This research work will give a new direction for applications of fuzzy set theory particularly in algebraic logics, non-classical logics, fuzzy finite state machines, fuzzy automata, fuzzy languages, cognitive modeling, multiagent decision analysis and mathematical morphology. Introducing (∈,∈ ∨q_k)-fuzzy ideals, (∈_γ,∈_γ ∨q_δ)-fuzzy ideals and (∈_γ,∈_γ ∨q_δ)-fuzzy soft ideals in a new non-associative algebraic structure called Abel-Grassmann’s groupoid, discuss severa...

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 ...

In this book we introduce a new procedure called α-Discounting Method for Multi-Criteria Decision Making (α-D MCDM), which is as an alternative and extension of Saaty’s Analytical Hierarchy Process (AHP). It works for any number of preferences that can be transformed into a system of homogeneous linear equations. A degree of consistency (and implicitly a degree of inconsistency) of a decision-making problem are defined. α-D MCDM is afterwards generalized to a set of pref...

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 a...

In this book, you will find algebraic exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone work. The book can also be used as an extracurricular source, as the reader shall find enclosed important theorems and formulas, standard definitions and notions that ar...

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized pal...

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 ...

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 spe...

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...

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. Likewi...

This work seeks to provide students and teachers with tools for assessing knowledge in all chapters of the current mathematics program. The tests included in this workbook have a complementary character as opposed to the many materials written with the purpose to support all those who prepare for such examinations and they refer to the entire subject matter included in the analytical mathematics syllabus of arithmetic in Romania, algebra and geometry from the lower secon...

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 ...

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 a...

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 typ...

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 neu...

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 g...

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling. Such study is new innovative and certainly lead to several special important applications in the field of science, engineering, techn...

The authors approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors’ aspiration and attraction, as a poet writes lyrics about spring according to his emotions.

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. The authors also give some applications of them in soft computing ...

The author introduces for the first time the concept of plithogeny in philosophy and, as a derivative, the concepts of plithogenic set / logic / probability / statistics in mathematics and engineering – and the degrees of contradiction (dissimilarity) between the attributes’ values that contribute to a more accurate construction of plithogenic aggregation operators and to the plithogenic relationship of inclusion (partial ordering). They resulted from practical needs in ...