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Graduate Textbook in Mathematics LINFAN MAO COMBINATORIAL GEOMETRY WITH APPLICATIONS TO FIELD THEORY Second Edition X A X B + E F G H - C X D X The Education Publisher Inc. 2011 Linfan MAO Academy of Mathematics and Systems Chinese Academy of Sciences Beijing 100190, P.R.China and Beijing Institute of Civil Engineering and Architecture Beijing 100044, P.R.China Email: [email protected] Combinatorial Geometry with Applications to Field Theory Second Edition The Education Publisher Inc. 2011 This book can be ordered from: The Educational Publisher, Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 E-mail: [email protected] Website: www.EduPublisher.com Peer Reviewers: F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Bei- jing 100190, P.R.China J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, Beijing 100083, P.R.China R.X.Hao and W.L.He, Department of Applied Mathematics, Beijing Jiaotong Uni- versity, Beijing 100044, P.R.China Tudor Sireteanu, Dinu Bratosin and Luige Vladareanu, Institute of Solid Mechanics of Romanian Academy, Bucharest, Romania Copyright 2011 by The Education Publisher Inc. and Linfan Mao Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htm ∼ ISBN: 978-1-59973-155-1 Printed in America Preface to the Second Edition Accompanied with humanity into the 21st century, a highlight trend for developing a science is its overlap and hybrid, and harmoniously with other sciences, which enables one to handle complex systems in the WORLD. This is also for developing mathematics. As a powerful tool for dealing with relations among objectives, combinatorics, including combinatorial theory and graphtheory mushroomed in last century. Its related with algebra, probability theory and geometry has made it to an important subject in mathematics and interesting results emerged in large number without metrics. Today, the time is come for applying combinatorial technique to other mathematics and other sciences besides just to find combinatorial behavior for objectives. That is the motivation of this book, i.e., to survey mathematics and fields by combinatorial principle. In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for developing mathematical sciences of 21st century, which means that we can combine different fields into a union one and then determines its behavior quantitatively. It is this notion that urges me to research mathematics and physics by combinatorics, i.e., mathematical combinatorics beginning in 2004 when I was a post-doctor of Chi- nese Academy of Mathematics and System Science. It finally brought about me one self-contained book, the first edition of this book, published by InfoQuest Publisher in 2009. This edition is a revisited edition, also includes the development of a few ii CombinatorialGeometrywithApplicationstoField topics discussed in the first edition. Contents in this edition are outlined following. Chapters 1 and 2 are the fundamental of this book. In Chapter 1, we briefly introduce combinatorial principle with graphs, such as those of multi-sets, Boolean algebra, multi-posets, countable sets, graphs and enumeration techniques, including inclusion-exclusion principle with applications, enumerating mappings, vertex-edge labeled graphs and rooted maps underlying a graph. The final section discusses the combinatorial principle in philosophy and the CC conjecture, also with its implica- tions for mathematics. All of these are useful in following chapters. Chapter 2 is essentially an algebraic combinatorics, i.e., an application of combinatorial principle to algebraic systems, including algebraic systems, multi-systems with diagrams. The algebraic structures, such as those of groups, rings, fields and modules were generalized to a combinatorial one. We also consider actions of multi- groups on finite multi-sets, which extends a few well-known results in classical permutation groups. Some interesting properties of Cayley graphs of finite groups can be also found in this chapter. Chapter3isasurveyoftopologywithSmarandachegeometry. Terminologiesin algebraic topology, such as those of fundamental groups, covering space, simplicial homology group and some important results, for example, the Seifert and Van- Kampen theorem are introduced. For extending application spaces of Seifert and Van-Kampen theorem, a generalized Seifert and Van-Kampen theorem can be also found in here. As a preparing for Smarandache n-manifolds, a popular introduction to Euclidean spaces, differential forms in Rn and the Stokes theorem on simplicial complexes are presented in Section 3.2. In Section 3.3-3.5, these pseudo-Euclidean spaces, Smarandache geometry, mapgeometry, Smarandachemanifoldwithdifferen- tial, principal fiber bundles and geometrical inclusions in pseudo-manifold geometry are seriously discussed. Chapters 4 6 are mainly on combinatorial manifolds motivated by the com- − binatorial principle on topological or smooth manifolds. In Chapter 4, we discuss topological behaviors of combinatorial manifolds with characteristics, such as Eu- clidean spaces and their combinatorial characteristics, topology on combinatorial manifolds, vertex-edge labeled graphs, Euler-Poincaré characteristic, fundamental groups, singular homology groups on combinatorial manifolds or just manifolds and Preface iii regular covering of combinatorial manifold by voltage assignment. Some well-known results in topology, for example, the Mayer-Vietoris theorem on singular homology groups can be found. Chapters 5 and 6 form the main parts of combinatorial differential geometry, which provides the fundamental for applying it to physics and other sciences. Chapter 5 discuss tangent and cotangent vector space, tensor fields and exterior dif- ferentiation on combinatorial manifolds, connections and curvatures on tensors or combinatorial Riemannian manifolds, integrations and the generalization of Stokes’ and Gauss’ theorem, and so on. Chapter 6 contains three parts. The first concen- trates on combinatorial submanifold of smooth combinatorial manifolds with fundamental equations. The second generalizes topological groups to multiple one, for example Lie multi-groups. The third is a combinatorial generalization of principal fiber bundles to combinatorial manifolds by voltage assignment technique, which provides the mathematical fundamental for discussing combinatorial gauge fields in Chapter 8. Chapters7and8introducetheapplicationsofcombinatorialmanifoldstofields. For this objective, variational principle, Lagrange equations and Euler-Lagrange equations in mechanical fields, Einstein’s general relativity with gravitational field, Maxwell field and Abelian or Yang-Mills gauge fields are introduced in Chapter 7. Applying combinatorial geometry discussed in Chapters 4 6, we then generalize − fields to combinatorial fields under the projective principle, i.e., a physics law in a combinatorial field is invariant under a projection on its a field in Chapter 8. Then, we show how to determine equations of combinatorial fields by Lagrange density, to solve equations of combinatorial gravitational fields and how to construct combinatorialgaugebasisandfields. Elementary applicationsofcombinatorialfields to many-body mechanics, cosmology, physical structure, economical or engineering fields can be also found in this chapter. This edition is preparing beginning from July, 2010. All of these materials are valuable for researchers or graduate students in topological graph theory with enumeration, topology, Smarandache geometry, Riemannian geometry, gravitational or quantum fields, many-bodysystem andgloballyquantifying economy. Forpreparing this book, many colleagues and friends of mine have given me enthusiastic support and endless helps. Without their help, this book will never appears today. Here I iv CombinatorialGeometrywithApplicationstoField must mention some of them. On the first, I would like to give my sincerely thanks to Dr.Perze for his encourage and endless help. Without his suggestion, I would do some else works, cannot investigate mathematical combinatorics for years and finish this book. Second, I would like to thank Professors Feng Tian, Yanpei Liu, Mingyao Xu, Fuji Zhang, Jiyi Yan and Wenpeng Zhang for them interested in my research works. Their encourage and warmhearted support advance this book. Thanks are also given to Professors Han Ren, Junliang Cai, Yuanqiu Huang, Rongxia Hao, Deming Li, Wenguang Zai, Goudong Liu, Weili He and Erling Wei for their kindly helps and often discussing problems in mathematics altogether. Partially research results of mine were reported at Chinese Academy of Mathematics & System Sci- ences, Beijing Jiaotong University, Beijing Normal University, East-China Normal University and Hunan Normal University in past years. Some of them were also re- portedatThe 2ndand3rd Conference on Graph Theory and Combinatorics of China in 2006 and 2008, The 3rd and 4th International Conference on Number Theory and Smarandache’s Problems of Northwest of China in 2007 and 2008. My sincerely thanks are also give to these audiences discussing mathematical topics with me in these periods. Of course, I am responsible for the correctness all of these materials presented here. Any suggestions for improving this book and solutions for open problems in this book are welcome. L.F.Mao July, 2011 Contents Preface to the Second Edition...............................................i Chapter 1. Combinatorial Principle with Graphs ........................1 1.1 Multi-sets with operations.................................................2 § 1.1.1 Set...................................................................2 1.1.2 Operation............................................................3 1.1.3 Boolean algebra......................................................5 1.1.4 Multi-Set.............................................................8 1.2 Multi-posets.............................................................11 § 1.2.1 Partially ordered set.................................................11 1.2.2 Multi-Poset .........................................................13 1.3 Countable sets...........................................................15 § 1.3.1 Mapping............................................................15 1.3.2 Countable set.......................................................16 1.4 Graphs .................................................................. 18 § 1.4.1 Graph...............................................................18 1.4.2 Subgraph ........................................................... 21 1.4.3 Labeled graph.......................................................22 1.4.4 Graph family........................................................22 1.4.5 Operation on graphs ................................................ 25 1.5 Enumeration techniques..................................................26 § vi CombinatorialGeometrywithApplicationstoField 1.5.1 Enumeration principle...............................................26 1.5.2 Inclusion-Exclusion principle ........................................ 26 1.5.3 Enumerating mappings..............................................28 1.5.4 Enumerating vertex-edge labeled graphs.............................30 1.5.5 Enumerating rooted maps...........................................34 1.5.6 Automorphism groups identity of trees ..............................36 1.6 Combinatorial principle..................................................37 § 1.6.1 Proposition in lgic...................................................37 1.6.2 Mathematical system................................................39 1.6.3 Combinatorial system...............................................41 1.7 Remarks.................................................................43 § Chapter 2. Algebraic Combinatorics .....................................47 2.1 Algebraic systems........................................................48 § 2.1.1 Algebraic system....................................................48 2.1.2 Associative and commutative law....................................48 2.1.3 Group...............................................................50 2.1.4 Isomorphism of systems.............................................50 2.1.5 Homomorphism theorem ............................................ 51 2.2 Multi-operation systems ................................................. 55 § 2.2.1 Multi-operation system..............................................55 2.2.2 Isomorphism of multi-systems.......................................55 2.2.3 Distribute law.......................................................58 2.2.4 Multi-group and multi-ring..........................................59 2.2.5 Multi-ideal..........................................................61 2.3 Multi-modules...........................................................62 § 2.3.1 Multi-module ....................................................... 62 2.3.2 Finite dimensional multi-module.....................................66 2.4 Action of multi-groups...................................................68 § Contents vii 2.4.1 Construction of permutation multi-group............................68 2.4.2 Action of multi-group...............................................71 2.5 Combinatorial algebraic systems ......................................... 79 § 2.5.1 Algebraic multi-system..............................................79 2.5.2 Diagram of multi-system............................................81 2.5.3 Cayley diagram ..................................................... 85 2.6 Remarks.................................................................89 § Chapter 3. Topology with Smarandache Geometry.....................91 3.1 Algebraic topology.......................................................92 § 3.1.1 Topological space....................................................92 3.1.2 Metric space ........................................................ 95 3.1.3 Fundamental group..................................................96 3.1.4 Seifert and Van-Kampen theorem .................................. 101 3.1.5 Space attached with graphs ........................................ 103 3.1.6 Generalized Seifert-Van Kampen theorem .......................... 106 3.1.7 Covering space.....................................................111 3.1.8 Simplicial homology group ......................................... 115 3.1.9 Surface.............................................................119 3.2 Euclidean geometry.....................................................122 § 3.2.1 Euclidean space....................................................122 3.2.2 Linear mapping....................................................126 3.2.3 Differential calculus on Rn .........................................128 3.2.4 Differential form ................................................... 131 3.2.5 Stokes’ theorem on simplicial complex..............................133 3.3 Smarandache manifolds.................................................135 § 3.3.1 Smarandache geometry.............................................135 3.3.2 Map geometry ..................................................... 138 3.3.3 Pseudo-Euclidean space............................................143 3.3.4 Smarandache manifold.............................................147