Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Uma abordagem para análise consensual de conjuntos fuzzy hesitantes típicos via agregações estendidas e implicações fuzzy com base em ordens admissíveis

The Typical Hesitant Fuzzy Logic (THFL) is founded on the theory of the Typical Hesitant Fuzzy Sets (THFS), which are defined by considering as membership degrees the finite and non-empty subsets of the unit interval, which are called as Typical Hesitant Fuzzy Elements (THFE). In such logical approach, not only a number but also subintervals, in the unitary interval are also THFE-representations, which can be applied in the decision-making process based on multiple criteria involving many specialists (ME-MCDM). In this context, THFL provides the modelling for situations where there exists not only data uncertainty, but also indecision or hesitation among experts about the possible values for preferences regarding collections of objects. In order to reduce the information collapse for comparison and/or ranking of alternatives in the preference relationships, this thesis firstly develops new ideas about THFL’s logical connectives, which are investigated within the scope of three admissible orders. In the set H of all hesitant fuzzy values, consider: (i) the lexicographic orders hH; Lex1i and hH; Lex2i, related to the occurrence of the smallest/largest element in an ascending/descending ordered THFS, respectively; (ii) the relevant class of order hH; Ai, satisfying the injective cardinality property. In particular, properties of negations and aggregations are studied, as t-norms and OWA operators, with special interest in the axiomatic structures defining the implications and preserving their algebraic properties and representability. Thus, these theoretical results are submitted to the ME-MCDM problem, in order to select the better support for multiple software alternatives. As a main contribution, in this thesis, we discuss consensus measures on THFE and present a model that formally builds consensus measures through extended aggregation functions and fuzzy negation, using admissible orders for comparison and further, differentiating an analysis of consistency over preference matrices. The action of automorphisms provides the generation of new conjugate operators, preserving the main consensual properties as proposed in the Beliakov’s research, including unanimity, minimum consensus, maximum dissension, symmetry and invariance for replication. The new CCAI-method is presented, by applying admissible orders and promoting the use of fuzzy consensus measures based on multi-valued fuzzy logics, and, then, this work enables comparisons even between THFE with different cardinalities. These new theoretical results are then applied to another ME-MCDM problem, obtaining CCAI-consensus in a group of experts which consider typical hesitant fuzzy sets to provide classifications for multiple styles of craft beers.Sem bolsaA Lógica Fuzzy Hesitante Típica (LFHT) está fundamentada na teoria dos Conjuntos Fuzzy Hesitantes Típicos (CFHT), os quais consideram como graus de pertinência os subconjuntos finitos e não vazios do intervalo unitário, chamados Elementos Fuzzy Hesitantes Típicos (EFHT). Nessa abordagem lógica, não apenas um número mas também subintervalos no intervalo unitário são também representações para EFHT, e podem ser aplicados no processo de tomada de decisão baseada em múltiplos critérios envolvendo muitos especialistas (TDMC-ME). Neste contexto, a LFHT provê a modelagem de situações onde ocorre não apenas incerteza de dados, mas também indecisão ou hesitação entre especialistas sobre os possíveis valores atribuídos às preferências referentes a coleções de objetos. Visando reduzir o colapso de informações para comparação e ranqueamento de alternativas nas relações de preferência, esta tese, primeiramente, desenvolve novas ideias sobre os conetivos lógicos da LFHT, as quais são investigadas no âmbito de três ordens admissíveis: (i) as ordens lexicográficas denominadas hH; Lex1i e hH; Lex2i, relacionadas a ocorrência do menor/maior elemento em um CFHT ordenado de forma crescente e decrescente, respectivamente; (ii) a classe relevante das ordens hH; i, satisfazendo a propriedade de cardinalidade injectiva. Estudamos propriedades das negações e agregações, como as t-normas e operadores OWA são considerados, com especial interesse nas estruturas axiomáticas que definem as implicações e preservam suas propriedades algébricas e representabilidade. Estes estudos teóricos são aplicados a problemas TDMC-ME, para seleção de suporte a múltiplas alternativas de software. Como principal contribuição, introduzimos uma análise consenso sobre EFHT que formalmente contrói medidas de consenso por meio de funções de agregação estendidas, implicações e negações fuzzy. Usamos ordens admissíveis para comparação e, ainda, fornecendo uma análise de consistência sobre matrizes de preferência. A ação de automorfismos mostra-se oportuna para geração de novos operadores, preservando as principais propriedades consensuais que incluem unanimidade, consenso mínimo, dissensão máxima, simetria e invariância para replicação. O modelo CCAI aplica ordens admissíveis para promover o uso de medidas de consenso fuzzy, viabilizando comparações mesmo entre EFHT com cardinalidades diferentes. E ainda, o CCAI-método é aplicado na análise consensual, via grupo de especialistas que consideram conjuntos fuzzy hesitantes típicos e fornecem classificações para múltiplos estilos de cervejas artesanais