ISBN 978-1-59973-308-1

VOLUME 3, 2014

MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES)

Edited By Linfan MAO

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

BEIJING UNIVERSITY OF CIVIL ENGINEERING AND ARCHITECTURE

September, 2014

Vol.3, 2014 ISBN 978-1-59973-308-1

MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES)

Edited By Linfan MAO

The Madis of Chinese Academy of Sciences and

Beijing University of Civil Engineering and Architecture

September, 2014

Aims and Scope: The Mathematical Combinatorics (International Book Series) (ISBN 978-1-59973-308-1) is a fully refereed international book series, published in USA quar- terly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, math- ematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are:

Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,---, etc.. Smarandache geometries;

Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration;

Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;

Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics.

Generally, papers on mathematics with its applications not including in above topics are also welcome.

It is also available from the below international databases:

Serials Group/Editorial Department of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106, USA Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535, USA Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075 http://www.gale.com Indexing and Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Refer- ativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), Academia. edu, International Statistical Institute (ISI), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by an email directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China

Email: maolinfan@163.com

Price: US$48.00

Editorial Board (3nd)

Editor-in-Chief

Shaofei Du Linfan MAO Cepital Normal el verely, P.R.China ; : Email: dushf@mail.cnu.edu.cn Chinese Academy of Mathematics and System Science, P.R.China Baizhou He and Beijing University of Civil Engineering and Beijing University of Civil Engineering and Architecture, P-.R.China Architecture, P.R.China Email: hebaizhou@bucea.edu.cn

Email: maolinfan@163.com Xiaodong Hu

Chinese Academy of Mathematics and System Science, P.R.China

Email: xdhu@amss.ac.cn

Deputy Editor-in-Chief

Guohua Song Beijing University of Civil Engineering and Yuangiu Huang

Architecture, P.R.China Hunan Normal University, P.R.China

Email: songguohua@bucea.edu.cn Email: hyqq@public.cs.hn.cn H.Iseri

Editors Mansfield University, USA

Email: hiseriQ@mnsfld.edu S.Bhattacharya

: ; ; Xueliang Li Deakin University

Nankai University, P.R.China

Geelong Campus at Waurn Ponds . : Email: lxl@nankai.edu.cn

Australia

Email: Sukanto.Bhattacharya@Deakin.edu.au Guodong Liu Huizhou University Email: Igd@hzu.edu.cn

Said Broumi

Hassan IT University Mohammedia Hay El Baraka Ben M’sik Casablanca W.B.Vasantha Kandasamy B.P.7951 Morocco Indian Institute of Technology, India

. , Email: vasantha@iitm.ac.in Junliang Cai

Beijing Normal University, P-R.China Ion Patrascu Email: caijunliang@bnu.edu.cn Fratii Buzesti National College

Craiova Romania Yanxun Chang

Beijing Jiaotong University, P.R.China Han Ren Email: yxchang@center.njtu.edu.cn East China Normal University, P.R.China

: : Email: hren@math.ecnu.edu.cn Jingan Cui

Beijing University of Civil Engineering and Ovidiu-Ilie Sandru Architecture, P.R.China Politechnica University of Bucharest

Email: cuijingan@bucea.edu.cn Romania

li International Journal of Mathematical Combinatorics

Mingyao Xu Y. Zhang Peking University, P.R.China Email: xumy@math.pku.edu.cn

Department of Computer Science Georgia State University, Atlanta, USA Guiying Yan

Chinese Academy of Mathematics and System

Science, P.R.China

Email: yanguiying@yahoo.com

Famous Words: Too much experience is a dangerous thing.

By Ooscar Wilde, A British dramatist.

Math.Combin. Book Ser. Vol.3(2014), 01-84

Mathematics on Non-Mathematics

— A Combinatorial Contribution

Linfan MAO

(Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China)

E-mail: maolinfan@163.com

Abstract: A classical system of mathematics is homogenous without contradictions. But it is a little ambiguous for modern mathematics, for instance, the Smarandache geometry. Let ¥ be a family of things such as those of particles or organizations. Then, how to hold its global behaviors or true face? Generally, ¥ is not a mathematical system in usual unless a set, ie., a system with contradictions. There are no mathematical subfields applicable. Indeed, the trend of mathematical developing in 20th century shows that a mathemati- cal system is more concise, its conclusion is more extended, but farther to the true face for its abandoned more characters of things. This effect implies an important step should be taken for mathematical development, i.e., turn the way to extending non-mathematics in classical to mathematics, which also be provided with the philosophy. All of us know there always exists a universal connection between things in #. Thus there is an underlying structure, i.e., a vertex-edge labeled graph G for things in ¥. Such a labeled graph G is invariant accompanied with ¥. The main purpose of this paper is to survey how to extend classical mathematical non-systems, such as those of algebraic systems with contradictions, algebraic or differential equations with contradictions, geometries with contradictions, and generally, classical mathematics systems with contradictions to mathematics by the under- lying structure G. All of these discussions show that a non-mathematics in classical is in fact a mathematics underlying a topological structure G, i.e., mathematical combinatorics,

and contribute more to physics and other sciences.

Key Words: Non-mathematics, topological graph, Smarandache system, non-solvable

equation, CC conjecture, mathematical combinatorics.

AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35

§1. Introduction

A thing is complex, and hybrid with other things sometimes. That is why it is difficult to know the true face of all things, included in “Name named is not the eternal Name; the unnamable is the eternally real and naming the origin of all things”, the first chapter of TAO TEH KING {9], a well-known Chinese book written by an ideologist, Lao Zi of China. In fact, all of things with

1Received February 16, 2014, Accepted August 8, 2014.

2 Linfan MAO

universal laws acknowledged come from the six organs of mankind. Thus, the words “existence” and “non-existence” are knowledged by human, which maybe not implies the true existence or not in the universe. Thus the existence or not for a thing is invariant, independent on human knowledge.

The boundedness of human beings brings about a unilateral knowledge for things in the world. Such as those shown in a famous proverb “the blind men with an elephant”. In this proverb, there are six blind men were asked to determine what an elephant looked like by feeling different parts of the elephant’s body. The man touched the elephant’s leg, tail, trunk, ear, belly or tusk respectively claims it’s like a pillar, a rope, a tree branch, a hand fan, a wall or a solid pipe, such as those shown in Fig.1 following. Each of them insisted on his own and not accepted others. They then entered into an endless argument.

All of you are right! A wise man explains to them: why are you telling it differently is because each one of you touched the different part of the elephant. So, actually the elephant has all those features what you all said. Thus, the best result on an elephant for these blind men is

Anelephant = {4 pillars} Ua rope} Ua tree branch} U {2 hand fans} Ua wall} Uta solid pipe}

What is the meaning of this proverb for understanding things in the world? It lies in that the situation of human beings knowing things in the world is analogous to these blind men. Usually, a thing T is identified with its known characters ( or name ) at one time, and this process is advanced gradually by ours. For example, let w11, f12,--- , Un be its known and 1,7 > 1 unknown characters at time t. Then, the thing 7 is understood by

2 (Wim) Ul Uw (1)

k>1

in logic and with an approximation T° = =U {ui} for T at time t. This also answered why

difficult for human beings knowing a thing really,

Mathematics on Non-Mathematics — A Combinatorial Contribution 3

Generally, let © be a finite or infinite set. A rule or a law on a set % is a mapping

ux d---x } — E for some integers n. Then, a mathematical system is a pair (©;R), where a

n R consists those of rules on © by logic providing all these resultants are still in U.

Definition 1.1([28]-[30]) Let (%1;R1), (H2;R2), +--+, (Um; Rm) be m mathematical system,

~ m as m different two by two. A Smarandache multi-system X& is a union (J dX; with rules R = U Ri i=1 i=1

on ¥, denoted by (E:R).

Consequently, the thing T is nothing else but a Smarandache multi-system (1.1). However, these characters 1%, k > 1 are unknown for one at time t. Thus, T » T° is only an approximation for its true face and it will never be ended in this way for knowing T, i.e., “Name named is not the eternal Name”, as Lao Zi said.

But one’s life is limited by its nature. It is nearly impossible to find all characters vz, k > 1 identifying with thing 7. Thus one can only understands a thing T' relatively, namely find invariant characters J on vz, k > 1 independent on artificial frame of references. In fact, this notion is consistent with Erlangen Programme on developing geometry by Klein [10]: given a manifold and a group of transformations of the same, to investigate the configurations belonging to the manifold with regard to such properties as are not altered by the transformations of the group, also the fountainhead of General Relativity of Einstein [2]: any equation describing the law of physics should have the same form in all reference frame, which means that a universal law does not moves with the volition of human beings. Thus, an applicable mathematical theory for a thing T should be an invariant theory acting on by all automorphisms of the artificial frame of reference for thing T.

All of us have known that things are inherently related, not isolated in philosophy, which implies that these is an underlying structure in characters p;, 1 <i<n for a thing T, namely, an inherited topological graph G. Such a graph G should be independent on the volition of human beings. Generally, a labeled graph G for a Smarandache multi-space is introduced following.

Definition 1.2([21]) For any integer m > 1, let (5: ) be a Smarandache multi-system con- sisting of m mathematical systems (X11; 71), (H2;R2), +++, (mj Rm). An inherited topological structure GS] of (5:8) is a topological vertex-edge labeled graph defined following:

V(G([S]) = {X1, Daye Um}, E(G[S]) = {(5,E;)|Xi QE; 40, 1<i4 5 <m} with labeling for integers 1<iAg<m. However, classical combinatorics paid attentions mainly on techniques for catering the need of other sciences, particularly, the computer science and children games by artificially giving

up individual characters on each system (X,7#). For applying more it to other branch sciences initiatively, a good idea is pullback these individual characters on combinatorial objects again,

4 Linfan MAO

ignored by the classical combinatorics, and back to the true face of things, i.e., an interesting

conjecture on mathematics following:

Conjecture 1.3(CC Conjecture, [15],[19]) A mathematics can be reconstructed from or turned

into combinatorization.

Certainly, this conjecture is true in philosophy. So it is in fact a combinatorial notion on

developing mathematical sciences. Thus:

(1) One can combine different branches into a new theory and this process ended until it has been done for all mathematical sciences, for instance, topological groups and Lie groups. (2) One can selects finite combinatorial rulers and axioms to reconstruct or make general-

izations for classical mathematics, for instance, complexes and surfaces.

From its formulated, the CC conjecture brings about a new way for developing mathematics , and it has affected on mathematics more and more. For example, it contributed to groups, rings and modules ([11]-[14]), topology ({23]-[24]), geometry ([16]) and theoretical physics ([17]- [18]), particularly, these 3 monographs [19]-[21] motivated by this notion.

A mathematical non-system is such a system with contradictions. Formally, let & be mathematical rules on a set ©. A pair (U;@) is non-mathematics if there is at least one ruler R € & validated and invalided on © simultaneously. Notice that a multi-system defined in Definition 1.1 is in fact a system with contradictions in the classical view, but it is cooperated with logic by Definition 1.2. Thus, it lights up the hope of transferring a system with contradictions to mathematics, consistent with logic by combinatorial notion.

The main purpose of this paper is to show how to transfer a mathematical non-system, such as those of non-algebra, non-group, non-ring, non-solvable algebraic equations, non-solvable or- dinary differential equations, non-solvable partial differential equations and non-Euclidean ge- ometry, mixed geometry, differential non- Euclidean geometry, ---, etc. classical mathematics systems with contradictions to mathematics underlying a topological structure G, i.e., math- ematical combinatorics. All of these discussions show that a mathematical non-system is a mathematical system inherited a non-trivial topological graph, respect to that of the classical underlying a trivial K, or Kz. Applications of these non-mathematic systems to theoretical physics, such as those of gravitational field, infectious disease control, circulating economical field can be also found in this paper.

All terminologies and notations in this paper are standard. For those not mentioned here, we follow [1] and [19] for algebraic systems, [5] and [6] for algebraic invariant theory, [3] and [32] for differential equations, [4], [8] and [21] for topology and topological graphs and [20], [28]-[31]

for Smarandache systems.

§2. Algebraic Systems

Notice that the graph constructed in Definition 1.2 is in fact on sets 4;, 1 <i < m with relations on their intersections. Such combinatorial invariants are suitable for algebraic systems. All operations o: & x M — PM on a set & considered in this section are closed and single valued, i.e., ao b is uniquely determined in &/, and it is said to be Abelian if aob = boa for

Mathematics on Non-Mathematics — A Combinatorial Contribution 5

Va,beE of. 2.1 Non-Algebraic Systems

An algebraic system is a pair (#;R) holds with aob € & for Va,b EE MW andoe R. A non-algebraic system =(/;R) on an algebraic system (&;R) is AS~!: there maybe exist an operation o € R, elements a,b € A with ao b undetermined.

Similarly to classical algebra, an isomorphism on —(&/;R) is such a mapping on & that for Vo € R, h(aob) = h(a) 0 h(b)

holds for Va,b € &@ providing ao b is defined in =(#;R) and h(a) = h(b) if and only if a = b. Not loss of generality, let o € R be a chosen operation. Then, there exist closed subsets @, i > 1 of &. For instance,

(a)° = {a,aoa,acaoa,::: ,€0G0+:-0a,-+-} — Ee k is a closed subset of & for Va € &. Thus, there exists a decomposition #°, Z>,---, &? of &

such that aob € &° for Va,b € &° for integers 1 <i <n. Define a topological graph G[7(.#/; 0)] following: VGH 2)]) = {0 Bs Gah E(G[-(#;0)]) = {(@°, F}) if AEP AOL Si Aj <n}

with labels

L: #2 EV(G(P;0))) > L(@P) = &, L: (2,2) € E(G[>(a#;0)]) > #2 GEA for integers 1<ifAj <n.

For example, let off = {a,b,c}, off = {a,d,f}, of? = {ede}, 2 = {a,e,f} and ae = {d,e, f}. Calculation shows that #°()@ = {a}, HI) MP = {c}, ML = {a}, de(\ade = 0, delle = {d}, Ala? = {a}, AE(\se = {af (\oe = {eh, As (|e = {d,e} and #P(\#e = {e,f}. Then, the labeled graph G[=(#;0)] is shown in Fig.2.

6 Linfan MAO

Let h : & — & be an isomorphism on -(.#;0). Then Va,b € &%°), h(a) o h(b) = h(a ob) € h(A?) and h(A?) 1) h(A$) = h(AP?() AZ) = 0 if and only if A?) A$ = 0 for integers 1<i#j<n. Whence, if G"[=(#;0)| defined by

V(G"[>(#;0)]) = {h( AP), R(@E), > RCM): E(G"[>(;0)]) = {(h(G?), h( A?) if WAP) h(@) AOL Si AG <n}

with labels

L': RA?) € V(G"[-(#;0)]) > L(h(e@?)) = h(@?), L': (h(x), (7) € E(G"[>(;0)]) > hf?) (| h@)

for integers 1 <i4# gj <

n. Thus h: & — & induces an isomorphism of graph h* : G[>(#;0)] > G"[-(#;0)]. We

therefore get the following result.

Theorem 2.1 A non-algebraic system =(/;0) in type AS~* inherits an invariant G[=(@; 0)| of labeled graph.

Let CHAR) = U Gh(%9)] o€R be a topological graph on 7=(</;R). Theorem 2.1 naturally leads to the conclusion for non- algebraic system —(.7; R) following.

Theorem 2.2 A non-algebraic system —(@;R) in type AS“! inherits an invariant G[>(@; R)] of topological graph.

Similarly, we can also discuss algebraic non-associative systems, algebraic non-Abelian sys- tems and find inherited invariants G[=(./; 0)] of graphs. Usually, we adopt different notations for operations in R, which consists of a multi-system (./;R). For example, R = {+,-} in an algebraic field (R;+,-). If we view the operation + is the same as -, throw out 0-a, a-0 and 1+a, a+1 for Va € Rin R, then (R;+,-) comes to be a non-algebraic system (R;-) with topological graph G[R;-] shown in Fig.3.

R\ {1} aE

R\ {0}

Fig.3

2.2 Non-Groups

A group is an associative system (Y; 0) holds with identity and inverse elements for all elements in Y. Thus, for a,b,c € GY, (aob)oc=ao (boc), dlg € such that lyoa=aolg =a and for Va € Y, Ja~! € AY such that aca~' =1g. A non-group ~(Y; 0) on a group (Y;0) is an

algebraic system in 3 types following:

Mathematics on Non-Mathematics — A Combinatorial Contribution 7

AG_': there maybe exist a1,b1,c1 and ag, be,co €Y such that (a, 0b,) oc, = ay 0 (b1 0c1) but (az 0 bz) 0 cg # az 0 (be 02), also holds with identity 1g and inverse element a~} for all

elements ina € G.

AG: there maybe exist distinct ly, 1y € G such that ajo lg = lgoa, = a, and a2 0 ly = 140 a2 = ag for a, F az EY, also holds with associative and inverse elements at on lg and ly for Va EG.

AG;?: there maybe exist distinct inverse elements a~',a~' for a € GY, also holds with

associative and identity elements.

Notice that (ao a) oa = ao (aoa) always holds with a € Y in an algebraic system. Thus there exists a decomposition 4,%,---,G, of Y such that (Y;;0) is a group for integers 1<i<n for Type AG{’.

Type AG; is true only if lyoly # ly and # lly. Thus ly and 1 are local, not a global identity on Y. Define

Glg)={aEe¥ ifacly=lyoa=a}.

Then Y(ly) 4 Y(1y) if ly 4 14. Denoted by I(Y) the set of all local identities on Y. Then Gilg), ly € I(Y) is a decomposition of Y such that (¥(1g);0°) is a group for Vlg € I(¥Y).

Type AG;’ is true only if there are distinct local identities ly on Y. Denoted by I(Y) the set of all local identities on Y. We can similarly find a decomposition of Y with group (Y(1g);0) holds for Vlg € I(¥) in this type.

Thus, for a non-group —(; 0) of AG] '-AG3 ', we can always find groups (4; 0), (;0),-* , (Gn; 0) for an integer n > 1 with Y¥ = LU Y. Particularly, if (Y; 0) is itself a group, then such i=1 a decomposition is clearly exists by its subgroups.

Define a topological graph G[=(G; °)] following:

V(GI(F;9))) = {A, 4° Gnd EGG; 0)]) ={GiGH) #fAl)G 401 < 1,45 <n}

with labels

L: YE V(GIA(G;0)]) — L(G) =, L: (GG) € E(G(G;0)]) + G [|G for integers 1<i Aj <n.

For example, let @ = (a, 3), @ = (a,7,9), #@ = (8,7), G% = (G,6,0) be 4 free Abelian groups witha #4 6 4 y #6 # O. Calculation shows that (1% = (a), ANS = (), G3(1G%. = (6), ANG = (GB) and @lI|% = (0). Then, the topological graph G[A(¥Y;0)] is shown in Fig.4.

8 Linfan MAO

G, (a) Gp

Ge (6) G3

Fig.4

For an isomorphism g : Y — Y on —~(G;0), it naturally induces a 1-1 mapping g* : V(GIF(Y;°)]) — V(G[A(Y; °)]) such that each g*(Y;) is also a group and g*(%) (19° (G;) #9 if and only if Y(|Y; # 0 for integers 1 < i 4 j <n. Thus g induced an isomorphism g* of graph from G[=(G; 0)] to g*(G[A(G; °)]), which implies a conclusion following.

Theorem 2.3. A non-group —(Y;0) in type AG] '-AG3' inherits an invariant G[-(Y;0)] of topological graph.

Similarly, we can discuss more non-groups with some special properties, such as those of non-Abelian group, non-solvable group, non-nilpotent group and find inherited invariants G|7>(G;0)]. Notice that({19]) any group Y can be decomposed into disjoint classes CH), C(A2),--- ,C(H,) of conjugate subgroups, particularly, disjoint classes Z(a1), Z(a2), +--+ , Z(az) of centralizers with |C(H;)| = |¥ : Ng(Ai)|, |Z(a;)| = |¥ : Zg(a;)|, 1<i<s,1<j <land |C(Ai)| + |C(A2)|+-+-+|CCHs)| = IYI, |2(a1)| + |2(a2)| +++» +|Z(ai)| = |¥|, where Ng (H), Z(a) denote respectively the normalizer of subgroup H and centralizer of element a in group

¢Y. This fact enables one furthermore to construct topological structures of non-groups with

special classes of groups following: Replace a verter GY, by s; (or 1,) isolated vertices labeled with C(H1), C(H2),--- , C(Hs,) (or Z(a,), Z(az), «+ ,Z(ai,)) in G[A(Y;0)] and denoted the resultant by G[A(G;0)].

We then get results following on non-groups with special topological structures by Theorem 2.3.

Theorem 2.4 A non-group —(Y;0) in type AG]'-AG3' inherits an invariant GIG; o)| of

topological graph labeled with conjugate classes of subgroups on its vertices.

Theorem 2.5 A non-group —(Y;0) in type AG]'-AG3' inherits an invariant GIG; o)| of topological graph labeled with Abelian subgroups, particularly, with centralizers of elements in GY

on its vertices.

Particularly, for a group the following is a readily conclusion of Theorems 2.4 and 2.5.

Corollary 2.6 A group (Y;0) inherits an invariant Gg; 0] of topological graph labeled with

ny

conjugate classes of subgroups (or centralizers) on its vertices, with E(G[Y;0]) =

Mathematics on Non-Mathematics — A Combinatorial Contribution 9

2.3 Non-Rings

A ring is an associative algebraic system (R;+,0) on 2 binary operations “+”, “o”, hold with an Abelian group (R; +) and for Vz,y,z € R, ro(y+z) =xoyt+aroz and (x+y)oz =xoz+yoz. Denote the identity by 0,, the inverse of a by —a in (R;+). A non-ring =(R;+,0°) on a ring

Rag 9

(R;+,0) is an algebraic system on operations “+”, “o” in 5 types following:

AR{?: there maybe exist a,b € R such that a+b #6+<a, but hold with the associative in (R;0) and a group (R;+);

AR;!: there maybe exist a1, b1,c1 and dg, be, c2 € R such that (a; 0b)) oc, = a, 0(b1 0¢1), (az 0 bz) 0 co F az O (b2 OC), but holds with an Abelian group (R; +).

AR;!: there maybe exist a1, bi, c1 and ag, bg, co € R such that (a1 +b1)+c1 = ai +(b1+c1), (az + be) + cg 4 a2 + (bo + C2), but holds with (ao b)oc = ao (boc), identity 04 and —a in (R; +) for Va,b,cE R.

ARj;': there maybe exist distinct 04,04 € R such that a+ 0; = 0; +a = a and b+0,.=0..+b=6 fora #b€ R, but holds with the associative in (R;+), (R; 0) and inverse elements —a on 0,, 04. in (R;+) for Va € R.

AR;?!: there maybe exist distinct inverse elements —a,—a for a € R in (R;+), but holds with the associative in (R;+), (R;0o) and identity elements in (R; +).

Notice that (a+a)+a=a+(a+a),a+a=a+a and aoa = ao always hold in non-ring -=(R;+,0°). Whence, for Types AR," and AR 1 there exists a decomposition R1, R2,--- ,Rn of R such that a+ b = b+a and (a0b)oc=a0 (boc) if a,b,c € Ri, ie., each (Ri; +, 0°) is a ring for integers 1 <i <n. A similar discussion for Types AG; *-AG3" in Section 2.2 also shows such a decomposition (R;;+,°), 1 <i <n of subrings exists for Types 3— 5. Define a topological graph G[=(R; +4, 0)] by

V(G[>(R; +,0)]) = {Ri, Ra,--+ Rn}; E(G[-(R; +,°)]) = {(Ri, Rj) if Rif] Ry #01 <4, 45 <n}

with labels

DL: R€ V(G[7(R; +,0)]) = L(R;) = Rj, L: (Ri, R;) € E(G[A(R; +,°)]) > Ri( Rj for integers 1<iAj <n.

Then, such a topological graph G[=(R; +, 0)] is also an invariant under isomorphic actions on -(R;+,0). Thus,

Theorem 2.7 A non-ring =(R;+,0) in types AR; ‘-AR;5' inherits an invariant G[-(R; +4, °)] of topological graph.

Furthermore, we can consider non-associative ring, non-integral domain, non-division ring, skew non-field or non-field, ---, etc. and find inherited invariants G[=(R;+,0)] of graphs. For example, a non-field =(F';+,0) on a field (F;+,0) is an algebraic system on operations “+”,

Oa

o” in 8 types following:

10 Linfan MAO

AF7?: there maybe exist aj, b1,c, and ag, b2, co € F such that (a1 0b) oc, = a1 0 (by 0c1), (az 0b) 0cg 4 ag 0 (b2 C2), but holds with an Abelian group (F; +), identity 1., a~+ fora € F in (F;0).

AF;’: there maybe exist a1, b1,¢1 and ag, be, co € F such that (a; +61)+¢e1 = ai +(bi+c1), (a2 + bg) + cg F ag + (b2 + ce), but holds with an Abelian group (F;0), identity 14, —a for a € F in (F;+).

AF;!: there maybe exist a,b € F such that aob #4 boa, but hold with an Abelian group (F; +), a group (F; 0);

AF;’: there maybe exist a,b € F such that a+b #b-+<a, but hold with a group (F; +), an Abelian group (F; 0);

AF;': there maybe exist distinct 04,0/, € F such that a+ 0, =04+a=aandb+04, = 04. +b=b fora #4 b € F, but holds with the associative, inverse elements —a on 04, 04, in (F;+) for Va € F, an Abelian group (F; 0°);

AF;": there maybe exist distinct 1.,15 € F such that aol, =1l,ca=aand bol} = 1,ob=b fora be F, but holds with the associative, inverse elements a~' on lo, 14, in (F;0) for Va € F, an Abelian group (F; +);

AF? '; there maybe exist distinct inverse elements —a,—a for a € F in (F;+), but holds with the associative, identity elements in (F;+), an Abelian group (F; 0).

AF, ': there maybe exist distinct inverse elements a~', a7! for a € F in (F;0), but holds with the associative, identity elements in (F;0), an Abelian group (F; +).

Similarly, we can show that there exists a decomposition (F;;+,0), 1 <i<n of fields for non-fields =(F;+,°) in Types AF; ‘-AFg* and find an invariant G[-(F;+,0)] of graph.

Theorem 2.8 A non-ring =(F;+,0) in types AF, '-AF* inherits an invariant G[-(F; +, °)] of topological graph.

2.4 Algebraic Combinatorics

All of previous discussions with results in Sections 2.1-2.3 lead to a conclusion alluded in philosophy that a non-algebraic system 7(</;R) constraint with property can be decomposed into algebraic systems with the same constraints, and inherits an invariant G[7=(@;R)| of topological graph labeled with those of algebraic systems, i.e., algebraic combinatorics, which is in accordance with the notion for developing geometry that of Klein’s. Thus, a more applicable approach for developing algebra is including non-algebra to algebra by consider various non-algebraic systems constraint with property, but such a process will never be ended if we do not firstly determine all algebraic systems. Even though, a more feasible approach is by its inverse, i.e., algebraic G-systems following:

Definition 2.9 Let (4;R1),(2%;R2),--+,(GriRn) be algebraic systems. An algebraic G-

system is a topological graph G with labeling L: v € V(G) > Li(v) € {GH,&h,--- , GH} and

L: (u,v) € E(G) > L(u)() Lv) with L(u) (1) L(v) 4 0, denoted by Gla, R], where fH = U & i=1

{=

Mathematics on Non-Mathematics — A Combinatorial Contribution 11

and R = U Ri. i=1 Clearly, if G[.a/, R] is prescribed, these algebraic systems (4; R1), (4; R2), ++: , (Haj Rn) with intersections are determined.

Problem 2.10 Characterize algebraic G-systems G[&@/,R], such as those of G-groups, G-rings, integral G-domain, skew G-fields, G'-fields, ---, etc., or their combination G — {groups, rings}, G — {groups, integral domains}, G — {groups, fields}, G — {rings, fields} ---. Particularly, characterize these G-algebraic systems for complete graphs G = Ko, K3,K4, path P3, P, or circuit C4 of order< 4.

In this perspective, classical algebraic systems are nothing else but mostly algebraic Ky-

systems, also a few algebraic K2-systems. For example, a field (F';+,-) is in fact a Ko-group prescribed by Fig.3.

§3. Algebraic Equations

All equations discussed in this paper are independent, maybe contain one or several unknowns, not an impossible equality in algebra, for instance 2*+¥** = 0.

3.1 Geometry on Non-Solvable Equations

Let (LES), (LES?) be two systems of linear equations following:

r=y r+y=1 =-y rt+ty=4 (LES%) (LES?) v= 2y g-y=l1 v= —2y xr—y=A4

Clearly, the system (LES}) is solvable with x = 0,y = 0 but (LES?) is non-solvable because x+y =1 is contradicts to that of x+y = 4 and so forr—y=1tox—y=A4. Even so, is the system (LES?) meaningless in the world? Similarly, is only the solution x = 0, y = 0 of system (LES}) important to one? Certainly NOT! This view can be readily come into being by all figures on R? of these equations shown in Fig.5. Thus, if we denote by

Ly = {(2,y) € R’la = y} Li ={(2,y) € Plz +y = 1} Lo = {(2,y) € R?|x = —y} =A Ly = {(z,y) € R’lz + y = 4} Ls = {(x,y) € R?|a = 2y} Lk = {(a,y) €R|a—-y=1} | La = {(2,y) € R?|x = —2y} Li, = {(2,y) € R’|x — y = 4}

12 Linfan MAO

(LES?)

Fig.5 the global behavior of (LES}), (LES?) are lines L1 — La, lines Li — L’4 on R? and

Lif )Lof \Ls(\L4 = {(0,0)} but L415) L404 = 0.

Generally, let

fi(t1,2,+++,%n) =0 (ESm) fo(a1, v2, ro )

fim(@1, £2,°° 7 , In) =0

be a system of algebraic equations in Euclidean space R” for integers m,n > 1 with non-empty

point set Sr, C R” such that fi(r1,22,--- ,¢n) = 0 for (a1, 2,---

,In) © Sp, 1 St <m. Clearly, the system (£S,,) is non-solvable or not dependent on

()Sr=0 or #90.

i=1 Conversely, let Y be a geometrical space consisting of m parts G%,,%,--- ,Ym in R”, where, each Y, is determined by a system of algebraic equations f(a, L2,°°° ln) — 0 fa, T2,°°° yn) = 0

fl (x1, 22, yo ,En) =0 Then, the system of equations

Mathematics on Non-Mathematics — A Combinatorial Contribution 13

is non-solvable or not dependent on

m

(\G=0 or 40.

i=1

Thus we obtain the following result.

Theorem 3.1 The geometrical figure of equation system (ES) is a space Y consisting of m parts GY, determined by equation fi(%1,22,--+,%n) =0, 1 <i<m in (ES,,), and is non- solvable if a G, =. Conversely, if a geometrical space Y consisting of m parts,4Y,,G, ++: ,Gm; each of thenias determined by a system of algebraic equations in R”, then all of these equations

m consist a system (ES,,), which is non-solvable or not dependent on (| Y; = or not. i=1

For example, let G be a planar graph with vertices vj, v2, v3, v4 and edges v1 V2, U1 U3, V2U3,

U3U4, U4U1, Shown in Fig.6.

Then, a non-solvable system of equations with figure G on R? consists of

nS.

y=8 (LEs)4 x =12

y=2

3a + 5y = 46.

Thus G is an underlying graph of non-solvable system (LEs). Definition 3.2 Let (ES»,) be a solvable system of m; equations

fl (a1, 22, -°- tre) =0 flan, 22,--- ti) =0

14 Linfan MAO

with a solution space S'r in R” for integers 1<i<m. A topological graph G|ES,] is defined by

V(G|ESm]) = {S ta, 1<i<m};

E(GIESm]) = {(Spa, Spur) if Spa (Span #9, 1 <i 4G <m}

with labels

LD: Spa € V(G[ES;,)]) > L(S pt) = Sra, L: (Spa, Spt) € E(G[ES;,)]|) => Sta) () Siu for integers 1<i4#j<m.

Applying Theorem 3.1, a conclusion following can be readily obtained.

Theorem 3.3 A system (ESm) consisting of equations in (ES, ), 1 <i <m is solvable if and only if GIESm] ~ Km withDASC 1) Spy. Otherwise, non-solvable, i.e., G[ESm] # Km, or i=1

G[ESm] ~ Km but (1) Sym = 9. i=l

Let T: (@1,%2,°++ ,4n) > (a},25,---,2/,) be linear transformation determined by an

invertible matrix [ajj],,...,> Le, ©; = i121 +aj2%2+-+-+Gjn%n, 1 <i < nand let T (Sim) = Sm

for integers 1 <k <™m. Clearly, T: {Spm, 1<i<m}— {Seta 1<i<m} and

Srtal Siu AO if and only if Spa (| Spun #0

for integers 1 <i #7 <_m. Consequently, if T: (ES) <— (‘ESy), then G[ES,,] ~ GES]. Thus T induces an isomorphism T* of graph from G/ES,,] to G[/ES,,], which implies the

following result:

Theorem 3.4 A system (ES,,) of equations f;(Z) = 0,1 <i < m inherits an invariant G[ES |

under the action of invertible linear transformations on R”.

Theorem 3.4 enables one to introduce a definition following for algebraic system (E'S,,,) of equations, which expands the scope of algebraic equations.

Definition 3.5 If G[ES,,] is the topological graph of system (ESi,) consisting of equations in (ESm,) for integers 1 <4 <m, introduced in Definition 3.2, then G[ES;,] is called a G-solution of system (E'S'y,).

Thus, for developing the theory of algebraic equations, a central problem in front of one should be:

Problem 3.6 For an equation system (ES), determine its G-solution G[ESn].

For example, the solvable system (£S,,,) in classical algebra is nothing else but a Ky,- solution with (] Sia A 0, as claimed in Theorem 3.3. The readers are refereed to references

i=l [22] or [26] for more results on non-solvable equations.

Mathematics on Non-Mathematics — A Combinatorial Contribution 15

3.2 Homogenous Equations

A system (E'S,,) is homogenous if each of its equations f;(%0,%1,--:,%n), 1 < i < m is homogenous, i.e., fi(Avo, AX1, eee jAtn) = r f;(z0, 21, ed i)

for a constant , denoted by (hES,,). For such a system, there are always existing a Ky,-

m solution with {x; = 0, 0 <i<n}C [) Sp and each fi(%o,21,--- ,@n) = 0 passes through i=1 O = (0,0,--- ,0) in R”. Clearly, an invertible linear transformation T action on such a K,,- — n+1

solution is also a Ky,-solution. However, there are meaningless for such a K,,-solution in projective space P” because O ¢ P”. Thus, new invariants for such systems under projective transformations (7p, 24,--- ,2},) =

[i] (41) x(n41) (0, %1,"** ,Xn) should be found, where [ajj] ) is invertible. In R’,

(n+1)x(n+41 two lines P(x, y), Q(x, y) are parallel if they are not intersect. But in P?, this parallelism will never appears because the Bézout’s theorem claims that any two curves P(x,y,z), Q(z,y, 2) of degrees m,n without common components intersect precisely in mn points. However, de- noted by I(P,Q) the set of intersections of homogenous polynomials P(%) with Q(%) with E = (Xo, 21,°+: , Xn). The parallelism in R” can be extended to P” following, which enables one

to find invariants on systems homogenous equations.

Definition 3.7 Let P(Z),Q(Z) be two complex homogenous polynomials of degree d with & = (%0,%1,°++,2n). They are said to be parallel, denoted by P || Q ifd > 1 and there are constants a,b,--+,¢ (not all zero) such that for VE € I(P,Q), avo + ba, +--+ + cap, = 0, te, all intersections of P(Z) with Q(Z) appear at a hyperplane on P’C, or d = 1 with all intersections at the infinite x, =0. Otherwise, P(Z) are not parallel to Q(Z), denoted by P | Q.

Definition 3.8 Let P,(Z) = 0, Po(%) =0,--- , Pm(¥) = 0 be homogenous equations in (RES). Define a topological graph G[IhES,,| in P” by

V(G[hESm]) = {Pi (2), Po(Z), +++, Pm(®)}; E(G|RESm]) = {(Pi(®), Pi(®))|Pi WP), 1S t,7 Sm} with a labeling L: Pi(@)—> PZ), (Pi(®), P)(®)) — 1B, Pj), where 1 <i fj <m.

For any system (hE‘S,,) of homogenous equations, G[hE'S,,] is an indeed invariant under the action of invertible linear transformations T on P”. By definition in [6], a covariant C(ag, Z) on homogenous polynomials P(%) is a polynomial function of coefficients ag and variables 7.

We furthermore find a topological invariant on covariants following.

Theorem 3.9 Let (hES,) be a system consisting of covariants C;(ag,%) on homogenous polynomials P;(Z) for integers 1<i<m. Then, the graph G[RES] is a covariant under the action of invertible linear transformations T, i.e., for VC;(az,) € (ESm), there is Cy (ag, Z) € (ESm) with

Cy (az, 7’) = APC; (az, Z)

16 Linfan MAO

holds for integers 1<1i<m, where p is a constant and A is the determinant of T.

Proof Let GT[hES,,| be the topological graph on transformed system T(hES,,) defined in Definition 3.8. We show that the invertible linear transformation TJ naturally induces an isomorphism between graphs G[hES,,] and G7 [hES;,]. In fact, T naturally induces a mapping T* : G[hESm] — GT[hES,,] on P". Clearly, T* : V(G[RESm]) —~ V(GT[hESm]) is 1-1, also onto by definition. In projective space P”, a line is transferred to a line by an invertible linear transformation. Therefore, C7 || C7 in T(ES,,) if and only if C, || Cy in (hRESin), which implies that (C7,C?) € E(GT[ES,,]) if and only if (Cu,C.) € E(G[hES,,]). Thus, G[hES im] ~ GT [RES in] with an isomorphism T* of graph.

Notice that I (C7,C7) = T (I(Cy,Cy)) for V(Cu, Cy) € E(G[RES;,]). Consequently, the induced mapping

T*: V(G[hESm]) > V(G" [RESm]), E(G[hES]) — E(G7 [hESm])

is commutative with that of labeling LD, i.e, T* oD = LoT*. Thus, T* is an isomorphism from topological graph G[hES\,] to G7 [hE Sn].

Particularly, let p = 0, ie., (ES;,) consisting of homogenous polynomials P;(%), P2(Z), - , P(X) in Theorem 3.9. Then we get a result on systems of homogenous equations following.

Corollary 3.10 A system (hES) of homogenous equations f;(Z) = 0,1 <i <m inherits an

invariant G[|RES;,] under the action of invertible linear transformations on P”.

Thus, for homogenous equation systems (hE'S;,), the G-solution in Problem 3.6 should be substituted by G[AES,,]-solution.

§4. Differential Equations

4.1 Non-Solvable Ordinary Differential Equations

For integers m, n > 1, let X=F,(X),1<i<m (DES)

, dX ee be a differential equation system with continuous fF, :R” — R”, X = ere such that F;(0) = 0, particularly, let

X= A,X,--- ,X = AgX,--- , X = Am X (LDES}) be a linear ordinary differential equation system of first order with , 3 é : dx, dx dxn, X= ++ dy)§ = (—, —,--- , (a a »v ) ( dt ’ dt ’ ’ dt ) and g®™ + al) g(m—1) +... ft ally = 0 n 0) (n- 0 a” 4 aXler—D 4...4 ale =0 pes

Mathematics on Non-Mathematics — A Combinatorial Contribution 17

a linear differential equation system of order n with

[k] [A [k]

Qi Ag Ain r1(t) k k k

ee oe eel Gt <td Seas: anh tn (t)

Lae 0<k<m, 1<i,j <n are numbers. Such a system (DES},) or

(LDES}.) (or (LDE®,)) are called non-solvable if there are no function X(t) (or x(t)) hold with (DES!) or (LDES},) (or (LDE®.)) unless constants. For example, the following differential equation system

where, «) = ae all a

— 3% + 2c =0 (

é —5é + 6x =0 (

i — Te + 122 = 0

(HDR ge noes ( & — 94 + 202 = 0 ( @—1lé+30r=0 (

(

&-—7Tz+ 6x =0

is a non-solvable system.

According to theory of ordinary differential equations ([32]), any linear differential equation system (LDES}) of first order in (LDES},) or any differential equation (LDE?) of order n with complex coefficients in (L DE") are solvable with a solution basis Z = { B;(t)| 1 <i<n} such that all general solutions are linear generated by elements in &.

Denoted the solution basis of systems (DES},) or (LDES},) (or (LDE™)) of ordinary dif- ferential equations by 41, F2,--- , Am and define a topological graph G[DES},] or G[LDES}] (or G[LDE?]) in R” by

V(G[DES,,]) = V(G[LDES,,]) = V(G[LDE},]) = {#, Ba,» , Bm}; E(G|DES),]) = E(G[LDES},]) = E(G|LDE®)) = {(Bi,B;) if Bl )B #9, 1<i,7<m} with a labeling L: BB, (Bi, B;) > Bil \B; for 1 <i#j<m.

Let T be a linear transformation on R” determined by an invertible matrix [a,;| Let

nxn’ T: {B, 1<i<m} 3 {BH, 1<i<m}.

It is clear that &; is the solution basis of the ith transformed equation in (DES},) or (LDES},) (or (LDEy,)), and B()\B; A 0 if and only if (| 4; # 0. Thus T naturally induces an isomorphism 7T* of graph with T* o L = Lo T™ on labeling L.

18 Linfan MAO

Theorem 4.1 A system (DES!) or (LDES}) (or (LDE™,)) of ordinary differential equations inherits an invariant G[DES},] or G[LDES}] (or G[LDE™]) under the action of invertible

linear transformations on