(x,y\right)\in {R}_{l}^{}*{R}_{j}^{}$.

Therefore, there is ${R}_{l}^{}*{R}_{i}\subseteq {R}_{l}^{}*{R}_{j},\forall {R}_{i},{R}_{j},{R}_{l}^{}\in {\Re}_{}^{m}$.

Similarly, it is proved that

${R}_{i}*{R}_{l}^{}\subseteq {R}_{j}*{R}_{l}^{},\forall {R}_{i},{R}_{j},{R}_{l}^{}\in {\Re}_{}^{m}.$

(4) Assume that ${R}_{i}\subseteq {R}_{j},\forall {R}_{i},{R}_{j}\in {\Re}_{}^{m}$.

For any one $\left(x,y\right)\in {\left({R}_{i}\right)}_{}^{-1}$, by the definition of the inverse relationship ${*}^{-1}$, there is a relation $\left(y,x\right)\in {R}_{i}^{}$.

Thus, by the definition of collections subordinate relations ${R}_{i}\subseteq {R}_{j}$, $\forall {R}_{i},{R}_{j}\in {\Re}_{}^{m}$, there is $\left(y,x\right)\in {R}_{j}^{}$, Thereby, by the definition of the inverse relationship ${*}^{-1}$ again, there is $\left(x,y\right)\in {\left({R}_{j}\right)}_{}^{-1}$. Therefore, there is ${\left({R}_{i}\right)}_{}^{-1}\subseteq {\left({R}_{j}\right)}_{}^{-1}$, $\forall {R}_{i},{R}_{j}\in {\Re}_{}^{m}$.

Similarly, it is proved that

${R}_{i}*{R}_{l}^{}\subseteq {R}_{j}*{R}_{l}^{},\forall {R}_{i},{R}_{j},{R}_{l}^{}\in {\Re}_{}^{m}.$

It completes the proof. #

Corollary 2.1. *For any a relation set
${\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$ *,* assume that *
${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i},{R}_{j},{R}_{k},{R}_{l}^{}\in {\Re}_{}^{m}$.

*Then the following statements are true*.

(1) ${R}_{j}^{-1}*{R}_{i}^{-1}\subseteq {R}_{k}^{-1}$.

(2) ${R}_{l}*\left({R}_{i}*{R}_{j}\right)\subseteq {R}_{l}*{R}_{k}^{},\left({R}_{i}*{R}_{j}\right)*{R}_{l}\subseteq {R}_{k}^{}*{R}_{l}$.

(3) $\begin{array}{l}{R}_{l}^{-1}*\left({R}_{j}^{-1}*{R}_{i}^{-1}\right)\subseteq {R}_{l}^{-1}*{R}_{k}^{-1}={\left({R}_{k}^{}*{R}_{l}^{}\right)}_{}^{-1}\\ \left({R}_{j}^{-1}*{R}_{i}^{-1}\right)*{R}_{l}^{-1}\subseteq {R}_{k}^{-1}*{R}_{l}^{-1}={\left({R}_{l}^{}*{R}_{k}^{}\right)}_{}^{-1}\end{array}$

Proof. (1) Assume that

${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i},{R}_{j},{R}_{k}\in {\Re}_{}^{m}.$

By (4) of Lamma 2.1, there is

${\left({R}_{i}*{R}_{j}\right)}_{}^{-1}\subseteq {\left({R}_{k}\right)}_{}^{-1},\forall {R}_{i},{R}_{j},{R}_{k}\in {\Re}_{}^{m}.$

By (2) of Lamma 2.1, there is

${\left({R}_{i}*{R}_{j}\right)}_{}^{-1}={\left({R}_{j}\right)}_{}^{-1}*{\left({R}_{i}\right)}_{}^{-1}\subseteq {\left({R}_{k}\right)}_{}^{-1},\forall {R}_{i},{R}_{j},{R}_{k}\in {\Re}_{}^{m}.$

(2) Assume that

${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i},{R}_{j},{R}_{k},{R}_{l}\in {\Re}_{}^{m}.$

By (3) of Lamma 2.1, there are

${R}_{l}*\left({R}_{i}*{R}_{j}\right)\subseteq {R}_{l}*{R}_{k}^{},\left({R}_{i}*{R}_{j}\right)*{R}_{l}\subseteq {R}_{k}^{}*{R}_{l},\forall {R}_{i},{R}_{j},{R}_{k},{R}_{l}\in {\Re}_{}^{m}.$

(3) Assume that

${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i},{R}_{j},{R}_{k},{R}_{l}\in {\Re}_{}^{m}.$

By the result in (1), there is

${\left({R}_{j}\right)}_{}^{-1}*{\left({R}_{i}\right)}_{}^{-1}\subseteq {\left({R}_{k}\right)}_{}^{-1},\forall {R}_{i},{R}_{j},{R}_{k},{R}_{l}\in {\Re}_{}^{m}.$

Because each of the relations ${R}_{i},{R}_{j},{R}_{k},{R}_{l}\in {\Re}_{}^{m}$ is reasonable, so ${\left({R}_{i}\right)}_{}^{-1},{\left({R}_{j}\right)}_{}^{-1},{\left({R}_{k}\right)}_{}^{-1},{\left({R}_{l}\right)}_{}^{-1}\in {\Re}_{}^{m}$.

Thus, by (3) of Lamma 2.1, there are

$\begin{array}{l}{R}_{l}^{-1}*\left({R}_{j}^{-1}*{R}_{i}^{-1}\right)\subseteq {R}_{l}^{-1}*{R}_{k}^{-1},\\ \left({R}_{j}^{-1}*{R}_{i}^{-1}\right)*{R}_{l}^{-1}\subseteq {R}_{k}^{-1}*{R}_{l}^{-1},\\ \forall {\left({R}_{i}\right)}_{}^{-1},{\left({R}_{j}\right)}_{}^{-1},{\left({R}_{k}\right)}_{}^{-1},{\left({R}_{l}\right)}_{}^{-1}\in {\Re}_{}^{m}.\end{array}$

By (2) of Lamma 2.1, there are

$\begin{array}{l}{R}_{l}^{-1}*{R}_{k}^{-1}={\left({R}_{k}^{}*{R}_{l}^{}\right)}_{}^{-1},{R}_{k}^{-1}*{R}_{l}^{-1}={\left({R}_{l}^{}*{R}_{k}^{}\right)}_{}^{-1},\\ \forall {\left({R}_{i}\right)}_{}^{-1},{\left({R}_{j}\right)}_{}^{-1},{\left({R}_{k}\right)}_{}^{-1},{\left({R}_{l}\right)}_{}^{-1}\in {\Re}_{}^{m}.\end{array}$

It completes the proof. #

By Lemma 2.1, the generalized relationship reasoning satisfies the associative law of reasoning, *i.e.*,

$\left({R}_{i}*{R}_{j}\right)*{R}_{l}={R}_{i}*\left({R}_{j}*{R}_{l}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {R}_{i},{R}_{j},{R}_{l}\in {\Re}_{}^{m}.$

The associative law is the basic requirement of reasoning in TCP or TCMath. But there are a lot of reasoning forms which do not satisfy the associative law of reasoning in Western Mathematics.

For example, in the true and false binary of proposition logic, the associative law of reasoning does not hold since

$\begin{array}{l}\left(false*false\right)*false=true*false=false\\ \ne true=false*true=false*\left(false*false\right).\end{array}$

2.2. Equivalence Relations of Non-Authigenic Logic

Let ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ be a non empty set (the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects) and ${R}_{0}$ be its a relation. It is called an equivalence relation, denoted by ~, if the following 3 conditions are all true:

(1) Reflexive:
$\left(x,x\right)\in {R}_{0}$ for all
$x\in {V}_{}^{n}$ ; *i.e.*,
$x~x$ *;*

(2) Symmetric: if
$\left(x,y\right)\in {R}_{0}$, then
$\left(y,x\right)\in {R}_{0}$ ; *i.e.*, if
$x~y$, then
$y~x$ ;

(3) Conveyable (Transitivity): if
$\left(x,y\right)\in {R}_{0},\left(y,z\right)\in {R}_{0}$, then
$\left(x,z\right)\in {R}_{0}$ ; *i.e.*, if
$x~y$ and
$y~z$, then
$x~z$.

A relation $R\subseteq {V}_{}^{n}\times {V}_{}^{n}$ is called a compatibility relation if there is a non empty subset ${R}_{}^{1}\subseteq R$ such that the non empty subset ${R}_{1}$ satisfies at least one of the conditions above.

A relation $R\subseteq {V}_{}^{n}\times {V}_{}^{n}$ is called an incompatibility relation if there is not any a non empty subset ${R}_{}^{1}\subseteq R$ such that the non empty subset ${R}_{}^{1}$ satisfies at least one of the conditions above.

Any one of compatibility relations can be expanded into an equivalence relation to some extent. That is to say, in a certain logic analysis macro or global level, any a compatible relation $R$ can be handled as an equivalence relation ${R}_{0}$.

Western Mathematics only considers the reasoning under one Axiom system such that only compatibility relation reasoning is researched. However there are many Axiom systems in nature. TCP or TCMath mainly researches the relation reasoning among many Axiom systems in nature. Of course, she also considers the relation reasoning under one Axiom system but she only expands the reasoning as the equivalence relation reasoning.

2.3. Two Kinds of Opposite Incompatibility Relations of Non-Authigenic Logic

Equivalence relations, even compatibility relations, can not portray the structure of the mathematical complex systems clearly. In the following, two incompatibility relations can be considered.

In TCP or TCMath, the non-authigenic logic having another name: image mathematics in Zhang & Shao (2012), any an Axiom system is not considered, but should first consider to use a logic system. Believe that the rule of Heaven and the behavior of Human can follow the same logic system (天人合一) in Zhang (2018b). This logic system is equivalent to a group of computation. The method is to abide by the selected logic system to the research objects classification, without considering the specific content of the research object, namely, classification taking images (比类取象). Analysis of the relationship between research objects, make relationships with a computational reasoning comply with the selected logic system operation. And then consider the research objects of the specific content of the conditions, according to the logic of the selected system operation to solve specific problems. In mathematics, the method of classification taking images is explained in the following Definition 2.1.

Definition 2.1. Suppose that there exists a finite group
${G}_{}^{m}=\left\{{g}_{0},{g}_{1},\cdots ,{g}_{m-1}\right\}$ of order *m* where
${g}_{0}$ is identity. Let
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ be a none empty set which can be classified into some classes satisfying that
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$ where the notation means that
${V}_{}^{n}={V}_{}^{m}={V}_{{g}_{0}}\cup \cdots \cup {V}_{{g}_{m-1}}$,
${V}_{{g}_{i}}\cap {V}_{{g}_{j}}=\varnothing $,
$\forall i\ne j$ (the following the same).

In image mathematics, the
${V}_{{g}_{j}}$ is first called a factor corresponding to the group element
${g}_{j}$ for any *j*, and
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$ is called a factor space (all “Hexagram” (卦)). People do not consider the factor size (as class variables) and only consider it as mathematical symbols (“Hexagram” (卦)), such as, −1 or 1, because the size is defined by a human behavior for
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}={V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$, but people have no assumption of
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}={V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$.

A mathematical index of the unknown multivariate function
$f\left({x}_{{g}_{0}},\cdots ,{x}_{{g}_{m-1}}\right),\forall {x}_{{g}_{i}}\in {V}_{{g}_{i}},i=0,\cdots ,m-1$, is called a function image of
$F\left({V}_{}^{m}\right)$. All mathematical indexes of the unknown multivariate function *f* compose of the formation of a new set, namely, the image space
$F\left({V}_{}^{m}\right)={F}_{{\omega}_{0}}\left({V}_{}^{m}\right)+\cdots +{F}_{{\omega}_{q-1}}\left({V}_{}^{m}\right)$ where
${G}^{q}=\left\{{\omega}_{0},{\omega}_{1},\cdots ,{\omega}_{q-1}\right\}$ is also a finite group of order *q* where
${\omega}_{0}$ is identity. The
${F}_{{\omega}_{j}}\left({V}_{}^{m}\right)$ is also called an Axiom system for any *j* if at least
${F}_{{\omega}_{j}}\left({V}_{}^{m}\right)\ne \varnothing $ because any an Axiom system is the assumption of
$F\left({V}_{}^{m}\right)$ in which there are only the compatibility relations, *i.e.*, pursuing the same mathematics index
$h\left({F}_{{\omega}_{j}}\left({V}_{}^{m}\right)\right)$.

The special multivariate function *f* (*i.e.*, special function image) is not considered and only the calculation way of the general mathematical indexes of *f* is considered from the factor space
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$ in order to know some causal relations, because of no assumption of *f*. But the size of the data image should be considered if specific issues are studied by the general rule of data images.

In other words, on one hand, a study of the hexagrams (“Hexagram” (卦)) in image mathematics is to learn the generalized properties of the inputs
${x}_{{g}_{0}},\cdots ,{x}_{{g}_{m-1}}$ of any a multivariate function *f* for the given factor space
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$, such as all inputs
${x}_{{g}_{0}},\cdots ,{x}_{{g}_{m-1}}$ are the non-size and non-order relations, but there are orthogonal relations among some inputs
${x}_{{g}_{0}},\cdots ,{x}_{{g}_{m-1}}$, there are also symmetrical relations among some inputs
${x}_{{g}_{0}},\cdots ,{x}_{{g}_{m-1}}$, and only there is an equivalence relation among all inputs
${x}_{{g}_{j}}\in {V}_{{g}_{j}}$ for any *j*, and so on.

On the other hand, a study of the image (“Xiang” (象)) in image mathematics is to learn the generalized properties of all outputs inputs
${f}_{{\omega}_{0}},\cdots ,{f}_{{\omega}_{q-1}}$ for the image space
$F\left({V}_{}^{m}\right)={F}_{{\omega}_{0}}\left({V}_{}^{m}\right)+\cdots +{F}_{{\omega}_{q-1}}\left({V}_{}^{m}\right)$, such as all outputs
${f}_{{\omega}_{0}},\cdots ,{f}_{{\omega}_{q-1}}$ are size specific meaning and a sequence of relationship, but there are the killing relations among some outputs
${f}_{{\omega}_{0}},\cdots ,{f}_{{\omega}_{q-1}}$, and there are also the loving relations among some outputs
${f}_{{\omega}_{0}},\cdots ,{f}_{{\omega}_{q-1}}$, and only there is a liking relation among all outputs
${f}_{{\omega}_{j}}\in {F}_{{\omega}_{j}}\left({V}_{}^{m}\right)$ for any *j*, and so on.

Without loss of generality, the function image space $F\left({V}_{}^{m}\right)$ and the factor space ${V}_{}^{m}$ are putted, still keep for ${V}_{}^{m}$ because of no assumption of ${V}_{}^{m}$ and $F\left({V}_{}^{m}\right)$. In order to study the generalized relations and generalized reasoning, image mathematics researches the following relations.

Assume that

${V}_{{g}_{i}}\times {V}_{{g}_{j}}=\left\{\left(x,y\right):x\in {V}_{{g}_{i}},y\in {V}_{{g}_{j}}\right\}$,

where the note $\times $ is the usual Decartesian product or cross join. Define relations

${R}_{{g}_{r}}={\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{r}},r=0,\cdots ,m-1.$

Here the note ${R}_{{g}_{0}}={R}_{{g}_{0}}^{-1}={R}_{{g}_{0}^{-1}}$ is called an equivalence relation of ${V}_{}^{m}$ if ${g}_{0}$ is identity; denoted by ~;

The note ${R}_{{g}_{s}}={R}_{{g}_{s}}^{-1}={R}_{{g}_{s}^{-1}}\ne {R}_{{g}_{0}}$ is called a symmetrical relation of ${V}_{}^{m}$ if ${g}_{s}={g}_{s}^{-1},s\ne 0$ ; denoted by $\stackrel{{R}_{{g}_{s}}}{\leftrightarrow}$ or $\leftrightarrow $ ;

The note ${R}_{{g}_{1}}={R}_{{g}_{1}^{-1}}^{-1}\ne {R}_{{g}_{1}^{-1}}$ is called a neighboring relation of ${V}_{}^{m}$ if ${g}_{1}\ne {g}_{1}^{-1}$ ; denoted by $\stackrel{{R}_{{g}_{1}}^{}}{\to}$ or $\to $ ;

The note
${R}_{{g}_{a}}={R}_{{g}_{a}^{-1}}^{-1}\ne {R}_{{g}_{a}^{-1}},{R}_{{g}_{1}},{R}_{{g}_{1}^{-1}}$ is called an alternate (or atavism) relation of *V* if
${g}_{a}\ne {g}_{a}^{-1},{g}_{1},{g}_{1}^{-1},a>1$ ; denoted by
$\stackrel{{R}_{{g}_{a}}}{\Rightarrow}$ or
$\Rightarrow $.#

In this case, the equivalence relations and symmetrical relations are compatibility relations but both neighboring relations and alternate relations are incompatibility relations. These relations are all reasoning relations since the relation ${R}_{{g}_{i}}^{-1}={R}_{{g}_{i}^{-1}}\in \Re =\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}$ if ${R}_{{g}_{i}}\in {\Re}_{}^{m}$.

The equivalence relation ${R}_{{g}_{0}}$, symmetrical relations ${R}_{{g}_{s}}$ ’s, neighboring relation ${R}_{{g}_{1}}$ and alternate relations ${R}_{{g}_{a}}$ ’s are all the possible relations for the method of classification taking images. In the following, the equivalence relation ${R}_{{g}_{0}}$, the neighboring relation ${R}_{{g}_{1}}$ and the alternate relations ${R}_{{g}_{a}}$ are mainly considered.

Assume there is an energy function on $V$ (see Zhang & Shao (2012)). In the future, the equivalence relation ${R}_{{g}_{0}}$ will be the liking relation, the symmetrical relations ${R}_{{g}_{s}}$ ’s will be the reciprocal causation relations, the neighboring relation ${R}_{{g}_{1}}$ will be the loving relation and the alternate relations ${R}_{{g}_{a}}$ ’s will be the killing relations. In the following, the liking relation ${R}_{{g}_{0}}$, the loving relation ${R}_{{g}_{1}}$ and the killing relations ${R}_{{g}_{a}}$ are mainly considered.

For example, there is a unique generalized logic reasoning model between the two kinds of opposite incompatibility relations for case $m=5$. Let ${V}_{}^{5}={V}_{0}+\cdots +{V}_{4}$ be a none empty set, there are two kinds of opposite relations: the neighboring relation ${R}_{1}$, denoted $\to $ and the alternate (or atavism) relation ${R}_{2}$, denoted $\Rightarrow $, having the following property:

(1) If
$x\to y,y\to z$, then
$x\Rightarrow z$, *i.e.*, if
$\left(x,y\right)\in {R}_{1},\left(y,z\right)\in {R}_{1}$, then
$\left(x,z\right)\in {R}_{2}$, or,
${R}_{1}*{R}_{1}\subseteq {R}_{2}$ ;

$\iff $ If
$x\to y,x\Rightarrow z$, then
$y\to z$, *i.e.*, if
$\left(x,y\right)\in {R}_{1},\left(x,z\right)\in {R}_{2}$, then
$\left(y,z\right)\in {R}_{1}$, or,
${R}_{1}^{-1}*{R}_{2}\subseteq {R}_{1}$ ;

$\iff $ If
$x\Rightarrow z,y\to z$, then
$x\to y$, *i.e.*, if
$\left(x,z\right)\in {R}_{2},\left(y,z\right)\in {R}_{1}$, then
$\left(x,y\right)\in {R}_{1}$, or,
${R}_{2}*{R}_{1}^{-1}\subseteq {R}_{1}$.

(2) If
$x\Rightarrow y,y\Rightarrow z$, then
$z\to x$, *i.e.*, if
$\left(x,y\right)\in {R}_{2},\left(y,z\right)\in {R}_{2}$, then
$\left(z,x\right)\in {R}_{1}$, or,
${R}_{2}*{R}_{2}\subseteq {R}_{1}^{-1}$ ;

$\iff $ If
$z\to x,y\Rightarrow z$, then
$x\Rightarrow y$, *i.e.*, if
$\left(z,x\right)\in {R}_{1},\left(y,z\right)\in {R}_{2}$, then
$\left(x,y\right)\in {R}_{2}$, or,
${R}_{2}*{R}_{1}\subseteq {R}_{2}^{-1}$ ;

$\iff $ If
$x\Rightarrow y,z\to x$, then
$y\Rightarrow z$, *i.e.*, if
$\left(x,y\right)\in {R}_{2},\left(z,x\right)\in {R}_{1}$, then
$\left(y,z\right)\in {R}_{2}$, or,
${R}_{1}*{R}_{2}\subseteq {R}_{2}^{-1}$.

Let ${R}_{3}={R}_{2}^{-1}$ and ${R}_{4}={R}_{1}^{-1}$. Then above reasoning is equivalent to the calculating as follows:

${R}_{i}*{R}_{j}\subseteq {R}_{mod\left(i+j,5\right)},\forall i,j\in \left\{1,2,3,4\right\},$

where the $mod\left(i+j,5\right)$ is the addition of module 5.

Two kinds of opposite relations can not exist separately. Such reasoning can be expressed in Figure 1 and Figure 2. The first triangle reasoning is known as a jumping-transition reasoning in Figure 1, while the second triangle reasoning is known as an atavism reasoning in Figure 2.

Figure 1. Triangle Reasoning (I).

Figure 2. Triangle Reasoning (II).

Reasoning method is a triangle on both sides decided to any a third side. Both neighboring relations and alternate relations are not compatibility relations, of course, not equivalence relations, called incompatibility or compatibility relations.

2.4. Genetic Reasoning of Non-Authigenic Logic

Let ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ be a none empty set (the collection ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects) with the equivalent relation ${R}_{0}$, the neighboring relation ${R}_{1}$ and the alternate relations ${R}_{a}\ne {R}_{1}^{-1},a>1$. Then a genetic reasoning is defined as follows:

(1) If
$x~y,y\to z$, then
$x\to z$,*i.e.*, if
$\left(x,y\right)\in {R}_{0},\left(y,z\right)\in {R}_{1}$, then
$\left(x,z\right)\in {R}_{1}$, or,
${R}_{0}*{R}_{1}={R}_{1}$ ;

(2) If
$x~y,y\Rightarrow z$, then
$x\Rightarrow z$, *i.e.*, if
$\left(x,y\right)\in {R}_{0},\left(y,z\right)\in {R}_{a}$, then
$\left(x,z\right)\in {R}_{a}$, or,
${R}_{0}*{R}_{a}={R}_{a}$ ;

(3) If
$x\to y,y\sim z$, then
$x\to z$, *i.e.*, if
$\left(x,y\right)\in {R}_{1},\left(y,z\right)\in {R}_{0}$, then
$\left(x,z\right)\in {R}_{1}$, or,
${R}_{1}*{R}_{0}={R}_{1}$ ;

(4) If
$x\Rightarrow y,y~z$, then
$x\Rightarrow z$, *i.e.*, if
$\left(x,y\right)\in {R}_{a},\left(y,z\right)\in {R}_{0}$, then
$\left(x,z\right)\in {R}_{a}$, or,
${R}_{a}*{R}_{0}={R}_{a}$.

The genetic reasoning is equivalent to that there is a group ${G}_{0}^{m}=\left\{0,1,\cdots ,m-1\right\}$ with the operation $*$ such that ${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ can be classified into

${V}_{}^{m}={V}_{0}+\cdots +{V}_{m-1}$ where
${V}_{i}$ may be an empty set and
${R}_{r}={\displaystyle \underset{i=0}{\overset{q-1}{\sum}}{V}_{i}}\times {V}_{i*r}$,
$\forall r\in {G}_{0}^{m}=\left\{0,1,\cdots ,m-1\right\}$, satisfying
${R}_{0}*{R}_{j}={R}_{j}*{R}_{0}={R}_{j}$,*
$\forall j\in {G}_{0}^{m}$ *.

2.5. Steady Multilateral Systems of Non-Authigenic Logic

For a none empty set
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ (the collection
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects) and its some relations
${R}_{0},\cdots ,{R}_{m-1}\subseteq {V}_{}^{n}\times {V}_{}^{n}=\left\{\left(x,y\right):x\in {V}_{}^{n},y\in {V}_{}^{n}\right\}$,*
${\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$ *, the form
$\left({V}_{}^{n},{\Re}_{}^{m}\right)$ (or simply,
${V}_{}^{n}$ ) is called a multilateral system, if
$\left({V}_{}^{n},{\Re}_{}^{m}\right)$ satisfies the following properties:

(a) (uniqueness) ${R}_{0}+\cdots +{R}_{m-1}={V}_{}^{n}\times {V}_{}^{n}$.

(b) (hereditary, or identity, or macro) ${R}_{0}*{R}_{j}={R}_{j}*{R}_{0}={R}_{j},\forall {R}_{j}\in {\Re}_{}^{m}$.

(c) (invertibility) For any ${R}_{i}\in {\Re}_{}^{m}=\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}$, there is ${R}_{i}^{-1}\in {\Re}_{}^{m}$.

(d) (generalized reasoning) For any ${R}_{i}*{R}_{j}\ne \varnothing $, there exists ${R}_{k}\in {\Re}_{}^{m}$ such that ${R}_{i}*{R}_{j}\subseteq {R}_{k},\forall {R}_{i},{R}_{j},{R}_{k}\in {\Re}_{}^{m}$.

The (d) is called the generalized reasoning, the (a) the uniqueness of the generalized reasoning, the (b) the hereditary (pass on from generation to generation) of the generalized reasoning (or genetic reasoning) and the (c) the equivalent property of the generalized reasoning of both relations
${R}_{i}$ and
${R}_{i}^{-1}$, *i.e.*, the reasoning of
${R}_{i}$ is equivalent to the reasoning of
${R}_{i}^{-1}$. In this case, the two-relation set
$\left\{{R}_{i},{R}_{i}^{-1}\right\}$ is a lateral relation of
${V}_{}^{n}$.

Furthermore, the ${V}_{}^{n}$ and ${\Re}_{}^{m}$ are called the state space and relationship set of $\left({V}_{}^{n},{\Re}_{}^{m}\right)$ considered, respectively. The mathematical complex system $\left({V}_{}^{n},{\Re}_{}^{m}\right)$ can be written as $\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right)$ where ${R}_{0}$ is called the equivalent relation or identity or macro-variable of $\left({V}_{}^{n},{\Re}_{}^{m}\right)$.

For a multilateral system $\left({V}_{}^{n},{\Re}_{}^{m}\right)$, it is called complete (or, perfect) if “ $\subseteq $ ” changes into “=”. And it is called complex if there exists at least a incompatibility relation ${R}_{i}\in {\Re}_{}^{m}$. In this case, the multilateral system is also called a logic analysis model of a mathematical complex system in Zhang & Shao (2012).

Let
${R}_{1}$ be a incompatibility relation. The multilateral system
$\left({V}_{}^{n},{\Re}_{}^{m}\right)=\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right)$ is said as a steady multilateral system (or, a stable multilateral system) if there exists a number *n* such that
${R}_{1}^{*n}={R}_{0}\ne \varnothing $ where
${R}_{1}^{*n}={R}_{1}*\stackrel{n}{\cdots}*{R}_{1}$.

The condition is equivalent to there is a the chain
${x}_{1},\cdots ,{x}_{n}\in {V}_{}^{n}$ such that
$\left({x}_{1},{x}_{2}\right),\cdots ,\left({x}_{n},{x}_{1}\right)\in {R}_{1}$, *i.e.*,
${x}_{1}\to {x}_{2}\to \cdots \to {x}_{n}\to {x}_{1}$.

The steady multilateral system is equivalent to the complete multilateral system. The stability definition given above, for a relatively stable system, is most essential. If there is not the chain or cycle, then there will be some elements without causes or some elements without results in a system. Thus, this system is to be in the persistent state of finding its results or causes, *i.e.*, this system will fall into an unstable state, and there is not any stability to say.

Lemma 2.2. *The mathematical complex system
$\left({V}_{}^{n},{\Re}_{}^{m}\right)$ is a multilateral system if and only if there exists a group
${G}_{}^{m}=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$ * *of order m* *where
${g}_{0}$ * *is identity such that the relation set
${\Re}_{}^{m}=\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}$ satisfies *
${R}_{{g}_{i}}*{R}_{{g}_{j}}\subseteq {R}_{{g}_{i}{g}_{j}}$,
$\forall i,j\in \left\{0,\cdots ,m-1\right\}$.* The state space can be classified into
${V}_{}^{n}={V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$ * *where
${V}_{{g}_{i}}$ * *may be an empty set such that *

${R}_{{g}_{i}}={\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}$, $\forall i\in \left\{0,\cdots ,m-1\right\}$.

Proof. Let a multilateral system

$\left({V}_{}^{n},{\Re}_{}^{m}\right)=\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right)$.

Because only the relation of the group of order *m *is considered, so can the serial number of the relationship with a group of Numbers to remember.

Let the group of order *m* is
$G=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$ where
${g}_{0}$ is identity. Then consider the all relations
${R}_{{g}_{i}}={\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}$,
$\forall i\in \left\{0,\cdots ,m-1\right\}$.

So, the classification of the space at least there should be a group of number. In other words, *n* is greater then or equal to *m*. In the following, it is proved that *n* is not greater than *m*. It is proved by using the reduction to absurdity.

If $n>m$, there is a space ${V}_{m+1}$ such that

${V}_{}^{n}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}+{V}_{m+1}+\cdots $

Thus ${V}_{}^{n}\times {V}_{}^{n}=\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}\right)\times \left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}\right)+{V}_{m+1}\times {V}_{{g}_{0}}+\cdots $.

Because by the definition of the relations, there is

${R}_{{g}_{0}}+\cdots +{R}_{{g}_{m+1}}\subseteq \left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}\right)\times \left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}\right)$,

so the condition (a) of the multilateral system is not true, *i.e.*,
${R}_{{g}_{0}}+\cdots +{R}_{{g}_{m+1}}\ne {V}_{}^{n}\times {V}_{}^{n}$. It is because
${V}_{m+1}\times {V}_{{g}_{0}}^{}+\cdots \ne \varnothing $. This is a contradiction since
${R}_{{g}_{0}}+\cdots +{R}_{{g}_{m+1}}={V}_{}^{n}\times {V}_{}^{n}$. Thus
$n\le m$.

But there is also the case $n<m$. For example, let

${V}_{}^{2}={V}_{0}+{V}_{1},{\Re}_{}^{3}=\left\{{R}_{0},{R}_{1},{R}_{2}\right\}.$

Taking ${V}_{2}=\varnothing $, can consider ${V}_{}^{3}={V}_{0}+{V}_{1}+{V}_{2}$. Directly calculating the relations of the steady multilateral system with three elements, it is to get

$\begin{array}{l}{R}_{0}={V}_{0}\times {V}_{0}+{V}_{1}\times {V}_{1}+{V}_{2}\times {V}_{2}={V}_{0}\times {V}_{0}+{V}_{1}\times {V}_{1},\\ {R}_{1}={V}_{0}\times {V}_{1}+{V}_{1}\times {V}_{2}+{V}_{2}\times {V}_{0}={V}_{0}\times {V}_{1},\\ {R}_{2}={V}_{0}\times {V}_{2}+{V}_{1}\times {V}_{0}+{V}_{2}\times {V}_{1}={V}_{1}\times {V}_{0}.\end{array}$

It satisfies the condition (a) of the multilateral system: ${R}_{0}+{R}_{1}+{R}_{2}=\left({V}_{0}+{V}_{1}+{V}_{2}\right)\times \left({V}_{0}+{V}_{1}+{V}_{2}\right)={V}_{}^{3}\times {V}_{}^{3}$.

Therefore, if $n<m$, people can add some empty set to make the multilateral system for the state space satisfying

${V}_{}^{n}={V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},{R}_{{g}_{0}}+\cdots +{R}_{{g}_{m+1}}={V}_{}^{m}\times {V}_{}^{m}.$

On the other hand, the definition of directly using the relationship between operations shows the following conclusions:

$\begin{array}{c}{R}_{{g}_{i}}*{R}_{{g}_{j}}=\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{j}}\right)\\ =\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\\ ={\displaystyle \underset{g\in G}{\sum}\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]}\\ ={\displaystyle \underset{g\in G}{\sum}\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({V}_{g{g}_{i}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]}\\ \subseteq \left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}{g}_{j}}\right)={R}_{{g}_{i}{g}_{j}}.\end{array}$

It is because for the multiplication of relations, the calculation of the relation

$\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]$ is only to consider the relation between ${V}_{g{g}_{i}}$ and $\underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}$ whether there is a common element. But there is a set relation ${V}_{g{g}_{i}}\subseteq {\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}$ and ${V}_{g{g}_{i}}\cap {V}_{{g}^{\prime}{g}_{i}}=\varnothing $, $\forall {g}^{\prime}\ne g$, ${g}^{\prime}\in {G}_{}^{m}=\left\{{g}_{0}^{},{g}_{1}^{},\cdots ,{g}_{m-1}^{}\right\}$. Thereby, whether there is a common element between ${V}_{g{g}_{i}}$ and $\underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}$ is

equivalent to that whether there is a common element between
${V}_{g{g}_{i}}$ and
${V}_{g{g}_{i}}$, *i.e.*, whether
${V}_{g{g}_{i}}$ is an empty set. Thus

$\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)=\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({V}_{g{g}_{i}}\times {V}_{g{g}_{i}{g}_{j}}\right)$.

Therefore, the mathematical complex system
$\left({V}_{}^{n},{\Re}_{}^{m}\right)$ is a multilateral system if and only if there exists a group
${G}_{}^{m}=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$ of order *m *where
${g}_{0}$ is identity such that the relation set
${\Re}_{}^{m}=\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}$ satisfies
${R}_{{g}_{i}}*{R}_{{g}_{j}}\subseteq {R}_{{g}_{i}{g}_{j}}$,*
$\forall i,j\in \left\{0,\cdots ,m-1\right\}$ *.

The state space*
${V}_{}^{n}={V}_{0}+\cdots +{V}_{n-1}$ *can be classified into
${V}_{}^{n}={V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}}$ where
${V}_{{g}_{i}}$ may be an empty set. It completes the proof. #

In this case, the multilateral system $\left({V}_{}^{n},{\Re}_{}^{m}\right)$ can be written as $\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}\right)=\left({V}_{}^{m},{\Re}_{}^{m}\right)$ satisfying

$\begin{array}{l}{R}_{{g}_{i}}={\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}},\forall {g}_{i}\in G=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\},\\ {R}_{{g}_{i}}*{R}_{{g}_{j}}\subseteq {R}_{{g}_{i}{g}_{j}},\forall {g}_{i},{g}_{j}\in G=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}.\end{array}$

Here ${V}_{{g}_{i}}$ may be an empty set. There is an Axiom system ${V}_{{g}_{i}}$ if at least ${V}_{{g}_{i}}\ne \varnothing $. #

Lemma 2.3. *If the following multilateral system is a steady multilateral system*:

*
$\left({V}_{}^{n},{\Re}_{}^{m}\right)=\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right),$ *

*Then
$n=m$ and
${\Re}_{}^{m}$ is a group of order m about multiplication
${R}_{k}*{R}_{l}={R}_{i}$ where
${V}_{i}$ must be at least a non empty set for any i*.* # *

Proof. By Lemma 2.2, there is

$\left({V}_{}^{m},{\Re}_{}^{m}\right)=\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}\right).$

If the following multilateral system is a steady multilateral system:

$\begin{array}{l}\left({V}_{}^{n},{\Re}_{}^{m}\right)=\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right)\\ =\left({V}_{}^{m},{\Re}_{}^{m}\right)=\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}\right)\end{array}$

then

(a) (uniqueness) ${R}_{{g}_{0}}+\cdots +{R}_{{g}_{m-1}}={V}_{}^{m}\times {V}_{}^{m}$.

(b) (hereditary, or identity, or macro) ${R}_{{g}_{0}}*{R}_{{g}_{j}}={R}_{{g}_{j}}*{R}_{{g}_{0}}={R}_{{g}_{j}}$, $\forall {R}_{{g}_{j}}^{}\in {\Re}_{}^{m}$.

(c) (invertibility) For any ${R}_{{g}_{i}}\in {\Re}_{}^{m}=\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}$, there is ${R}_{{g}_{i}}^{-1}\in {\Re}_{}^{m}$, $\forall {R}_{{g}_{i}}^{}\in {\Re}_{}^{m}$.

(d) (generalized reasoning) For any ${R}_{{g}_{i}}*{R}_{{g}_{j}}\ne \varnothing $, there exists ${R}_{{g}_{i}{g}_{j}}\in {\Re}_{}^{m}$ such that

${R}_{{g}_{i}}*{R}_{{g}_{j}}={R}_{{g}_{i}{g}_{j}},\forall {R}_{{g}_{i}},{R}_{{g}_{j}}^{}\in {\Re}_{}^{m}.$

By the definition of group, the*
${\Re}_{}^{m}$ * is a group of order *m* about multiplication

${R}_{{g}_{i}}*{R}_{{g}_{j}}={R}_{{g}_{i}{g}_{j}},\forall {R}_{{g}_{i}},{R}_{{g}_{j}}^{}\in {\Re}_{}^{m}.$

Here, ${R}_{{g}_{0}}$ is identity of the group ${\Re}_{}^{m}$ and ${R}_{{g}_{i}}^{-1}$ is the inverse of ${R}_{{g}_{i}}$ of the group ${\Re}_{}^{m}$ for any a ${R}_{{g}_{i}}^{}\in {\Re}_{}^{m}$.

In the following, it just need to be proved that if the multilateral system

$\begin{array}{l}\left({V}_{}^{n},{\Re}_{}^{m}\right)=\left({V}_{0}+\cdots +{V}_{n-1},\left\{{R}_{0},\cdots ,{R}_{m-1}\right\}\right)\\ =\left({V}_{}^{m},{\Re}_{}^{m}\right)=\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}\right)\end{array}$

is a steady multilateral system, then ${V}_{{g}_{i}}\ne \varnothing $ must be at least a non empty set for any a ${R}_{{g}_{i}}^{}\in {\Re}_{}^{m}$.

It is because, if ${V}_{{g}_{i}}\ne \varnothing $ is a non empty set for all ${R}_{{g}_{i}}^{}\in {\Re}_{}^{m}$, then $n=m$. It is proved by using the reduction to absurdity.

In fact, if only ${V}_{{g}_{k}}=\varnothing $ for some ${g}_{k}^{}\in {G}_{}^{m}=\left\{{g}_{0}^{},{g}_{1}^{},\cdots ,{g}_{m-1}^{}\right\}$, there are ${g}_{i}^{},{g}_{j}^{}\in {G}_{}^{m}=\left\{{g}_{0}^{},{g}_{1}^{},\cdots ,{g}_{m-1}^{}\right\}$ such that ${g}_{k}{g}_{i}^{-1}\ne {g}_{k}$, ${g}_{k}{g}_{j}^{}\ne {g}_{k}$, thus

${V}_{{g}_{k}{g}_{i}^{-1}}\ne \varnothing $, ${V}_{{g}_{k}{g}_{j}}\ne \varnothing $. Thereby ${V}_{{g}_{k}{g}_{i}^{-1}}\times {V}_{{g}_{k}{g}_{j}}\ne \varnothing $.

The definition of directly using the multiplication relationship between relation operations shows the following conclusions:

$\begin{array}{c}{R}_{{g}_{i}}*{R}_{{g}_{j}}=\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{j}}\right)\\ =\left({\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\\ ={\displaystyle \underset{g\in G}{\sum}\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]}\\ ={\displaystyle \underset{g\in G}{\sum}\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({V}_{g{g}_{i}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]}\text{\hspace{0.05em}}\text{\hspace{0.05em}}.\end{array}$

It is because for the multiplication of relations, the calculation of the relation
$\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]$ is only to consider the relation between
${V}_{g{g}_{i}}$ and
$\underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}$ whether there is a common element. But there is a set relation
${V}_{g{g}_{i}}\subseteq {\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}$ and
${V}_{g{g}_{i}}\cap {V}_{{g}^{\prime}{g}_{i}}=\varnothing $,
$\forall {g}^{\prime}\ne g$,
${g}^{\prime}\in {G}_{}^{m}=\left\{{g}_{0}^{},{g}_{1}^{},\cdots ,{g}_{m-1}^{}\right\}$. Thereby, whether there is a common element between
${V}_{g{g}_{i}}$ and
$\underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}$ is equivalent to that whether there is a common element between
${V}_{g{g}_{i}}$ and
${V}_{g{g}_{i}}$, *i.e.*, whether
${V}_{g{g}_{i}}$ is an empty set. Thus
$\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({\displaystyle \underset{g{g}_{i}\in G}{\sum}{V}_{g{g}_{i}}}\times {V}_{g{g}_{i}{g}_{j}}\right)=\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({V}_{g{g}_{i}}\times {V}_{g{g}_{i}{g}_{j}}\right)$.

Continue to calculate the relationship of ${R}_{{g}_{i}}*{R}_{{g}_{j}}$, there is

$\begin{array}{c}{R}_{{g}_{i}}*{R}_{{g}_{j}}={\displaystyle \underset{g\in G}{\sum}\left[\left({V}_{g}\times {V}_{g{g}_{i}}\right)*\left({V}_{g{g}_{i}}\times {V}_{g{g}_{i}{g}_{j}}\right)\right]}\\ ={\displaystyle \underset{g\ne {g}_{k},{g}_{k}{g}_{i}^{-1},g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}{g}_{j}}+\left[\left({V}_{{g}_{k}}\times {V}_{{g}_{k}{g}_{i}}\right)*\left({V}_{{g}_{k}{g}_{i}}\times {V}_{{g}_{k}{g}_{i}{g}_{j}}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}+\left[\left({V}_{{g}_{k}{g}_{i}^{-1}}\times {V}_{{g}_{k}}\right)*\left({V}_{{g}_{k}}\times {V}_{{g}_{k}{g}_{j}}\right)\right]\\ \subset {\displaystyle \underset{g\ne {g}_{k},{g}_{k}{g}_{i}^{-1},g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}{g}_{j}}+{V}_{{g}_{k}{g}_{i}^{-1}}\times {V}_{{g}_{k}{g}_{j}}\\ \subseteq {\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}{g}_{j}}={R}_{{g}_{i}{g}_{j}}.\end{array}$

This is a contradiction to the definition of the steady multilateral system. It is because the definition of the steady multilateral system should be
${R}_{{g}_{i}}*{R}_{{g}_{j}}={R}_{{g}_{i}{g}_{j}}$,*
$\forall {R}_{{g}_{i}},{R}_{{g}_{j}}^{}\in {\Re}_{}^{m}$ *. It completes the proof. #

Definition 2.2. *Let the multilateral system
$\left({V}_{}^{m},{\Re}_{}^{m}\right)$ can be written as *

*
$\left({V}_{}^{m},{\Re}_{}^{m}\right)=\left({V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\left\{{R}_{{g}_{0}},\cdots ,{R}_{{g}_{m-1}}\right\}\right)$ *

*satisfying
${R}_{{g}_{i}}={\displaystyle \underset{g\in G}{\sum}{V}_{g}}\times {V}_{g{g}_{i}}$ *,
$\forall {g}_{i}\in G=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$,
${R}_{{g}_{i}}*{R}_{{g}_{j}}\subseteq {R}_{{g}_{i}{g}_{j}}$,

$\forall {g}_{i},{g}_{j}\in G=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$. *The group
${G}^{m}=\left\{{g}_{0},\cdots ,{g}_{m-1}\right\}$ of order m*,* where
${g}_{0}$ is identity*, *is called the representation group of the multilateral system
$\left({V}_{}^{m},{\Re}_{}^{m}\right)$ *.

*Denoted the generalized difference in*,* set
$I\left({R}_{{g}_{i}}\right)=\left\{\left(x,y\right):{x}^{-1}y={g}_{i},x,y\in G\right\}$ **based on the group
$G$ *,* called the representing function of relation
${R}_{{g}_{i}}$ *.* The representing function has nothing to do with the collection
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects*.

*Let multilateral systems
$\left({\left({V}_{}^{m}\right)}^{i},{\left({\Re}_{}^{m}\right)}^{i}\right),i=1,2$ be with two representation groups
${\left({G}_{}^{m}\right)}^{i},i=1,2$ *,* respectively*.* Both multilateral systems *
$\left({\left({V}_{}^{m}\right)}^{i},{\left({\Re}_{}^{m}\right)}^{i}\right),i=1,2$ *are called isomorphic if the two representation groups
${\left({G}_{}^{m}\right)}^{i},i=1,2$ are isomorphic*. #

Lemmas 2.2 and 2.3 and Definitions 2.1 and 2.2 are the key to concepts in multilateral system theory because they show the classification taking images as the basic method. In the following, introduce two basic models to illustrate the method.

Lemma 2.4. *Suppose that
${G}_{0}^{2}=\left\{0,1\right\}$ with multiplication table*

$\begin{array}{ccc}{G}_{0}^{2}& 0& 1\\ 0& 0& 1\\ 1& 1& 0\end{array}$

*i.e.*,* the multiplication of
${G}_{0}^{2}$ is the addition of module* 2. *In other words*,*
$i*r=mod\left(i+r,2\right),\forall i,j\in {G}_{0}^{2}$ *.

*And assume that
${V}_{}^{2}={V}_{0}+{V}_{1},{\Re}_{}^{2}=\left\{{R}_{0},{R}_{1}\right\}$ *, *where
${R}_{r}={\displaystyle \underset{i=0}{\overset{1}{\sum}}{V}_{i}}\times {V}_{mod\left(i+r,2\right)}$ *,*
$\forall r\in {G}_{0}^{2}=\left\{0,1\right\}$ * *satisfying
${R}_{i}*{R}_{j}={R}_{mod\left(i+j,2\right)},\forall i,j\in {G}_{0}^{2}$ *.

*Then
$\left({V}_{}^{2},{\Re}_{}^{2}\right)=\left({V}_{0}+{V}_{1},\left\{{R}_{0},{R}_{1}\right\}\right)$ * *is a steady multilateral system with one equivalent relation
${R}_{0}$ and one symmetrical relation
${R}_{1}$ *.

*The system is simple since there is not any incompatibility relation*.* In other words*,* the relations
${R}_{i}$ *’*s are the simple forms as follows*:

$I\left({R}_{0}\right)=\left\{\left(0,0\right),\left(1,1\right)\right\},I\left({R}_{1}\right)=\left\{\left(0,1\right),\left(1,0\right)\right\},$

*where
$\left(i,j\right)$ is corresponding to
${V}_{i}\times {V}_{j}$ *.

*In the sense of isomorphism*,* there is only one unique steady multilateral system
${V}_{}^{2}$ *. *# *

Proof. Because in the sense of isomorphism, dual group only Lemma 2.4 defined in a structure, so Definition 2.2, up to isomorphism only Lemma 2.4 defined in this a multilateral system.

In fact, for the steady multilateral system $\left({V}_{}^{2},{\Re}_{}^{2}\right)=\left({V}_{0}+{V}_{1},\left\{{R}_{0},{R}_{1}\right\}\right)$, if $I\left({R}_{0}\right)=\left\{\left(0,0\right),\left(1,1\right)\right\}$, $I\left({R}_{1}\right)=\left\{\left(0,1\right),\left(1,0\right)\right\}$, then ${R}_{i}*{R}_{j}={R}_{mod\left(i+j,2\right)}$, $\forall i,j\in {G}_{0}^{2}$. Thus in the sense of isomorphism, $\left\{{R}_{0},{R}_{1}\right\}$ is isomorphic to the group ${G}_{0}^{2}=\left\{0,1\right\}$.

The relation ${R}_{0}$ is an equivalent relation and ${R}_{1}$ is a symmetrical relation since ${R}_{1}={\left({R}_{1}\right)}_{}^{-1}$. Of course, they are the compatibility relations.

Because in the sense of isomorphism, there is only one unique group ${G}_{0}^{2}=\left\{0,1\right\}$, so in the sense of isomorphism, there is only one unique steady multilateral system ${V}^{2}={V}_{0}+{V}_{1}$. It completes the proof. #

The steady multilateral system in Lemma 2.4 is the reasoning model of “*Tao*” (道) or “Yin-Yang” (阴阳) in TCP or TCMath if there are two energy functions
$\phi \left({V}_{0}\right)$ and
$\phi \left({V}_{1}\right)$ satisfying
$\phi \left({V}_{0}\right)<\phi \left({V}_{1}\right)$, called *Tao* model, denoted by

${V}^{2}={V}_{0}+{V}_{1},{R}_{0}={V}_{0}\times {V}_{0}+{V}_{1}\times {V}_{1},{R}_{1}={V}_{0}\times {V}_{1}+{V}_{1}\times {V}_{0},$

where ${V}_{0}$ is called Yin (阴) state of ${V}^{2}$ and ${V}_{1}$ is called Yang (阳) state of ${V}^{2}$.

Lemma 2.5. *Suppose that
${G}_{0}^{5}=\left\{0,1,2,3,4\right\}$ with multiplication table*

$\begin{array}{cccccc}{G}_{0}^{5}& 0& 1& 2& 3& 4\\ 0& 0& 1& 2& 3& 4\\ 1& 1& 2& 3& 4& 0\\ 2& 2& 3& 4& 0& 1\\ 3& 3& 4& 0& 1& 2\\ 4& 4& 0& 1& 2& 3\end{array}$

*i.e.*,* the multiplication of
${G}_{0}^{5}$ is the addition of module *5. *In other words*,*
$i*r=mod\left(i+r,5\right)$ *.* And assume that
${V}_{}^{5}={V}_{0}+\cdots +{V}_{4}$ *,*
${\Re}_{}^{5}=\left\{{R}_{0}{}_{}^{5},\cdots ,{R}_{4}{}_{}^{5}\right\}$ *,

*where
${R}_{r}{}_{}^{5}={\displaystyle \underset{i=0}{\overset{4}{\sum}}{V}_{i}}\times {V}_{mod\left(i+r,5\right)},\forall r\in {G}_{0}^{5}=\left\{0,1,2,3,4\right\}$ satisfying *
${R}_{i}{}_{}^{5}*{R}_{j}{}_{}^{5}={\left({R}_{mod\left(i+j,5\right)}\right)}_{}^{5},\forall i,j\in {G}_{0}^{5}$.

*Then
$\left({V}_{}^{5},{\Re}_{}^{5}\right)=\left({V}_{0}+\cdots +{V}_{4},\left\{{R}_{0}{}_{}^{5},\cdots ,{R}_{4}{}_{}^{5}\right\}\right)$ is a steady multilateral system with one equivalent relation
${R}_{0}{}_{}^{5}$ and one loving relation
${R}_{1}{}_{}^{5}={\left({R}_{4}^{}{}_{}^{5}\right)}_{}^{-1}$ * *and one **killing relation
${R}_{2}^{}{}_{}^{5}={\left({R}_{3}^{}{}_{}^{5}\right)}_{}^{-1}$ *.* The relations
${R}_{1}{}_{}^{5}={\left({R}_{4}^{}{}_{}^{5}\right)}_{}^{-1}$ and
${R}_{2}^{}{}_{}^{5}={\left({R}_{3}^{}{}_{}^{5}\right)}_{}^{-1}$ are two incompatibility relationships*.

*The system is complex since there are two incompatibility relationships*.* In other words*,* the relations
${R}_{i}{}_{}^{5}$ *’*s are the simple forms as follows*:

$\begin{array}{l}I\left({R}_{0}{}_{}^{5}\right)=\left\{\left(0,0\right),\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,5\right)\right\},\\ I\left({R}_{1}{}_{}^{5}\right)=\left\{\left(0,1\right),\left(1,2\right),\left(2,3\right),\left(3,4\right),\left(4,0\right)\right\},\\ I\left({R}_{2}{}_{}^{5}\right)=\left\{\left(0,2\right),\left(1,3\right),\left(2,4\right),\left(3,0\right),\left(4,1\right)\right\},\\ I\left({R}_{3}{}_{}^{5}\right)=\left\{\left(0,3\right),\left(1,4\right),\left(2,0\right),\left(3,1\right),\left(4,2\right)\right\},\\ I\left({R}_{4}{}_{}^{5}\right)=\left\{\left(0,4\right),\left(1,0\right),\left(2,1\right),\left(3,2\right),\left(4,3\right)\right\},\end{array}$

*where
$\left(i,j\right)$ is corresponding to
${V}_{i}\times {V}_{j}$ *.

*In the sense of isomorphism*,* there is only one unique steady multilateral system
${V}_{}^{5}$ *. *# *

Proof. Because in the sense of isomorphism, dual group only Lemma 2.5 defined in a structure, so Definition 2.2, up to isomorphism only Lemma 2.5 defined in this a multilateral system.

In fact, for the steady multilateral system $\left({V}_{}^{5},{\Re}_{}^{5}\right)=\left({V}_{0}+\cdots +{V}_{4},\left\{{R}_{0}{}_{}^{5},\cdots ,{R}_{4}{}_{}^{5}\right\}\right)$, if

$\begin{array}{l}I\left({R}_{0}{}_{}^{5}\right)=\left\{\left(0,0\right),\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,5\right)\right\},\\ I\left({R}_{1}{}_{}^{5}\right)=\left\{\left(0,1\right),\left(1,2\right),\left(2,3\right),\left(3,4\right),\left(4,0\right)\right\},\\ I\left({R}_{2}{}_{}^{5}\right)=\left\{\left(0,2\right),\left(1,3\right),\left(2,4\right),\left(3,0\right),\left(4,1\right)\right\},\\ I\left({R}_{3}{}_{}^{5}\right)=\left\{\left(0,3\right),\left(1,4\right),\left(2,0\right),\left(3,1\right),\left(4,2\right)\right\},\\ I\left({R}_{4}{}_{}^{5}\right)=\left\{\left(0,4\right),\left(1,0\right),\left(2,1\right),\left(3,2\right),\left(4,3\right)\right\},\end{array}$

then ${R}_{i}{}_{}^{5}*{R}_{j}{}_{}^{5}={\left({R}_{mod\left(i+j,5\right)}\right)}_{}^{5},\forall i,j\in {G}_{0}^{5}$.

The relation ${R}_{0}{}_{}^{5}$ is an equivalent relation. Of course, it is a compatibility relationship. And ${R}_{1}{}_{}^{5}={\left({R}_{4}{}_{}^{5}\right)}_{}^{-1}$ is an incompatibility relation as the loving relationship since $\left(x,x\right)\notin {R}_{2}^{5}$, $\forall x\in {V}^{5}$ ; ${R}_{1}^{5}\ne {\left({R}_{1}^{5}\right)}_{}^{-1}$ ; ${R}_{1}^{5}*{R}_{1}^{5}\overline{)\subseteq}{R}_{1}^{5}$ and ${g}_{1}^{}=1\ne {g}_{1}^{-1}={1}_{}^{-1}=4$, $\forall {g}_{1}^{}\in {G}_{0}^{5}=\left\{{g}_{0}^{}=0,{g}_{1}^{}=1,\cdots ,{g}_{4}^{}=4\right\}$.

And ${R}_{2}{}_{}^{5}={\left({R}_{3}{}_{}^{5}\right)}_{}^{-1}$ is an incompatibility relation which can be as the killing relationship of the multilateral system $\left({V}_{}^{5},{\Re}_{}^{5}\right)=\left({V}_{0}+\cdots +{V}_{4},\left\{{R}_{0}{}_{}^{5},\cdots ,{R}_{4}{}_{}^{5}\right\}\right)$ since $\left(x,x\right)\notin {R}_{2}^{5}$, $\forall x\in {V}^{5}$ ; ${R}_{2}^{5}\ne {\left({R}_{2}^{5}\right)}_{}^{-1}$ ; ${R}_{2}^{5}*{R}_{2}^{5}\overline{)\subseteq}{R}_{1}^{5}$ and ${g}_{2}^{}=2\ne {g}_{2}^{-1}={2}_{}^{-1}=3$, ${g}_{1}^{}=1$, ${g}_{1}^{-1}={1}_{}^{-1}=4$, $\forall {g}_{1}^{},{g}_{2}^{}\in {G}_{0}^{5}=\left\{{g}_{0}^{}=0,{g}_{1}^{}=1,\cdots ,{g}_{4}^{}=4\right\}$.

Thus in the sense of isomorphism, ${\Re}_{}^{5}=\left\{{R}_{0}{}_{}^{5},\cdots ,{R}_{4}{}_{}^{5}\right\}$ is isomorphic to the group ${G}_{0}^{5}=\left\{0,1,2,3,4\right\}$.

Because in the sense of isomorphism, there is only one unique group ${G}_{0}^{5}=\left\{0,1,2,3,4\right\}$, so in the sense of isomorphism, there is only one unique steady multilateral system ${V}_{}^{5}={V}_{0}+\cdots +{V}_{4}$. It completes the proof. #

2.6. Yin Yang Wu Xing Model of Non-Authigenic Logic

In order to apply the reasoning to other fields rather than the health of a mathematical multilateral complex system, Zhang & Shao (2012) have started a steady multilateral system imitating a mathematical multilateral complex system. A most basic steady multilateral system is as follows.

Theorem 2.1. Feng et al. (2005) *For each element **x in a steady multilateral system
$V$ * *with two incompatibility relations*,* there exist five equivalence classes below*:

$X=\left\{y\in V|y~x\right\},{X}_{S}=\left\{y\in V|x\to y\right\},{X}_{K}=\left\{y\in V|x\Rightarrow y\right\},$

${K}_{X}=\left\{y\in V|y\Rightarrow x\right\},{S}_{X}=\left\{y\in V|y\to x\right\},$

*which the five equivalence classes have relations in *Figure 3. #

The Yin Yang Wu Xing model can be written as follows: Define

${V}_{0}=X,{V}_{1}={X}_{S},{V}_{2}={X}_{K},{V}_{3}={K}_{X},{V}_{4}={S}_{X},$

Figure 3*.* Finding Yin Yang Wu Xing Model.

corresponding to wood, fire, earth, metal, water, respectively, and assume that ${V}_{}^{5}={V}_{0}+{V}_{1}+{V}_{2}+{V}_{3}+{V}_{4}$ where ${V}_{i}\cap {V}_{j}=\varnothing ,\forall i\ne j$.

And assume that ${\Re}_{}^{5}=\left\{{R}_{0},{R}_{1},\cdots ,{R}_{4}\right\}$ satisfying

$\begin{array}{l}{R}_{r}={\displaystyle \underset{i=0}{\overset{4}{\sum}}{V}_{i}}\times {V}_{mod\left(i+r,5\right)},\forall r\in {G}_{0}^{5}=\left\{0,1,\cdots ,4\right\},\\ {R}_{i}*{R}_{j}\subseteq {R}_{mod\left(i+j,5\right)},\forall i,j\in {G}_{0}^{5}=\left\{0,1,\cdots ,4\right\},\end{array}$

where ${V}_{i}\times {V}_{j}=\left\{\left(x,y\right):x\in {V}_{i},y\in {V}_{j}\right\}$ is the Descartesian product in set theory and the following note ${R}_{i}*{R}_{j}=\left\{\left(x,y\right):\exists u\in {V}_{}^{5}\text{\hspace{0.17em}}\text{suchthat}\text{\hspace{0.17em}}\left(x,u\right)\in {R}_{i},\left(u,y\right)\in {R}_{j}\right\}$ is the relation multiplication operation. The relation multiplication of $*$ is isomorphic to the addition of module 5.

Then ${V}^{5}$ is a steady multilateral system with one equivalent relation ${R}_{0}$ and two incompatibility relations ${R}_{1}={R}_{4}^{-1}$ and ${R}_{2}={R}_{3}^{-1}$ where the note ${R}_{i}^{-1}=\left\{\left(x,y\right):\left(y,x\right)\in {R}_{i}\right\}$ is the relation inverse operation.

For the reasoning model of Yin Yang Wu Xing in TCP or TCMath, the words “Yin Yang” means there are two incompatibility relations ${R}_{1}={R}_{4}^{-1}$ and ${R}_{2}={R}_{3}^{-1}$, where the neighboring relation ${R}_{1}={R}_{4}^{-1}$ is the loving relation, also called Yang relation or interpromoting relation (相生) among five subsets ${V}_{j}$ ’s, and the alternate relation ${R}_{2}={R}_{3}^{-1}$ is the killing relation, also called Yin relation or restriction relation (相克) among five subsets ${V}_{j}$ ’s; The words “Wu Xing” means there are five subsets ${V}_{j}$ ’s such that ${V}_{}^{5}={V}_{0}+{V}_{1}+{V}_{2}+{V}_{3}+{V}_{4},$ where there is only the equivalence relation ${R}_{0}$ among interior for each of five subsets ${V}_{j}$ ’s, and there are Yin or Yang relations among exterior for all five subsets ${V}_{j}$ ’s. The five subsets ${V}_{j}$ ’s do not means the five elements (metal, wood, water, fire and earth), but with philosophical meaning of five aspects corresponding to the five subsets ${V}_{j}$ ’s.

It can be proved that the steady multilateral system in Theorem 2.1 is the reasoning model of Yin Yang Wu Xing in TCP or TCMath if there is an energy function $\phi (*)$ satisfying

$\begin{array}{l}\frac{\Delta \phi \left(X\right)}{\Delta}\to \frac{\text{d}\phi \left(X\right)}{\text{d}X}=\left(1-{\rho}_{2}{\rho}_{3}\right)=\left(1-c\rho {\left(x\right)}^{3}\right)>0;\\ \frac{\Delta \phi \left({X}_{S}\right)}{\Delta}\to \frac{\text{d}\phi \left({X}_{S}\right)}{\text{d}X}=\left({\rho}_{1}+{\rho}_{2}{\rho}_{3}\right)=\rho \left(x\right)\left(1+c\rho {\left(x\right)}^{2}\right)>0;\\ \frac{\Delta \phi \left({X}_{K}\right)}{\Delta}\to \frac{\text{d}\phi \left({X}_{K}^{}\right)}{\text{d}X}=-\left({\rho}_{1}^{}-{\rho}_{3}\right)=-\rho \left(x\right)\left(1-c\right)<0;\\ \frac{\Delta \phi \left({K}_{X}\right)}{\Delta}\to \frac{\text{d}\phi \left({K}_{X}^{}\right)}{\text{d}X}=-\left({\rho}_{2}^{}-{\rho}_{1}^{}{\rho}_{3}\right)=-\rho {\left(x\right)}^{2}\left(1-c\right)<0;\end{array}$

$\begin{array}{l}\frac{\Delta \phi \left({S}_{X}\right)}{\Delta}\to \frac{\text{d}\phi \left({S}_{X}^{}\right)}{\text{d}X}=\left({\rho}_{2}^{}-{\rho}_{1}^{}{\rho}_{3}\right)=\rho {\left(x\right)}^{2}\left(1-c\right)>0,\\ \text{if}\text{\hspace{0.17em}}\text{increase}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{energy}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}X\text{\hspace{0.17em}}\left(\forall \Delta \phi \left(X\right)=\Delta >0\right),\\ \text{where}\text{\hspace{0.17em}}\text{\hspace{0.05em}}{\rho}_{1}=\rho \left(x\right),{\rho}_{2}=\rho {\left(x\right)}_{}^{2},{\rho}_{3}=c\rho \left(x\right),0<\rho \left(x\right)<1,0\le c\le 1.\end{array}$

The parameter ${\rho}_{v}^{}={\rho}_{1}+{\rho}_{2}{\rho}_{3}$ is called the coefficient of the vital or righteousness energy. The parameter ${\rho}_{e}^{}=1-{\rho}_{2}{\rho}_{3}$ is called the coefficient of the evil energy. A mathematical complex system is called healthy if the vital or righteousness coefficient ${\rho}_{v}^{}={\rho}_{1}+{\rho}_{2}{\rho}_{3}$ is greater than or equal to the evil coefficient ${\rho}_{e}^{}=1-{\rho}_{2}{\rho}_{3}$. Otherwise, the mathematical complex system is called unhealthy. For a healthy mathematical complex system, the transfer law of the Yang vital or righteousness energy in the Yin Yang Wu Xing Model is

$\begin{array}{l}Wood\left(X\right)\to Fire\left({X}_{S}^{}\right)\to Earth\left({X}_{K}^{}\right)\to Metal\left({K}_{X}^{}\right)\\ \to Water\left({S}_{X}^{}\right)\to Wood\left(X\right)\to \cdots .\end{array}$

Figure 3 in Theorem 2.1 is the figure of Yin Yang Wu Xing theory in Ancient China. The steady multilateral system ${V}_{}^{5}$ with two incompatibility relations is equivalent to the logic architecture of reasoning model of Yin Yang Wu Xing theory in Ancient China. What describes the general method of the steady multilateral system ${V}_{}^{5}$ with two incompatibility relations can be used in any a mathematical complex system.

Non-authigenic logic of TCP or TCMath is similar to a group which has nothing to do with the research objects in Zhang & Shao (2012). In order to ensure the reproducibility such that the analysis conclusion can be applicable to any a mathematical complex system, a logic analysis model can be chosen which has nothing to do with the object of study. The*Tao* model of Yin and Yang is a generalized one which means that two is basic. But the*Tao* model of Yin Yang is simple in which there is not incompatibility relation. The analysis conclusion of*Tao* model of Yin Yang cannot be applied to an incompatibility relation model. Thus the Yin Yang Wu Xing model with two incompatibility relations of Theorem 2.1 will be selected as the logic analysis model in this paper.

Study of a mathematical complex system is the key to pathological diagnosis and treatment of a mathematical complex system, also known as interference. Any steady multilateral system with an energy function, an intervention response capacity and a self-protection capacity, is a logic to study the specific object of non-authigenic logic. According to non-authigenic logic thought, the stand or fall of the intervention method for a steady multilateral system imitating a mathematical complex system, depends on a mathematical complex system’s both side effects issue and mathematical intervention resistance problem from the intervention. Whether it can promote the health of the mathematical complex system without side effects issue? Whether it can promote the response ability to the intervention of the mathematical complex systems and self-protection ability without mathematical intervention resistance problem? These standards have nothing to do with mathematical researches. The intervention methods of this mathematical complex system are in Zhang (2011a, 2011b, 2020), Zhang & Zhang (2019) and Zhang et al. (2021a, 2021b) in general, are as follows:

(1) “Virtual disease with a healthy mathematical complex system is to fill its mother but real disease with a healthy mathematical complex system rushes down its son” (虚则补其母，实则泄其子) if only one subsystem of the mathematical complex system falls ill or if only two subsystems with the loving relation of a mathematical complex system encounter usual sick for a healthy mathematical complex system in Zhang (2011a, 2011b).

(2) “Virtual disease with an unhealthy mathematical complex system is to fill itself but real disease with an unhealthy mathematical complex system rushes down itself” (虚则补之，实则泄之) if only one subsystem of the mathematical complex system falls ill or if only two subsystems with the killing relation of a mathematical complex system encounter usual sick for an unhealthy mathematical complex system in Zhang (2011b) or Zhang & Zhang (2019).

(3) “Strong inhibition of the same time, support the weak” (抑强补弱，或者，泄南补北) if only two subsystems with a killing relation of a mathematical complex system encounter usual sick, which destroyed the killing relation of the mathematical complex system in Zhang (2011a), or Zhang (2020).

(4) “Do not treat a disease after it has occurred. But treat the disease before it will occur” (不治已病治未病) if there are a lot of subsystems of a mathematical complex system encountering sick for a long time in Zhang (2011b) or Zhang (2020).

(5) “Searching for the primary cause of disease in treatment and cure both symptoms and root-cause at the same time” (治病求本，标本兼治) if there are a lot of subsystems of a mathematical complex system encountering sick for a long time in Zhang & Zhang (2019).

(6) “Even if all changed, it is hard to change one’s nature” (江山易改，本性难移) if there are a lot of subsystems of a mathematical complex system encountering sick for a long time in Zhang et al. (2021a).

(7) “Seize the momentum of development, hasten lucky avoids disaster” (顺势而为，趋吉避凶) if there are a lot of subsystems of a mathematical complex system encountering sick for a long time in Zhang et al. (2021b).

The combination of Yin Yang Wu Xing theory with the problem in question of TCP is not quite well-founded. The key is the philosophical meaning of five aspects of non-authigenic logic. The steady multilateral complex system is used to simulate the complex system of human body. The philosophical meaning of five aspects of non-authigenic logic can be used to TCP itself. The following conclusions are drawn.

All healing methods represent cognition, equivalent to wood. These treatments include the following.

For example, “Virtual disease with a healthy mathematical complex system is to fill its mother but real disease with a healthy mathematical complex system rushes down its son” (虚则补其母，实则泄其子)

“Virtual disease with an unhealthy mathematical complex system is to fill itself but real disease with an unhealthy mathematical complex system rushes down itself” (虚则补之，实则泄之).

“Strong inhibition of the same time, support the weak” (抑强补弱，或者，泄南补北), and so on.

Similarly, all logic problems represent cognition, equivalent to wood. The non-authigenic thinking or no Axiom system (无公理系统) is the key to the non-authigenic logic, liking wood.

All disease transmission methods represent development, equivalent to fire. These disease transmission methods include the following.

There are healthy people and unhealthy people about the development of disease. The methods of disease transmission vary from body to body. Treatment should be tailored to different people. Do not treat the disease that has already occurred, and treat the disease that has not occurred but will occur by following the disease transmission method.

For example, “Do not treat a disease after it has occurred. But treat the disease before it will occur” (不治已病治未病), and so on.

Similarly, all loving or interpromoting relation problems represent developing, equivalent to fire. The loving or interpromoting analysis (相生分析) is the preconception principle of the non-authigenic logic, liking fire.

All zangxiang and jingluo methods represent combining, equivalent to earth. These zangxiang and jingluo methods include the following.

For example, “Searching for the primary cause of disease in treatment and cure both symptoms and root-cause at the same time” (治病求本，标本兼治), and so on.

Similarly, all killing or restriction or checking relation problems represent combining, equivalent to fire. The killing or restriction or checking analysis (相克分析) is the integration coordination combination principle of the non-authigenic logic, liking earth.

All eight palaces or eight veins methods represent accepted, equivalent to metal. These eight palaces or eight veins methods include the following.

For example, “Even if all changed, it is hard to change one’s nature” (江山易改，本性难移), and so on.

Similarly, all killing or restriction or checking reproducibility problems represent accepted, equivalent to metal. The killing or restriction or checking reproducibility (相克再现性) is the logic layering principle of the non-authigenic logic, liking metal.

All sixty JiaZi methods represent risk, equivalent to water. These sixty JiaZi methods include the following.

For example, “Seize the momentum of development，hasten lucky avoids disaster” (顺势而为，趋吉避凶), and so on.

Similarly, all loving or interpromoting reproducibility problems represent risk, equivalent to metal. The lovling or interpromoting reproducibility (相生再现性) is the automation principle of the non-authigenic logic, liking water.

The steady multilateral complex system is used to simulate the complex system of TCP. The combination of Yin Yang Wu Xing theory with the problem in question of TCP is obtained by the philosophical meaning of five aspects of non-authigenic logic. A lot of ideas can be obtained similar to the philosophical meaning of five aspects of non-authigenic logic.

2.7. Some Examples of the Steady Multilateral Systems of Non-Authigenic Logic

For the steady multilateral systems, there are a lot of models which satisfy the generalized relations and the generalized reasoning. The following are some examples.

(1) Suppose that ${G}_{0}^{3}=\left\{0,1,2\right\}$ where ${g}_{0}=0$ is identity. Its multiplication table is as follows:

$\begin{array}{cccc}{G}_{0}^{3}& 0& 1& 2\\ 0& 0& 1& 2\\ 1& 1& 2& 0\\ 2& 2& 0& 1\end{array}$

The steady multilateral system $\left({V}^{3},{\Re}_{}^{3}\right)=\left({V}_{0}^{2}+{V}_{1}^{2}+{V}_{2}^{3},\left\{{R}_{0}^{3},{R}_{1}^{3},{R}_{2}^{3}\right\}\right)$ is called the three types of talent model, denoted simply by ${V}^{3}=\left\{1,2,3\right\}=\left\{1\right\}+\left\{2\right\}+\left\{3\right\}={V}_{0}^{2}+{V}_{1}^{2}+{V}_{2}^{3}$, if it satisfies the following conditions:

$\begin{array}{l}{R}_{r}^{3}={\displaystyle \underset{i=0}{\overset{1}{\sum}}{V}_{i}^{3}}\times {V}_{mod\left(i+r,3\right)}^{3},\forall r\in {G}_{0}^{3}=\left\{0,1,2\right\},\\ {R}_{i}^{3}*{R}_{j}^{3}={R}_{mod\left(i+j,3\right)}^{3},\forall i,j\in {G}_{0}^{3}=\left\{0,1,2\right\}.\end{array}$

The system is complex since there is only one incompatibility relationship. In other words, the relations ${R}_{i}^{3}$ ’s are the simple forms as follows:

$\begin{array}{l}I\left({R}_{0}^{3}\right)=\left\{\left(0,0\right),\left(1,1\right),\left(2,2\right)\right\},\\ I\left({R}_{1}^{3}\right)=\left\{\left(0,1\right),\left(1,2\right),\left(2,0\right)\right\},\\ I\left({R}_{2}^{3}\right)=\left\{\left(0,2\right),\left(1,0\right),\left(2,1\right)\right\}.\end{array}$

The relation ${R}_{0}^{3}$ is the equivalent relation. Of course, it is a compatibility relation. Only there is one incompatibility relationship ${R}_{1}^{3}={\left({R}_{2}^{3}\right)}_{}^{-1}$ since $I\left({R}_{1}^{3}\right)=\left\{\left(0,1\right),\left(1,2\right),\left(2,0\right)\right\}={\left[I\left({R}_{2}^{3}\right)\right]}_{}^{-1}$ and $\left(x,x\right)\notin {R}_{1}^{3}$, $\forall x\in {V}^{3}$ ; ${R}_{1}^{3}\ne {\left({R}_{1}^{3}\right)}_{}^{-1}$ ; ${R}_{1}^{3}*{R}_{1}^{3}\overline{)\subseteq}{R}_{1}^{3}$.

The incompatibility relation ${R}_{1}^{3}={\left({R}_{2}^{3}\right)}_{}^{-1}$ is a loving relationship of the steady multilateral system $\left({V}^{3},{\Re}_{}^{3}\right)=\left({V}_{0}^{2}+{V}_{1}^{2}+{V}_{2}^{3},\left\{{R}_{0}^{3},{R}_{1}^{3},{R}_{2}^{3}\right\}\right)$ since ${g}_{1}^{}=1\ne {g}_{1}^{-1}={1}_{}^{-1}=2,\forall {g}_{1}^{}\in {G}_{0}^{3}=\left\{{g}_{0}^{}=0,{g}_{1}^{}=1,{g}_{2}^{}=2\right\}$.

Here, the number 1 or 2 or 3 is called the tengen (天元), the earth material (地元), the people ability (人元), corresponding to heaven (1), earth (2) and people (3), respectively. The set of {1, 2, 3} is called the three types of talent or material set.

In TCMath, any material can be found, not Yang is Yin. No matter of Yin and Yang are unable to see, known as dark matter, or nonphysical. Therefore, the *Tao* force is often exist in the physical world. Any steady multilateral system only force under the action of the *Tao*, may be to be perceived.

In TCMath, there is a generalized Yin Yang Wu Xing complex system model: Nature, *Tao*, Heaven, Earth, People. {Nature, Tao} is called as Yang. And {Heaven, Earth, People}, namely, three types of talent or material, is called as Yin.

It is believed that any a Yin Yang Wu Xing complex system is made up of three types of talent or material to combined changes. The three types come from the Yin energy in it’s a layer Yin Yang Wu Xing system. It is because a lot of complex systems can be seen as a Yin Yang Wu Xing system. However, any Yin Yang Wu Xing system is a mathematical complex system observation of the objective fact in one logic level, it will be a layer of the Yin Yang Wu Xing system of restriction and generation. In the Yin Yang Wu Xing system, both wood and fire are Yang; three types including earth, metal and water are Yin. So any Yin Yang Wu Xing system is generated by the three talents (earth, gold and water) at the upper logical level. The three types are generated from an upper layer of the Yin Yang Wu Xing system. For example, an upper layer of controlling on the Yin Yang Wu Xing system of mathematical complex system is the nature, *Tao*, heaven, earth and people system, so the formation of the mathematical complex system three materials are heaven (1), earth (2) and people (3).

(2) Suppose that the following set

${G}_{}^{6}=\left\{1=e,2=\left(12\right),3=\left(13\right),4=\left(23\right),5=\left(123\right),6=\left(132\right)\right\}$

is a group of order 6, where ${g}_{0}=1=e$ is identity. Its multiplication table is as follows:

$\begin{array}{ccccccc}i*r& 1=e& 2=\left(12\right)& 3=\left(13\right)& 4=\left(23\right)& 5=(123)& 6=(132)\\ 1=e& 1=e& 2=\left(12\right)& 3=\left(13\right)& 4=\left(23\right)& 5=\left(123\right)& 6=\left(132\right)\\ 2=\left(12\right)& 2=\left(12\right)& 1=e& 5=\left(123\right)& 6=\left(132\right)& 3=\left(13\right)& 4=\left(23\right)\\ 3=\left(13\right)& 3=\left(13\right)& 6=\left(132\right)& 1=e& 5=\left(123\right)& 4=\left(23\right)& 2=\left(12\right)\\ 4=\left(23\right)& 4=\left(23\right)& 5=\left(123\right)& 6=\left(132\right)& 1=e& 2=\left(12\right)& 3=\left(13\right)\\ 5=\left(123\right)& 5=\left(123\right)& 4=\left(23\right)& 2=\left(12\right)& 3=\left(13\right)& 6=\left(132\right)& 1=e\\ 6=\left(132\right)& 6=\left(132\right)& 3=\left(13\right)& 4=\left(23\right)& 2=\left(12\right)& 1=e& 5=(\; 123\; )\end{array}$

The steady multilateral system

$\left({V}^{6},{\Re}_{}^{6}\right)=\left({V}_{1}^{6}+\cdots +{V}_{6}^{6},\left\{{R}_{1}^{6},\cdots ,{R}_{6}^{6}\right\}\right)$

is called the *Telluric effluvium* model, denoted simply by
${V}^{6}=\left\{e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)\right\}$, if it satisfies the following conditions:

$\begin{array}{l}{R}_{r}^{6}={\displaystyle \underset{i=1}{\overset{6}{\sum}}{V}_{i}^{6}}\times {V}_{i*r}^{6},\forall r\in \left\{1,2,\cdots ,6\right\},\\ {R}_{i}^{6}*{R}_{j}^{6}={R}_{i*j}^{6},\forall i,j\in \left\{1,2,\cdots ,6\right\}.\end{array}$

The number 1 or 2 or 3 is called the tengen (天元), the earth material (地元), the people ability (人元), corresponding to heaven (1), earth (2) and people (3), respectively. The set of {1, 2, 3} is called three types of talent or material group. The three types heaven (1), earth (2) and people (3) to any change combine forming the elements of *Telluric effluvium* model as follows:

$\begin{array}{l}e=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\end{array}\right),\\ \left(12\right)=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{earth}\left(\text{2}\right)& \text{heaven}\left(\text{1}\right)& \text{people}\left(\text{3}\right)\end{array}\right),\\ \left(13\right)=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{people}\left(\text{3}\right)& \text{earth}\left(\text{2}\right)& \text{heaven}\left(\text{1}\right)\end{array}\right),\\ \left(23\right)=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{heaven}\left(\text{1}\right)& \text{people(3)}& \text{earth}\left(\text{2}\right)\end{array}\right),\\ \left(123\right)=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)& \text{heaven}\left(\text{1}\right)\end{array}\right),\\ \left(132\right)=\left(\begin{array}{ccc}\text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)& \text{people}\left(\text{3}\right)\\ \text{people}\left(\text{3}\right)& \text{heaven}\left(\text{1}\right)& \text{earth}\left(\text{2}\right)\end{array}\right).\end{array}$

It is with elements, $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$. The each of elements, $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$, is called the primordial energy (元气), essence derived from food (谷气), defensive energy (卫气), essential substance circulating in the Meridians and blood Meridians (营气), genuine energy (真气), pectoral energy (宗气), respectively.

Another name of the elements, $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$, is respectively

$\begin{array}{l}\text{shaoyang}\left(e\right)\left(\u5c11\u9633\right),\text{yangming}\left(\left(12\right)\right)\left(\u9633\u660e\right),\text{taiyang}\left(\left(13\right)\right)\left(\u592a\u9633\right),\\ \text{jueyin}\left(\left(23\right)\right)\left(\u53a5\u9634\right),\text{shaoyin}\left(\left(123\right)\right)\left(\u5c11\u9634\right),\text{taiyin}\left(\left(132\right)\right)\left(\u592a\u9634\right).\end{array}$

Generally positive or Yang material, they are able to be perceived, but few can see the material itself, can only use signs. Therefore, the Yang energy symptoms of the set ${M}_{1}=\left\{e,\left(12\right),\left(13\right)\right\}$ is call the marrow energy (髓); The Yin energy of the set ${M}_{2}=\left\{\left(123\right)\right\}$ is call the blood energy (血); The Yin energy of the set ${M}_{3}=\left\{\left(132\right)\right\}$ is call the saliva energy (津); The Yin energy of the set ${M}_{4}=\left\{\left(23\right)\right\}$ is call the essence of water and grain (水谷精微).

Growth and conveyance in the six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ known as the six roots (根); As the fruit of these six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ known as the six fruits (结); Storage of these four energies ${M}_{1}^{},{M}_{2}^{},{M}_{3}^{},{M}_{4}^{}$ known as the four seas (四海); Energy exchange of the four kinds of ${M}_{1}^{},{M}_{2}^{},{M}_{3}^{},{M}_{4}^{}$ known as the four streets (四街). Of course, for a healthy mathematical complex system, the transfer law of each of the six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ is from its root (root-causes) (根) to its fruit (symptoms) (结).

(3) Suppose that the following set

$\begin{array}{c}{G}_{}^{8}={G}_{0}^{2}\times {G}_{0}^{2}\times {G}_{0}^{2}\\ =\{1=\left(1,1,1\right),2=\left(0,1,1\right),3=\left(1,0,1\right),4=\left(0,0,1\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5=\left(1,1,0\right),6=\left(0,1,0\right),7=\left(1,0,0\right),8=\left(0,0,0\right)\}\end{array}$

is a group of order 8, where ${g}_{0}=8=\left(000\right)$ is identity. Its multiplication table is as follows:

The three *Tao* model can combine forming a steady multilateral system

$\left({V}^{8},{\Re}_{}^{8}\right)=\left({V}_{1}^{8}+\cdots +{V}_{8}^{8},\left\{{R}_{1}^{8},\cdots ,{R}_{8}^{8}\right\}\right)$

is called the Eight-Hexagram (八卦) model, denoted simply by

$\begin{array}{l}{V}^{8}=\{1=\left(1,1,1\right),2=\left(0,1,1\right),3=\left(1,0,1\right),4=\left(0,0,1\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5=\left(1,1,0\right),6=\left(0,1,0\right),7=\left(1,0,0\right),8=\left(0,0,0\right)\}\end{array}$

which satisfies the following conditions:

${R}_{r}^{8}={\displaystyle \underset{i=1}{\overset{8}{\sum}}{V}_{i}^{8}}\times {V}_{i*r}^{8},\forall r\in \left\{1,2,\cdots ,8\right\},{R}_{i}^{8}*{R}_{j}^{8}={R}_{i*j}^{8},\forall {R}_{i}^{8},{R}_{j}^{8}\in {\Re}_{}^{8}.$

The number $1,2,3,4,5,6,7,8$ is called the Qian (乾), Dui (兑), Li (离), Zhen (震), Xun (巽), Kan (坎), Gen (艮), Kun (坤) respectively. The set of $\left\{1,2,3,4,5,6,7,8\right\}$ is called the Eight-Hexagram (八卦) system.

(4) Zhang & Zhang (2019) Suppose that the following set

$\begin{array}{l}{G}_{}^{10}={G}_{0}^{2}\times {G}_{0}^{5}\text{\hspace{0.05em}}\text{\hspace{0.05em}}=\{1=\left(1,0\right),2=\left(0,0\right),3=\left(1,1\right),4=\left(0,1\right),5=\left(1,2\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6=\left(0,2\right),7=\left(1,3\right),8=\left(0,3\right),9=\left(1,4\right),10=\left(0,4\right)\}\end{array}$

is a group of order 8, where ${g}_{0}=2=\left(0,0\right)$ is identity. Its multiplication table is as follows:

Assume the Yin Yang Wu Xing model
${V}_{}^{5}$ is implemented by the *Tao *force of the *Tao *model
${V}_{}^{2}$. Then the steady multilateral system
${V}_{}^{2}\times {V}_{}^{5}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{5}\right\}$ is called the Zangxiang or Five Zang-organs model of the steady multilateral system.

*The Zangxiang model satisfies as follows*:

$\begin{array}{l}{R}_{\left(r,{r}^{\prime}\right)}^{\left(2,5\right)}={\displaystyle \underset{\left(i,{i}^{\prime}\right)=\left(0,0\right)}{\overset{\left(1,4\right)}{\sum}}{V}_{\left(i,{i}^{\prime}\right)}^{\left(2,5\right)}}\times {V}_{\left(mod\left(i+r,2\right),mod\left({i}^{\prime}+{r}^{\prime},5\right)\right)}^{\left(2,5\right)},\forall \left(r,{r}^{\prime}\right)\in {V}^{2}\times {V}^{5},\\ {R}_{\left(i,{i}^{\prime}\right)}^{\left(2,5\right)}*{R}_{\left(j,{j}^{\prime}\right)}^{\left(2,5\right)}={R}_{\left(mod\left(i+j,2\right),mod\left({i}^{\prime}+{j}^{\prime},5\right)\right)}^{\left(2,5\right)},\forall \left(i,{i}^{\prime}\right),\left(j,{j}^{\prime}\right)\in {V}^{2}\times {V}^{5}.\end{array}$

*Each of the elements*,

$\begin{array}{l}1\left(1,0\right),2\left(0,0\right),3\left(1,1\right),4\left(0,1\right),5\left(1,2\right),\\ 6\left(0,2\right),7\left(1,3\right),8\left(0,3\right),9\left(1,4\right),10\left(0,4\right),\end{array}$

*is called respectively *

$\begin{array}{l}1Jia\left(1,0\right),2Yi\left(0,0\right),3Bing\left(1,1\right),4Ding\left(0,1\right),5Wu\left(1,2\right),\\ 6Ji\left(0,2\right),7Geng\left(1,3\right),8Xin\left(0,3\right),9\mathrm{Re}n\left(1,4\right),10Gui\left(0,4\right),\end{array}$

*corresponding to the Chinese words respectively*:

$\begin{array}{l}\text{1}\u7532\left(1,0\right),\text{2}\u4e59\left(0,0\right),\text{3}\u4e19\left(1,1\right),\text{4}\u4e01\left(0,1\right),\text{5}\u620a\left(1,2\right),\\ \text{6}\u5df1\left(0,2\right),\text{7}\u5e9a\left(1,3\right),\text{8}\u8f9b\left(0,3\right),\text{9}\u58ec\left(1,4\right),\text{10}\u7678\left(0,4\right),\end{array}$

*also corresponding to the notations in Theorem* 3.1 *respectively*:

$\begin{array}{l}1{X}_{}^{+}\left(1,0\right),\text{2}{X}_{}^{-}\left(0,0\right),\text{3}{X}_{S}^{+}\left(1,1\right),4{X}_{S}^{-}\left(0,1\right),5{X}_{K}^{+}\left(1,2\right)\text{,}\\ 6{X}_{K}^{-}\left(0,2\right),\text{7}{K}_{X}^{+}\left(\text{1,3}\right),8{K}_{X}^{-}\left(0,3\right),9{S}_{X}^{+}\left(1,4\right),10{S}_{X}^{-}\left(0,4\right)\text{.}\end{array}$

*Here*,*each of the elements*:*
$X,{X}_{S}^{},{X}_{K}^{},{K}_{X}^{},{S}_{X}^{}$ * *is called wood*,*fire*, *earth*, *metal*, *water*,*respectively*,*and this is
${*}^{+}$ Yang*,*
${*}^{-}$ is Yin*.

In TCMath, another name of the Zangxiang system is the ten Heavenly Stems (天干). It means in the changes of energy transmission, plays a major role, is under the power of positive (Yang), although it is invisible, the reason of behind the sign.

(5) Zhang & Zhang (2019) Suppose that the following set

${G}_{}^{12}={G}_{0}^{2}\times {G}_{}^{6}=\left\{\begin{array}{cccccc}\begin{array}{l}1=\\ \left(0,e\right)\end{array}& \begin{array}{l}12=\\ \left(1,e\right)\end{array}& \begin{array}{l}5=\\ \left(0,\left(12\right)\right)\end{array}& \begin{array}{l}4=\\ \left(1,\left(12\right)\right)\end{array}& \begin{array}{l}9=\\ \left(0,\left(13\right)\right)\end{array}& \begin{array}{l}8=\\ \left(1,\left(13\right)\right)\end{array}\\ \begin{array}{l}2=\\ \left(0,\left(23\right)\right)\end{array}& \begin{array}{l}11=\\ \left(1,\left(23\right)\right)\end{array}& \begin{array}{l}6=\\ \left(0,\left(132\right)\right)\end{array}& \begin{array}{l}3=\\ \left(1,\left(132\right)\right)\end{array}& \begin{array}{l}10=\\ \left(0,\left(123\right)\right)\end{array}& \begin{array}{l}7=\\ \left(1,\left(123\right)\right)\end{array}\end{array}\right\}$

is a group of order 12, where ${g}_{0}=1=\left(0,e\right)$ is identity. Its multiplication table is as follows:

Assume the Telluric effluvium model
${V}_{}^{6}$ is implemented by the *Tao *force of the *Tao *model
${V}_{}^{2}$. Then the steady multilateral system
${V}_{}^{2}\times {V}_{}^{6}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{6}\right\}$ is called the Jingluo or the Six Fu-organs model of the steady multilateral system.

The Jingluo model satisfies as follows:

$\begin{array}{l}{R}_{\left(r,{r}^{\prime}\right)}^{\left(2,6\right)}={\displaystyle \underset{\left(i,{i}^{\prime}\right)=\left(0,1\right)}{\overset{\left(1,6\right)}{\sum}}{V}_{\left(i,{i}^{\prime}\right)}^{\left(2,6\right)}}\times {V}_{\left(mod\left(i+r,2\right),{i}^{\prime}\text{*}{r}^{\prime}\right)}^{\left(2,6\right)},\forall \left(r,{r}^{\prime}\right)\in {V}^{2}\times {V}^{6},\\ {R}_{\left(i,{i}^{\prime}\right)}^{\left(2,6\right)}*{R}_{\left(j,{j}^{\prime}\right)}^{\left(2,6\right)}={R}_{\left(mod\left(i+j,2\right),{i}^{\prime}*{j}^{\prime}\right)}^{\left(2,6\right)}\forall \left(i,{i}^{\prime}\right),\left(j,{j}^{\prime}\right)\in {V}^{2}\times {V}^{6}.\end{array}$

Each of the elements,

$\begin{array}{cccccc}1\left(0,e\right)& 12\left(1,e\right)& 5\left(0,\left(12\right)\right)& 4\left(1,\left(12\right)\right)& 9\left(0,\left(13\right)\right)& 8\left(1,\left(13\right)\right)\\ 2\left(0,\left(23\right)\right)& 11\left(1,\left(23\right)\right)& 6\left(0,\left(132\right)\right)& 3\left(1,\left(132\right)\right)& 10\left(0,\left(123\right)\right)& 7\left(1,\left(123\right)\right)\end{array}$

is called respectively

$\begin{array}{cccccc}1\text{Zi}\left(0,e\right)& 12\text{Hai}\left(1,e\right)& 5\text{Chen}\left(0,\left(12\right)\right)& 4\text{Mao}\left(1,\left(12\right)\right)& 9\text{Shen}\left(0,\left(13\right)\right)& 8\text{Wei}\left(1,\left(13\right)\right)\\ 2\text{Chou}\left(0,\left(23\right)\right)& 11\text{Xu}\left(1,\left(23\right)\right)& 6\text{Si}\left(0,\left(132\right)\right)& 3\text{Yin}\left(1,\left(132\right)\right)& 10\text{You}\left(0,\left(123\right)\right)& 7\text{Wu}\left(1,\left(123\right)\right)\end{array}$

Corresponding to the Chinese words respectively:

$\begin{array}{cccccc}1\u5b50\left(0,e\right)& 12\u4ea5\left(1,e\right)& 5\u8fb0\left(0,\left(12\right)\right)& 4\u536f\left(1,\left(12\right)\right)& 9\u7533\left(0,\left(13\right)\right)& 8\u672a\left(1,\left(13\right)\right)\\ 2\u4e11\left(0,\left(23\right)\right)& 11\u620c\left(1,\left(23\right)\right)& 6\u5df3\left(0,\left(132\right)\right)& 3\u5bc5\left(1,\left(132\right)\right)& 10\u9149\left(0,\left(123\right)\right)& 7\u5348\left(1,\left(123\right)\right)\end{array}$

Also corresponding to the notations in Theorem 3.1 as the five viscera and six entrails respectively:

$\begin{array}{cccccc}1{X}_{}^{+}\left(0,e\right)& 12{X}_{S}^{x+}\left(1,e\right)& 5{X}_{K}^{+}\left(0,\left(12\right)\right)& 4{K}_{X}^{+}\left(1,\left(12\right)\right)& 9{S}_{X}^{+}\left(0,\left(13\right)\right)& 8{X}_{S}^{j+}\left(1,\left(13\right)\right)\\ 2{X}_{}^{-}\left(0,\left(23\right)\right)& 11{X}_{S}^{x-}\left(1,\left(23\right)\right)& 6{X}_{K}^{-}\left(0,\left(132\right)\right)& 3{K}_{X}^{-}\left(1,\left(132\right)\right)& 10{S}_{X}^{-}\left(0,\left(123\right)\right)& 7{X}_{S}^{j-}\left(1,\left(123\right)\right)\end{array}$

Here, each of the elements: $X,{X}_{S}^{x},{X}_{K}^{},{K}_{X}^{},{S}_{X}^{},{X}_{S}^{j}$ is called wood, xiang-fire (相火), earth, metal, water, jun-fire (君火) respectively, and this is ${*}^{+}$ Yang, ${*}^{-}$ is Yin.

In TCM, an other name of the Jingluo system is the twelve Earthly Branches (地支). It means in the changes of energy transmission, plays a secondly role, is under the power of negative (Yin), although it is tangible, a sign of strength can be observed directly.

(6) Zhang et al. (2021a) Suppose that the following set

$\begin{array}{c}{G}_{}^{64}={G}_{}^{8}\times {G}_{}^{8}\\ =\{\left(1,1\right),\left(1,5\right),\left(1,7\right),\left(1,8\right),\left(5,8\right),\left(7,8\right),\left(3,8\right),\left(3,1\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(2,2\right),\left(2,6\right),\left(2,8\right),\left(2,7\right),\left(6,7\right),\left(8,7\right),\left(4,7\right),\left(4,2\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(3,3\right),\left(3,7\right),\left(3,5\right),\left(3,6\right),\left(7,6\right),\left(5,6\right),\left(1,6\right),\left(1,3\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(4,4\right),\left(4,8\right),\left(4,6\right),\left(4,5\right),\left(8,5\right),\left(6,5\right),\left(2,5\right),\left(2,4\right),\end{array}$

$\begin{array}{l}\left(5,5\right),\left(5,1\right),\left(5,3\right),\left(5,4\right),\left(1,4\right),\left(3,4\right),\left(7,4\right),\left(7,5\right),\\ \left(6,6\right),\left(6,2\right),\left(6,4\right),\left(6,3\right),\left(2,3\right),\left(4,3\right),\left(8,3\right),\left(8,6\right),\\ \left(7,7\right),\left(7,3\right),\left(7,1\right),\left(7,2\right),\left(3,2\right),\left(1,2\right),\left(5,2\right),\left(5,7\right),\\ \left(8,8\right),\left(8,4\right),\left(8,2\right),\left(8,1\right),\left(4,1\right),\left(2,1\right),\left(6,1\right),\left(6,8\right)\}\end{array}$

is a group of order 64, where ${g}_{0}=\left(8,8\right)=\left(\left(000\right),\left(000\right)\right)$ is identity. Its multiplication table is with that of the Descartesian product ${G}_{}^{8}\times {G}_{}^{8}$ of the group ${G}_{}^{8}$ in set theory.

Assume that the Eight-Hexagram model ${V}_{}^{8}$ is implemented by the Eight- Hexagram force of the Eight-Hexagram model ${V}_{}^{8}$. Then the steady multilateral system ${V}_{}^{8}\times {V}_{}^{8}=\left\{\left(i,j\right)|i\in {V}_{}^{8},j\in {V}_{}^{8}\right\}$ is called the model of eight palaces or eight veins of the steady multilateral system.

The model satisfies as follows:

$\begin{array}{l}{R}_{\left(r,{r}^{\prime}\right)}^{\left(8,8\right)}={\displaystyle \underset{\left(i,{i}^{\prime}\right)=\left(1,1\right)}{\overset{\left(8,8\right)}{\sum}}{V}_{\left(i,{i}^{\prime}\right)}^{\left(8,8\right)}}\times {V}_{\left(i*r,{i}^{\prime}*{r}^{\prime}\right)}^{\left(8,8\right)},\forall \left(r,{r}^{\prime}\right)\in {V}^{8}\times {V}^{8},\\ {R}_{\left(i,{i}^{\prime}\right)}^{\left(8,8\right)}*{R}_{\left(j,{j}^{\prime}\right)}^{\left(8,8\right)}={R}_{\left(i*j,{i}^{\prime}*{j}^{\prime}\right)}^{\left(8,8\right)},\forall \left(i,{i}^{\prime}\right),\left(j,{j}^{\prime}\right)\in {V}^{8}\times {V}^{8}.\end{array}$

Here, the multiplication operation * is that of the Eight-Hexagram Model.

Each of the elements,

$\begin{array}{l}\left(1,1\right),\left(1,5\right),\left(1,7\right),\left(1,8\right),\left(5,8\right),\left(7,8\right),\left(3,8\right),\left(3,1\right),\\ \left(2,2\right),\left(2,6\right),\left(2,8\right),\left(2,7\right),\left(6,7\right),\left(8,7\right),\left(4,7\right),\left(4,2\right),\\ \left(3,3\right),\left(3,7\right),\left(3,5\right),\left(3,6\right),\left(7,6\right),\left(5,6\right),\left(1,6\right),\left(1,3\right),\\ \left(4,4\right),\left(4,8\right),\left(4,6\right),\left(4,5\right),\left(8,5\right),\left(6,5\right),\left(2,5\right),\left(2,4\right),\end{array}$

$\begin{array}{l}\left(5,5\right),\left(5,1\right),\left(5,3\right),\left(5,4\right),\left(1,4\right),\left(3,4\right),\left(7,4\right),\left(7,5\right),\\ \left(6,6\right),\left(6,2\right),\left(6,4\right),\left(6,3\right),\left(2,3\right),\left(4,3\right),\left(8,3\right),\left(8,6\right),\\ \left(7,7\right),\left(7,3\right),\left(7,1\right),\left(7,2\right),\left(3,2\right),\left(1,2\right),\left(5,2\right),\left(5,7\right),\\ \left(8,8\right),\left(8,4\right),\left(8,2\right),\left(8,1\right),\left(4,1\right),\left(2,1\right),\left(6,1\right),\left(6,8\right)\end{array}$

is called respectively

$\begin{array}{l}Qian\left(1,1\right),Gou\left(1,5\right),Dun\left(1,7\right),Fou\left(1,8\right),Guan\left(5,8\right),Bo\left(7,8\right),\\ Jin\left(3,8\right),You\left(3,1\right);\\ Dui\left(2,2\right),Kun\left(2,6\right),Cui\left(2,8\right),Xian\left(2,7\right),Jian\left(6,7\right),Qian\left(8,7\right),\\ Xiao\left(4,7\right),Mei\left(4,2\right);\end{array}$

$\begin{array}{l}Li\left(3,3\right),Lv\left(3,7\right),Ding\left(3,5\right),Wei\left(3,6\right),Meng\left(7,6\right),Huan\left(5,6\right),\\ Song\left(1,6\right),Ren\left(1,3\right);\\ Zhen\left(4,4\right),Yu\left(4,8\right),Jie\left(4,6\right),Heng\left(4,5\right),Sheng\left(8,5\right),Jing\left(6,5\right),\\ Da\left(2,5\right),Sui\left(2,4\right);\end{array}$

$\begin{array}{l}Xun\left(5,5\right),Xu\left(5,1\right),Jia\left(5,3\right),Yi\left(5,4\right),Wang\left(1,4\right),He\left(3,4\right),\\ Yi\left(7,4\right),Gu\left(7,5\right);\\ Kan\left(6,6\right),Jie\left(6,2\right),Tun\left(6,4\right),Ji\left(6,3\right),Ge\left(2,3\right),Feng\left(4,3\right),\\ Ming\left(8,3\right),Shi\left(8,6\right);\end{array}$

$\begin{array}{l}Gen\left(7,7\right),Bi\left(7,3\right),Xu\left(7,1\right),Sun\left(7,2\right),Gui\left(3,2\right),Lv\left(1,2\right),\\ Fu\left(5,2\right),Jian\left(5,7\right);\\ Kun\left(8,8\right),Fu\left(8,4\right),Lin\left(8,2\right),Tai\left(8,1\right),Zhuang\left(4,1\right),Guai\left(2,1\right),\\ Xu\left(6,1\right),Bi\left(6,8\right);\end{array}$

corresponding to the Chinese words respectively:

乾(1, 1), 姤(1, 5), 遁(1, 7), 否(1, 8), 观(5, 8), 剥(7, 8), 晋(3, 8), 有(3, 1);

兑(2, 2), 困(2, 6), 萃(2, 8), 咸(2, 7), 蹇(6, 7), 谦(8, 7), 小(4, 7), 妹(4, 2);

离(3, 3), 旅(3, 7), 鼎(3, 5), 未(3, 6), 蒙(7, 6), 涣(5, 6), 讼(1, 6), 人(1, 3);

震(4, 4), 豫(4, 8), 解(4, 6), 恒(4, 5), 升(8, 5), 井(6, 5), 大(2, 5), 随(2, 4);

巽(5, 5), 畜(5, 1), 家(5, 3), 益(5, 4), 妄(1, 4), 嗑(3, 4), 颐(7, 4), 蛊(7, 5);

坎(6, 6), 节(6, 2), 屯(6, 4), 既(6, 3), 革(2, 3), 丰(4, 3), 明(8, 3), 师(8, 6);

艮(7, 7), 贲(7, 3), 畜(7, 1), 损(7, 2), 睽(3, 2), 履(1, 2), 孚(5, 2), 渐(5, 7);

坤(8, 8), 复(8, 4), 临(8, 2), 泰(8, 1), 壮(4, 1), 夬(2, 1), 需(6, 1), 比(6, 8).

Also corresponding to the notations in Theorem 3.1 respectively:

$\begin{array}{l}{K}_{X}^{+}\left(1,1\right),{K}_{X}^{+}\left(1,5\right),{K}_{X}^{+}\left(1,7\right),{K}_{X}^{+}\left(1,8\right),{K}_{X}^{+}\left(5,8\right),{K}_{X}^{+}\left(7,8\right),{K}_{X}^{+}\left(3,8\right),{K}_{X}^{+}\left(3,1\right);\\ {K}_{X}^{-}\left(2,2\right),{K}_{X}^{-}\left(2,6\right),{K}_{X}^{-}\left(2,8\right),{K}_{X}^{-}\left(2,7\right),{K}_{X}^{-}\left(6,7\right),{K}_{X}^{-}\left(8,7\right),{K}_{X}^{-}\left(4,7\right),{K}_{X}^{-}\left(4,2\right);\end{array}$

$\begin{array}{l}{X}_{S}^{-}\left(3,3\right),{X}_{S}^{-}\left(3,7\right),{X}_{S}^{-}\left(3,5\right),{X}_{S}^{-}\left(3,6\right),{X}_{S}^{-}\left(7,6\right),{X}_{S}^{-}\left(5,6\right),{X}_{S}^{-}\left(1,6\right),{X}_{S}^{-}\left(1,3\right);\\ {X}^{+}\left(4,4\right),{X}^{+}\left(4,8\right),{X}^{+}\left(4,6\right),{X}^{+}\left(4,5\right),{X}^{+}\left(8,5\right),{X}^{+}\left(6,5\right),{X}^{+}\left(2,5\right),{X}^{+}\left(2,4\right);\end{array}$

$\begin{array}{l}{X}^{-}\left(5,5\right),{X}^{-}\left(5,1\right),{X}^{-}\left(5,3\right),{X}^{-}\left(5,4\right),{X}^{-}\left(1,4\right),{X}^{-}\left(3,4\right),{X}^{-}\left(7,4\right),{X}^{-}\left(7,5\right);\\ {S}_{X}^{+}\left(6,6\right),{S}_{X}^{+}\left(6,2\right),{S}_{X}^{+}\left(6,4\right),{S}_{X}^{+}\left(6,3\right),{S}_{X}^{+}\left(2,3\right),{S}_{X}^{+}\left(4,3\right),{S}_{X}^{+}\left(8,3\right),{S}_{X}^{+}\left(8,6\right);\end{array}$

$\begin{array}{l}{X}_{K}^{+}\left(7,7\right),{X}_{K}^{+}\left(7,3\right),{X}_{K}^{+}\left(7,1\right),{X}_{K}^{+}\left(7,2\right),{X}_{K}^{+}\left(3,2\right),{X}_{K}^{+}\left(1,2\right),{X}_{K}^{+}\left(5,2\right),{X}_{K}^{+}\left(5,7\right);\\ {X}_{K}^{-}\left(8,8\right),{X}_{K}^{-}\left(8,4\right),{X}_{K}^{-}\left(8,2\right),{X}_{K}^{-}\left(8,1\right),{X}_{K}^{-}\left(4,1\right),{X}_{K}^{-}\left(2,1\right),{X}_{K}^{-}\left(6,1\right),{X}_{K}^{-}\left(6,8\right).\end{array}$

Here, each of the elements: $X,{X}_{S}^{},{X}_{K}^{},{K}_{X}^{},{S}_{X}^{}$ is called wood, fire, earth, metal, water, respectively, and this is ${*}^{+}$ Yang, ${*}^{-}$ is Yin.

Each of sets:

${K}_{X}^{+}\left(\text{*,*}\right),{K}_{X}^{-}\left(\text{*,*}\right),{X}_{S}^{-}\left(\text{*,*}\right),{X}^{+}\left(\text{*,*}\right),{X}^{-}\left(\text{*,*}\right),{S}_{X}^{+}\left(\text{*,*}\right),{X}_{K}^{+}\left(\text{*,*}\right),{X}_{K}^{-}(\; *,*\; )$

is called Qian palace (乾宫), Dui palace (兑宫), Li palace (离宫), Zhen palace (震宫), Xun palace (巽宫), Kan palace (坎宫), Gen palace (艮宫), Kun palace (坤宫), respectively.

(7) Zhang et al. (2021b) suppose that the ten heavenly stems model and the twelve earthly branches model are respectively

$\begin{array}{l}{V}_{}^{2}\times {V}_{}^{5}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{5}\right\},\\ {V}_{}^{2}\times {V}_{}^{6}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{6}\right\}.\end{array}$

Then the following model in Table 1 is called the sixty JiaZi model.

In Table 1, each of the elements (1, 0), (0, 0), ..., (0, 4) is that of group

${V}_{}^{2}\times {V}_{}^{5}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{5}\right\}$

as the ten heavenly stems model or Zangxiang Model

In Table 1, each of the elements (0, e), (0, (23)), ..., (1, e) is that of group

${V}_{}^{2}\times {V}_{}^{6}=\left\{\left(i,j\right)|i\in {V}_{}^{2},j\in {V}_{}^{6}\right\},$

as the twelve earthly branches model or Jingluo Model.

In Table 1, each of the elements of the set

${V}_{}^{2}\times {V}_{}^{5}\times {V}^{6}=\left\{\left(i,j,k\right)|i\in {V}_{}^{2},j\in {V}_{}^{5},k\in {V}^{6}\right\}$

Table 1. Sixty JiaZi for name, ten heavenly stems, twelve earthly branches and Yin Yang attribute.

is called the number of Sixty JiaZi. Another way to write the number sixty JiaZi is as follows:

1 = (1, 1, 1) 2 = (0, 2, 2) 3 = (1, 3, 3) 4 = (0, 4, 4)

5 = (1, 5, 5) 6 = (0, 6, 6) 7 = (1, 7, 7) 8 = (0, 8, 8)

9 = (1, 9, 9) 10 = (0, 10, 10) 11 = (1, 1, 11) 12 = (0, 2, 12)

13 = (1, 3, 1) 14 = (0, 4, 2) 15 = (1, 5, 3) 16 = (0, 6, 4)

17 = (1, 7, 5) 18 = (0, 8, 6) 19 = (1, 9, 7) 20 = (0, 10, 8)

21 = (1, 1, 9) 22 = (0, 2, 10) 23 = (1, 3, 11) 24 = (0, 4, 12)

25 = (1, 5, 1) 26 = (0, 6, 2) 27 = (1, 7, 3) 28 = (0, 8, 4)

29 = (1, 9, 5) 30 = (0, 10, 6) 31 = (1, 1, 7) 32 = (0, 2, 8)

33 = (1, 3, 9) 34 = (0, 4, 10) 35 = (1, 5, 11) 36 = (0, 6, 12)

37 = (1, 7, 1) 38 = (0, 8, 2) 39 = (1, 9, 3) 40 = (0, 10, 4)

45 = (1, 5, 9) 46 = (0, 6, 10) 47 = (1, 7, 11) 48 = (0, 8, 12)

49 = (1, 9, 1) 50 = (0, 10, 2) 51 = (1, 1, 3) 52 = (0.2, 4)

53 = (1, 3, 5) 54 = (0, 4, 6) 55 = (1, 5, 7) 56 = (0, 6, 8)

57 = (1, 7, 9) 58 = (0, 8, 10) 59 = (1, 9, 11) 60 = (0, 10, 12).

There exists a one-to-one correspondence between the above two sets. #

3. The Negative Philosophical Answer of Hilbert Sixth Mathematical Conjecture

3.1. Philosophical Meaning of Each of Five Aspects of Non-Authigenic Logic

The following cognitive logic of five aspects philosophical meaning is due to *Taoism*, called cognitive logic of non-authigenic or cognitive logic of automation or cognitive logic of “Wu Xing”, etc., it applies to any a steady multilateral system imitating a mathematical complex cognitive system. The basic idea is classified into five steps, called wood, fire, earth, metal, water. Every step has a principle and three rules. Principle provides basic idea. The rules provide specific practices. Specific practices include how to think, how to do, how to keep continuously developing.

In fact, it is also the logical system of TCP. The wood, fire, earth, metal, water are states of qi or Chi or energy of the system. The idea of obtaining image from classification or classification taking image technology comes from the states of qi or Chi or energy of the system. The logical system of TCP contains the following contents:

No suppose principle (无假设原则), don’t assume that teaching (不言施教), the criteria for comparison came from a third party (双方比较的标准来自于第三方), the non-authigenic thinking rule (不自生思维规则), the resource limited rule (资源有限规则), the fault-tolerant rule (容错规则), and so on, they’re always called the wood state of the Chi or the wood state of the energy of the system. That is equivalent to no single axiom postulates the system. Complex systems cannot be assumed.

The first impression is strongest (先入为主), preconception principle (先入为主原则), the objective consistency rule (客观一致性规则), and the repeatability rule (重复性规则), the default rule (默认规则), and so on, they’re always called the fire state of the Chi or the fire state of the energy of the system. That is equivalent to that there are multiple incompatible systems of axioms. Objective truth can only be recognized by using reproducibility.

The integration coordination combination principle (整体协调化合原则), the searching null composition rule (零成分搜索规则), integration capability (整合能力), the decomposition rule (分解规则), and so on, they’re always called the earth state of the Chi or the earth state of the energy of the system. That is equivalent to how to recognize the whole, adhere to the whole thinking. Zero component search is a necessary measure to identify the whole.

Logic layering principle (逻辑分层原则), Heaven and People as a whole (天人合一), the Global-Local thinking rule (从整体到局部思维), logic downward compatibility (逻辑向下兼容能力), the logic causal cycle rule (逻辑因果圈), or the logic causal chain (逻辑因果链), the seriousness of logic rule (逻辑的严肃性规则), and so on, they’re always called the metal state of the Chi or the metal state of the energy of the system. That is equivalent to how to obtain cognitive logic, adhere to the global to the local thinking. First the whole and then the part is the key technique of eastern thinking.

Automation principle (自动化原则), doing nothing, everything is done (无为而无不为), the intervention reaction rule (干预响应规则), the self-protection rule (自保护规则). the second physiologic system rule (第二生理系统规则), and so on, they’re always called the water state of the Chi or the water state of the energy of the system. That is equivalent to that the highest goal pursued by TCP’s logic is to do nothing without doing anything.

Among the five principles, the basic principles that play an important role as a necessary condition for the existence of the whole are the principle of no hypothesis and the principle of integration and coordination.

Among the five principles, the basic principles that play an important role as sufficient conditions for the existence or the normal running of the whole are the principle of no hypothesis, the principle of integration and coordination and the principle of automation.

Among the five principles, the basic principle that plays a leading role as a necessary and sufficient condition for the existence of the whole is a preconceived principle, which can be divided into two categories. One is Xiang-fire, which performs the normal functions of fire, and the other is Jun-fire, which performs the overall functions of the five elements of leadership.

(a)* Wood* (
$X$ ):* No assumption principle* (无假设原则)

“No suppose principle” has another name: Don’t assume that teaching (不言施教). It means that without a priori axiom hypothesis conditions, only there is cognition, reasoning and analysis on the relationship of a steady multilateral system imitating a mathematical complex system. In the traditional Chinese saying is: not afraid not perceive afraid goods than goods (不怕不识货就怕货比货).

The ideology includes three aspects of content: one is how to think as non-authigenic thinking rule (不自生思维规则). All assumptions and inspection standards must come from the actual problem itself or the third party standard (第三方标准). In other words, the criteria for comparison came from a third party (双方比较的标准来自于第三方). The assumption or standard or criteria is not relevant to the analysis methods or the analysis scholars (忘我能力). Not accord to the observed data and the hypothesized research object itself. It is the core of non-authigenic thinking rule. Using put forward the assumption of non-authigenic thinking rule can guarantee the processing method is a general problem. According to the rule of non-authigenic thinking, any a non-authigenic hypothesis, the principle of no violation, is no assumption.

Even with the assumptions and inspection standards, so they are not able to be determined in the field of Mathematics, both the western Mathematics and the oriental Mathematics cannot define these assumptions and inspection standards in the areas themselves, they must be determined by unrelated third parties to mathematical fields.

Second is how to do as the resource limited rule (资源有限规则). It is recognized that resources are limited, not admit unlimited resources, all the way to solve the problem that must be solved within limited resources (有限能力). Within the scope of the limited resources to solve the problem, and guarantee that all things will be successful. According to the rule of limited resource, any a resource hypothesis, the principle of no violation, is no assumption.

Resource limited rule is due to that one doesn’t have to look at the collection
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects, in which
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ can be classified into*
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\forall {g}_{i}\in {G}_{}^{m}$ *, but to focus on how the cognition of the basic elements of a mathematical complex system exist happen? First of all, don’t think that the ability of mathematical researchers is infinite, infinite resources are not available. Any mathematical complex system just can be understood only to rely on very small limited resources to solve the problem. This is the basic point of non-authigenic logic.

Third is how to keep continuously developing as the fault-tolerant rule (容错规则). It is to allow the people to solve the problem within a certain range to make mistakes, including the assumption error (容错能力). Only permissible maximum human to make mistakes, what to do can continue. According to the rule of fault tolerance, any a fault tolerant hypothesis, the principle of no violation, is no assumption.

The mathematical researchers themselves, to solve the problem of a steady multilateral system imitating a mathematical complex system, also have many insufficient, cannot assume the ability of these people is very high, can make no mistake. It is understood only under the condition of assuming that these people often make mistakes, to try to solve the problem of a mathematical complex system. The basic conclusion of non-authigenic logic, therefore, cannot has too strong dependence with resources and hypothesis.

No assumption rule is the thinking way for the “wood” properties of a steady multilateral system imitating a mathematical complex system, and to consider the beginning problem of the mathematical complex system. It belongs to the Wood ( $X$ ) subsystem of the mathematical complex system since it cognizes the structure of the mathematical complex system which is the beginning observation or birth stage of all things, just like in the Spring of a year. In the initial stages of a mathematical complex system, and in the absence of birth assumptions, the purpose of both the greatest possible to restrict the behavior of the mathematical researchers and achieving the preliminary cognition of the mathematical complex system is to make a mathematical complex system can be able to generate. In order to explain the occurrence state of all things, researchers must be in the absence of cognitive assumptions, with the above ways of thinking, to cognize the structure of the mathematical complex system first. The structure has nothing to do with the observer behavior, and only related to the birth state of the mathematical complex system. Only in this way, can understand the birth cause of a mathematical complex system, get real Wood ( $X$ ) properties of a mathematical complex system, and get the most cognitive goals.

Under the principle of no assumption, the articles can refer to see Zhang (1993, 2007a, 2011a, 2011b, 2012), Zhang & Shao (2012) and Zhang & Zhang (2013). #

(b)* Fire* (
${X}_{S}$ ):* preconception Principle* (先入为主原则

The principle of that “The first impression is strongest” is also namely preconception. The word preconception, spelling for “Xian Ru Wei Zhu” (先入为主) in TCP or TCMath, useful refers to listen or to get the first impression often dominant in researchers mind, later to meet different opinions, is not easy to accept.

This principle contains three aspects: one is how to think as the objective consistency rule (客观一致性规则). The objective consistency (客观一致性), *i.e.*, what researchers observed the conclusions and the objective facts are consistent? Respect for the objective facts is the core of the human mind. Only to respect the objective facts as an idea, can assure to observe it is correct. According to the objective rule of consistency, any an objective method to deal with problems, the principle of no violation, is preconception.

The preconception principle is also due to that the researcher doesn’t have to look at the collection
$V=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ of the research objects, in which
${V}_{}^{n}=\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ can be classified into*
${V}_{}^{m}={V}_{{g}_{0}}+\cdots +{V}_{{g}_{m-1}},\forall {g}_{i}\in {G}_{}^{m}$ *, but to focus on how to accelerate the development of a mathematical complex system to eliminate interference? First of all, don’t think that the researchers observed the conclusions and the objective facts are consistent if researchers do not know the objective facts. But under the condition of objective conclusion assume that researchers know that, researchers can simulate test with researchers observation analysis method to the conclusions and the objective facts are consistent? In many simulation test under the condition of reasonable, can think of researchers’ observation analysis method is reasonable, with the objective consistency. Simulation is not proof, but at least it means that researchers used in the method has certain rationality. In TCP or TCMath, the simulation method is considered to be the key method of a mathematical analysis for determining the reproducibility. This is the basic point of non-authigenic logic.

Second is how to do as the repeatability rule (重复性规则). That is, what the relationship between the observed conclusions, which are obtained by different observers and different reasonable ways, are consistent? Observations of the fact that is objective, regardless of the observer, no matter what observation method, after fully observed, the conclusion there will be no big changes. This is because the objective conclusion is only one. Adhere to the repetitive rule, can guarantee that can do things correctly, because people’s doing things is difficult to know the objective conclusion. According to the rule of repeatability, any repeatability method to deal with problems, the principle of no violation, is preconception.

The repeatability means under the condition of the same work, measured in the same input values in the same direction between the continuous measurements of the output value consistent with each other. Although researchers don’t check that all observed conclusions, obtained by different observers, different reasonable ways and different the objective facts, are consistent if researchers do not know the objective facts, researchers can at least check whether all observed conclusions obtained by both different observers and different ways are consistent or not. Some of them will be not the objective facts if they are not consistent. In other words, always the repeatability can be checked. In order to check the objective consistency, if researchers do not know the objective facts, researchers have to check whether the way researchers deal with data being objective consistency or not. The way is reasonable if researchers check a lot of times by simulation, according to the third party standard. Reproducibility also make the mean smaller difference between the mathematical measuring systems and the mathematical measuring conditions if the reasonable ways include many environmental conditions.

The analysis conclusion of non-authigenic logic is not relevant to the mathematical scholars. That is to say, no matter what mathematical scholars, according to the similar mathematical data, adopted with the reproducibility of statistic in Mathematics methods, analysis of the conclusion must be the similar to each other. This is because, the problem of a mathematical study has nothing to do with mathematical scholars, just is the only objective of law science.

Both the objective consistency and the repeatability are called reproducibility (再现性).

Third is how to keep continuously developing as the default rule (默认规则). It means that the implied terms of the rule before there is not new information are legal and can ensure the normal operation of a mathematical complex system. In other words, always respect the original hypothesis (尊重原假设). Everything is right in the world (存在的就是合理的)! The default rule is the fact that is one of the most reproducibility methods. Respect the default rule, can guarantee the things on the basis of the reproducibility being to keep continuously developing. According to the rule of the default, any default method to deal with problems, the principle of no violation, is preconception.

The default rule should be as complete as possible. In no other intervention conditions, the default rule can make the mathematical complex system running normally. For the complete default rule, researchers should strictly abide by as much as possible. If a mathematical researcher first has obtained the observations of some data of the object of study, non-authigenic logic will think that this data is a true reflection of the research object, the other data should be repeated observations of these data, so with the same type and form (reproducibility). The analysis conclusion of non-authigenic logic, in the data to the form of a limited number, will stabilize. At this time, must ensure no relationship between the analysis conclusion and the number of data of a small amount of increase and decrease.

What the “The first impression is strongest” is the thinking way for the “fire” properties of a steady multilateral system imitating a mathematical complex system, and to consider the controlling-development problems. It belongs to the Fire ( ${X}_{S}$ ) subsystem of the mathematical complex system since it controls the fluctuations of the mathematical complex system under the condition of a lot of unknown disturbances, which is