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The authors define MOD transformations and discuss several interesting features about them. So chapter one of this book serves the purpose of only recalling these properties. Various properties of MOD interval transformation of varying types are discussed. Next the mod planes are introduced mainly to make this book a self contained one. Finally the new notion of higher and mixed multidimensional MOD planes are introduced and their properties analysed. Several problems are discussed some of which are open conjectures....

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In this book authors for the first time elaborately study the notion of MOD vector spaces and MOD pseudo linear algebras. This study is new, innovative and leaves several open conjectures. In the first place as distributive law is not true one can define only MOD pseudo linear algebras. Secondly most of the classical theorems true in case of linear algebras are not true in case of MOD pseudo linear algebras. Finding even eigen values and eigen vectors happens to be a challenging problem. Further the notion of multidimensional MOD pseudo linear algebras are defined using the notion of MOD mixed matrices. Several innovative and interesting results are given in this book. Many open problems are proposed....

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In this book for the first time authors study MOD planes using modulo intervals. These planes unlike the real plane have only one quadrant so the study is carried aut in a compact space but infinite in dimension. The authors have given seven MOD planes viz real MOD planes (MOD real plane) finite complex MOD plane, neutrosophic MOD plane, fuzzy MOD plane, (or MOD fuzzy plane), MOD dual number plane, MOD special dual like number plane and MOD special quasi dual number plane. These MOD planes unlike real plane or complex plane or neutrosophic plane or dual number plane or special dual like number plane or special quasi dual number plane are infinite in numbers. Further for the first time the authors give a plane structure to the fuzzy product set [0, 1) × [0, 1); where 1 is not included; this is defined as the MOD fuzzy plane. Several properties are derived....

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In this book authors answer the question proposed by Florentin Smarandache “Does there exist neutrosophic numbers which are such that they take values differently and behave differently from I; the indeterminate?”. The authors have constructed a class of natural neutrosophic numbers. Further there are more natural neutrosophic numbers in the MOD interval [0, m). This concept is thoroughly analysed in the second chapter. Using all types of MOD planes and MOD intervals, the authors have generated both natural neutrosophic numbers and MOD neutrosophic numbers. Further the MOD intervals and MOD planes have a special type of zero divisors. Such type of zero divisors are termed as special pseudo zero divisors leading to the definition of special pseudo zero divisors and MOD neutrosophic numbers apart from natural (MOD) neutrosophic nilpotents, zero divisors and idempotents. Lots of open problems are suggested in this book. Certainly this paradigm of shift will give a new approach to the notion of neutrosophy....

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In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. These drawbacks becomes a challenging issue. So study in this direction is open. In fact several open conjectures are proposed in this book. However this paradigm of shift will give new dimension to mathematics....

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In this book the authors for the first time study special type of Euclid squares in the real plane, complex plane, neutrosophic plane, dual number plane and their specializations. This study can be visualized as a blend of algebra, geometry and analysis. There are six such planes and they behave distinctly. From the study it is revealed that each type of squares behave in a different way depending on the plane. The authors define several types of algebraic structures on them. Such study is new, innovative and interesting. However for some types of squares; one is not in a position to define product. Further under addition these squares from a group. One of the benefits is addition of point squares to any square of type I and II is an easy translation of the square without affecting the area or length of the squares. There are over 150 graphs which makes the book more understandable....

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Study of MOD planes happens to a very recent one. Authors have studied several properties of MOD real planes. In fact unlike the real plane R × R which is unique MOD real planes are infinite in number. Further three other new types of MOD planes constructed using dual numbers, special dual number like numbers and special quasi dual numbers are introduced. Systematically algebraic structures on MOD planes like, MOD semigroups, MOD groups and MOD rings of different types are defined and studied. Such study is very new and innovative for a large four quadrant planes are made into a small MOD planes. Several distinct features enjoyed by these MOD planes are defined, developed and described in this book....

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In this book authors for the first time introduce the notion of strong neutrosophic graphs. They are very different from the usual graphs and neutrosophic graphs. Using these new structures special subgraph topological spaces are defined. Further special lattice graph of subgraphs of these graphs are defined and described. Several interesting properties using subgraphs of a strong neutrosophic graph are obtained. Several open conjectures are proposed. These new class of strong neutrosophic graphs will certainly find applications in NCMs, NRMs and NREs with appropriate modifications. These new notions are interesting and researchers can find lots of applications where neutrosophic graphs find their applications. Apart from some open conjectures several problems at research level are also suggested for the readers....

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In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical fixed points. A study of this is carried out in this book. Several interesting properties are developed in this regard. The notion of these fixed points find many applications in the mathematical models which are dealt systematically by the authors in the forth coming books. These special type of fixed points or special realized limit cycles are always guaranteed as we use only MOD matrices as operators with its entries from modulo integers. However this sort of results are NP hard problems if we use reals or complex numbers....

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In this book the notion of semigroups under + is constructed using the MOD natural neutrosophic integers or MOD natural neutrosophic-neutrosophic numbers or MOD natural neutrosophic finite complex modulo integer or MOD natural neutrosophic dual number integers or MOD natural neutrosophic special dual like number or MOD natural neutrosophic special quasi dual numbers are analysed in a systematic way. Secondly, the operation of × is defined and under × also these sets form only a semigroup. Several important properties about them are analysed. Several open conjectures are proposed....

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In this book authors for the first time give several types of problems on MOD structures happens to be an interesting field of study as it makes the whole 4 quadrant plane into a single quadrant plane and the infinite line into a half closed open interval. So study in this direction will certainly yield several interesting results. The law of distributivity is not true. Further the MOD function in general do not obey all the laws of integration or differentiation. Likewise MOD polynomials in general do not satisfy the basic properties of polynomials like its roots etc. Thus over all this study is not only innovative and interesting but challenging. So this book which is only full of problems based on MOD structures will be a boon to researchers. Further the MOD series books of the authors will certainly be an appropriate guide to solve these problems....

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In this book authors for the first time construct a MOD Relational Maps model analogous to Fuzzy Relational Maps (FRMs) model or Neutrosophic Relational Maps (NRMs) model using the MOD rectangular or relational matrices. The advantage of using these models is that we are sure to get the MOD fixed point pair or a MOD limit cycle pair after a finite number of iterations. However we as in case of FRMs or NRMs need not at each stage threshold the resultant state vectors. We do updating when the on state which we started becomes zero. This is yet another advantage of using these newly constructed models. Finally the resultant state vector pair can be real MOD values or finite complex or neutrosophic or a dual number or a special quasi dual number or special dual like number....

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In this book authors for the first time develop the notion of MOD natural neutrosophic subset special type of topological spaces using MOD natural neutrosophic dual numbers or MOD natural neutrosophic finite complex number or MOD natural neutrosophic-neutrosophic numbers and so on to build their respective MOD semigroups. Later they extend this concept to MOD interval subset semigroups and MOD interval neutrosophic subset semigroups. Using these MOD interval semigroups and MOD interval natural neutrosophic subset semigroups special type of subset topological spaces are built. Further using these MOD subsets we build MOD interval subset matrix semigroups and MOD interval subset matrix special type of matrix topological spaces. Likewise using MOD interval natural neutrosophic subsets matrices semigroups we can build MOD interval natural neutrosophic matrix subset special type of topological spaces. The final chapter mainly proposes several open conjectures about the validity of the Kakutani’s fixed point theorem for all MOD special type of subset topological spaces....

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In this book authors for the first time introduce the notion of MOD subsets. The S-semigroups enjoy several interesting properties. The notion of MOD universal subset and MOD absorbing subsets are defined and developed. MOD natural neutrosophic subsets forms only a semigroup under ‘+’. In fact the main feature enjoyed by this structure is they have subset idempotents with respect to ‘+’. They are S-semigroups under ‘+’. These MOD natural neutrosophic subsets of all 6 types are only semigroups under ‘x’. These enjoy distinct properties depending on the subset that is used. Further in these cases two types of products can be used one zero dominated product and the other MOD natural neutrosophic zero dominated product. Both the product are different. Finally using these MOD subsets matrices and polynomials are defined and developed. The new notion of MOD subset matrices are defined and these collections also under ‘+’ (or ‘x’) are only semigroups. On similar lines MOD subset polynomials are defined and they also form only a semigroup under ‘+’ (or ‘x’). These new notions are interesting and researchers can find lots of applications whe...

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In this book authors for the first time introduce new mathematical models analogous to FCMs and NCMs. They have constructed 12 types of MOD Cognitive Maps models, such as: MOD Cognitive Maps model, MOD dual number Cognitive Maps model, MOD neutrosophic Cognitive Maps model, MOD finite complex number Cognitive Maps model, MOD special dual like number Cognitive Maps model, MOD special quasi dual number Cognitive Maps model. Apart from this they have defined MOD natural neutrosophic Cognitive Maps model, MOD natural neutrosophic-neutrosophic Cognitive Maps model, MOD natural neutrosophic dual number Cognitive Maps model, MOD natural neutrosophic special dual like number Cognitive Maps model, MOD natural neutrosophic special quasi dual number model and MOD natural neutrosophic finite complex number Cognitive Maps model. These model certainly are more appropriate than the existing ones. Finally they construct MOD interval Cognitive Maps model and describes it. The final chapter suggests a collection of problems for the interested reader....

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In this book authors for the first time introduce study and develop the notion of MOD graphs, MOD directed graphs, MOD finite complex number graphs, MOD neutrosophic graphs, MOD dual number graphs and so on. Likewise MOD directed natural neutrosophic graphs are defined. Further type I, type II and type III. MOD directed graphs and MOD natural neutrosophic graphs are defined and developed. This book has over 185 examples and over 250 figures. The notion of MOD bipartite graphs and MOD natural neutrosophic bipartite graphs are described. This book gives the probable applications of these concepts to MOD mathematical models like MOD Cognitive Maps model and MOD Relational Maps model which have been introduced by the authors. There are open conjectures which can help the researchers in graph theory. Several innovative results are obtained....

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In this book authors introduce the new notion of MOD rectangular planes. The functions on them behave very differently when compared to MOD planes (square). These are different from the usual MOD planes. Algebraic structures on these MOD rectangular planes are defined and developed. These semigroups have special and distinct features. Certainly these finite structures can find applications in several important problems related to automaton theory. The authors also mention about matrix and polynomial MOD rectangular natural neutrosophic semigroups. Several innovative results are obtained....

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In this book authors for the first time introduce the new notion of special subset vertex subgraph of subset vertex graphs. These subset vertex graphs takes the vertex set values from the power set P(X) of any set X. The main speciality of these subset vertex graphs is that once a set of subsets from P(X) is given, the edges of the graph are fixed in a unique way, so for a given collection of subset vertices the graph is always unique. The special subset vertex subgraphs of G are the ones, which have the same number of vertices as that of the subset vertex graph G. This special property enables one to use these subgraphs as fault tolerant graphs for fault tolerant networks. In this book authors define the notion of special subset vertex hyper subgraphs, which are better suited for fault tolerant graphs....

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In this book authors for the first time introduce the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. Theey have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and innovative features of these graphs are defined, described and developed in this book. The set S considered can be real or complex or indeterminate or dual numbers depending on the problem in hand. However by the definitions of the subset vertex graphs of type I and type II the edges or relation between two vertices are always real. Based on the property that in case of subset vertex graphs for a given set of vertices the graph is unique, these graphs are well suited for social networking. This is one of the special features enjoyed by these subset vertex graphs. The applications of these new types of subset vertex graphs are de...

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In this book authors study special type of subset vertex multi subgraphs; these multi subgraphs can be directed or otherwise. Another special feature of these subset vertex multigraphs is that we are aware of the elements in each vertex set and how it affects the structure of both subset vertex multisubgraphs and edge multisubgraphs. It is pertinent to record at this juncture that certain ego centric directed multistar graphs become empty on the removal of one edge, there by theorising the importance, and giving certain postulates how to safely form ego centric multi networks. Given any subset vertex multigraph we are not always guaranteed of getting the special subset vertex multisubgraphs. However, if we have subset vertex multigraphs which has special subset vertex multisubgraphs then we can have corresponding to them multinet works, these multi networks are fault tolerant networks, hence the study of these special type of multisubgraphs has become mandatory. Further in this book we introduce a new notion of super special subset vertex multigraphs which are built using the subset vertex multigraphs, especially when the subset ver...