LINFAN MAO SMARANDACHE MULTI-SPACE THEORY (Partially post-doctoral research for the Chinese Academy of Sciences) HEXIS Phoenix, AZ 2006 Linfan MAO Academy of Mathematics and Systems Chinese Academy of Sciences Beijing 100080, P.R.China [email protected] SMARANDACHE MULTI-SPACE THEORY HEXIS Phoenix, AZ 2006 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.Zeeb Road P.O.Box 1346, Ann Arbor MI 48106-1346, USA Tel:1-800-521-0600(Customer Service) http://wwwlib.umi.com/bod Peer Reviewers: Y.P.Liu, Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China. F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Bei- jing 100080, P.R.China. J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, Beijing 100083, P.R.China. E.L.Wei, Information school, China Renmin University, Beijing 100872, P.R.China. Copyright 2006 by Hexis and Linfan Mao Phoenix United States Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/˜ smarandache/eBooks-otherformats.htm ISBN: 1-931233-14-4 Standard Address Number: 297-5092 Printed in the United States of America Preface A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n 2, which can be used both for discrete or ≥ connected spaces, particularly for geometries and spacetimes in theoretical physics. We are used to the idea that our space has three dimensions: length, breadth and height; with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. In 21th century, the- oretical physicists believe the 11-dimensional M-theory is the best candidate for the Theory of Everything, i.e., a fundamental united theory of all physical phenomena, but the bottleneck is that 21st century mathematics has not yet been discovered. Today, we think the Smarandache multi-space theory is the best candidate for 21st century mathematics and a new revolution for mathematics has come. Although it is important, only a few books can be found in the libraries research about these spaces and their relations with classical mathematics and theoretical physics. The purpose of this book is to survey this theory, also to establish its relation with physics and cosmology. Many results and materials included in this book are now inspired by the Smarandache’s notion. Now we outline the content of this book. Three parts are included in this book altogether. Part one consists of Chapters 1 and 2 except the Section 1.4, may be called Algebraic multi-spaces. In Chapter 1, we introduce various algebraic Smarandache multi-spaces including those such as multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds and get some el- ementary results on these multi-spaces. ii Linfan Mao: Smarandache Multi-Spaces Theory Chapter 2 concentrates on multi-spaces on graphs. In an algebraic view, each multi-space is a directed graph and each graph is also a multi-space. Many new conceptions are introduced in this chapter, such as multi-voltage graphs, graphs in an n-manifold, multi-embeddings in an n-manifold, Cayley graphs of a multi-group and many results in graph theory, combinatorial map theory are generalized. These new conceptions and results enlarge the research fields of combinatorics. A graph phase is introduced and discussed in Section 2.5, which is a generalization of a particle and can be used to construct a model of p-branes in Chapter 6. The second part of this book consists of Chapters 3-5 including the Section 1.4 concentrates on these multi-metric spaces, particularly, these Smarandache ge- ometries. In fact, nearly all geometries, such as Finsler geometry, Riemann geom- etry, Euclid geometry and Lobachevshy-Bolyai-Gauss geometry are particular case of Smarandache geometries. In Chapter 3, we introduce a new kind geometry, i.e., map geometries, which is a generalization of Iseri’s s-manifolds. By applying map geometries with or without boundary, paradoxist geometries, non-geometries, counter-projective geometries and anti-geometries are constructed. The enumeration of map geometries with or without boundary underlying a graph are also gotten in this chapter. Chapter 4 is on planar map geometries. Those fundamental elements such as points, lines, polygons, circles and line bundles are discussed. Since we can investi- gate planar map geometries by means of Euclid plane geometry, some conceptions for general map geometries are extend, which enables us to get some interesting results. Those measures such as angles, curvatures, areas are also discussed. Chapter 5 is a generalization of these planar map geometries. We introduce the conception of pseudo-planes, which can be seen as the limitation case of planar map geometries when the diameter of each face tends to zero but more general than planar map geometries. On a pseudo-plane geometry, a straight line does not always exist again. These pseudo-plane geometries also relate with differential equations and plane integral curves. Conditions for existing singular points of differential equations are gotten. In Section 5.4, we define a kind of even more general spaces called metric pseudo-spaces or bounded metric pseudo-spaces. Applying these bounded or unbounded metric pseudo-spaces, bounded pseudo-plane geometries, pseudo-surface geometries, pseudo-space geometries and pseudo-manifold geometries are defined. Preface iii By choice different smooth function ω, such as ω = a Finsler or a Riemann norm, we immediately get the Finsler geometry or Riemann geometry. For the further research of interested readers, each of the final sections of Chap- ters 1 5 contains a number of open problems and conjectures which can be seen − as additional materials. Part three only consists of Chapter 6 which is concentrated on applications of multi-spaces to theoretical physics. The view that every observation of human beings for cosmos is only a pseudo-face of our space is discussed in the first section by a mathematical manner. A brief introduction to Einstein’s relative theory and M-theory is in Section 6.2 and 6.3.1. By a view of multi-spaces, models for p-branes and cosmos are constructed in Section 6.3. It is very interesting that the multi- space model of cosmos contains the shelf structure as a special case. The later is a fundamental structure in the modern algebraic geometry in recent years. This book is began to write in the July, 2005 when I finished my post-doctor report: On automorphisms of maps, surfaces and Smarandache geometries for the Chinese Academy of Sciences. Many colleagues and friends of mine have given me enthusiastic support and endless helps in preparing this book. Without their help, this book will never appears today. Here I 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 encourage and suggestion, I would do some else works, can not investigate multi-spaces and finish this book. Second, I would like to thank Prof. Feng Tian, Yanpei Liu and Jiyi Yan for them interested in my post-doctor report: On automorphisms of maps, surfaces and Smarandache geometries. Their encourage and warmhearted support advance this book. Thanks are also given to Professor Mingyao Xu, Xiaodong Hu, Yanxun Chang, Han Ren, Rongxia Hao, Weili He and Erling Wei for their kindly helps and often discussing problems in mathematics altogether. 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 AMSS, Beijing February, 2006 Contents Preface ................................................... ......................i Chapter 1 Smarandache Multi-Spaces ....................................1 1.1 sets.............................................. ..........................1 § 1.1.1 sets .............................................. .........................1 1.1.2 Partially ordered sets .............................. .......................4 1.1.3 Neutrosophic sets .................................. .......................6 1.2 Algebraic Structures ............................... .......................8 § 1.2.1 Groups ............................................ ........................8 1.2.2 Rings ............................................. .......................10 1.2.3 Vector spaces ...................................... ......................12 1.3 Algebraic Multi-Spaces ............................. ......................14 § 1.3.1 Algebraic multi-spaces............................. .......................14 1.3.2 Multi-Groups ...................................... ......................22 1.3.3 Multi-Rings ....................................... .......................29 1.3.4 Multi-Vector spaces ................................ ......................35 1.4 Multi-Metric Spaces................................ ......................39 § 1.4.1 Metric spaces...................................... .......................39 1.4.2 Multi-Metric spaces................................ .......................41 1.5 Remarks and Open Problems............................ .................48 § Chapter 2 Multi-Spaces on Graphs ......................................55