MOD Planes: A New Dimension to Modulo Theory

In this book for the first time authors study mod planes using modulo intervals [0, m); 2 ≤ m ≤ ∞. These planes unlike the real plane have only one quadrant so the study is carried out in a compact space but infinite in dimension. We 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 we 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

MOD Natural Neutrosophic Subset Topological Spaces and Kakutani’s Theorem

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

MOD Graphs

Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Atanassov introduced the degree of nonmembership/ falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/ neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood): http://fs.gallup.unm.edu/FlorentinSmarandache.htm Etymology. The words “neutrosophy” and “neutrosophic” were coined/ invented by F. Smarandache in his 1998 book. Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 1980, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity, A in relation to its opposite, Anti-A and that which is not A, Non-A , and that which is neither A nor Anti-A , denoted by Neut-A . Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics. {From: The Free Online Dictionary of Computing, edited by Denis Howe from England. Neutrosophy is an extension of the Dialectics.} Neutrosophic Logic is a general framework for unification of many existing logics, such as fuzzy logic (especially intuitionistic fuzzy logic), paraconsistent logic, intuitionistic logic, etc

New Techniques to Analyse the Prediction of Fuzzy Models

For the first time authors have ventured to study, analyse and investigate the properties of the fuzzy models, the experts opinion and so on. Here the concept of merged Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps are carried out, which are based on merged graphs and merged matrices. This concept is better than the usual combined Fuzzy Cognitive Maps. Further by this new technique we are able to give equal importance to all the experts who work with the problem. Here the new concept of New Average Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps is defined and described. This new tool helps in saving time and economy. Another new tool called Kosko Hamming distance of FCMs and NCMs are defined which measures the closeness or otherwise of the experts

SPECIAL FUZZY MATRICES FOR SOCIAL SCIENTISTS

This book is a continuation of the book, Elementary fuzzy matrix and fuzzy models for socio-scientists by the same authors. This book is a little advanced because we introduce a multi-expert fuzzy and neutrosophic models. It mainly tries to help social scientists to analyze any problem in which they need multi-expert systems with multi-models. To cater to this need, we have introduced new classes of fuzzy and neutrosophic special matrices. The first chapter is essentially spent on introducing the new notion of different types of special fuzzy and neutrosophic matrices, and the simple operations on them which are needed in the working of these multi expert models. In the second chapter, new set of multi expert models are introduced; these special fuzzy models and special fuzzy neutrosophic models that can cater to adopt any number of experts. The working of the model is also explained by illustrative examples. However, these special fuzzy models can also be used by applied mathematicians to study social and psychological problems. These models can also be used by doctors, engineers, scientists and statisticians. The SFCM, SMFCM, SNCM, SMNCM, SFRM, SNRM, SMFRM, SMNRM, SFNCMs, SFNRMs, etc. can give the special hidden pattern for any given special input vector

Complex Valued Graphs for Soft Computing

In this book authors for the first time introduce in a systematic way the notion of complex valued graphs, strong complex valued graphs and complex neutrosophic valued graphs. Several interesting properties are defined, described and developed. Most of the conjectures which are open in case of usual graphs continue to be open problems in case of both complex valued graphs and strong complex valued graphs. We also give some applications of them in soft computing and social networks. At this juncture it is pertinent to keep on record that Dr. Tohru Nitta was the pioneer to use complex valued graphs in neural networks in particular and soft computing in general. However in this book authors define and develop mathematical models using the complex valued directed graphs akin to Fuzzy Cognitive Maps model and Fuzzy Relational Maps models

Neutrosophic Triplet Groups and their Applications to Mathematical Modelling

The innovative notion of neutrosophic triplet groups, introduced by Smarandache and Ali in 2014-2016, happens to yield the anti-element and neutral element once the element is given. It is established that the neutrosophic triplet group collection forms the classical group under product for Zn, for some specific n. However the collection is not even closed under sum. These neutrosophic triplet groups are built using only modulo integers or Cayley tables. Several interesting properties related with them are defined. It is pertinent to record that in Zn, when n is a prime number, we cannot get a neutral element which can contribute to nontrivial neutrosophic triplet groups. Further, all neutral elements in Zn are only nontrivial idempotents. Using neutrosophic triplet groups authors have defined the notion of neutrosophic triplet group matrices

MOD Pseudo Linear Algebras

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 we 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. These function only under the natural product ×n as the usual product × cannot be even defined on these mixed MOD matrices