Mathematics

Research

7 pages, 224 KiB

Open AccessArticle

Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality

by Taechang Byun, Ji Eun Lee, Keun Young Lee and Jin Hee Yoon

Mathematics 2020, 8(4), 571; https://doi.org/10.3390/math8040571 - 11 Apr 2020

Cited by 6 | Viewed by 2426

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to [...] Read more.

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

18 pages, 340 KiB

Open AccessArticle

A Study on Cubic H-Relations in a Topological Universe Viewpoint

by Jeong-Gon Lee, Kul Hur and Xueyou Chen

Mathematics 2020, 8(4), 482; https://doi.org/10.3390/math8040482 - 1 Apr 2020

Cited by 1 | Viewed by 1758

Abstract

We introduce the concrete category

{CRel}_{P} (H)

[resp.

{CRel}_{R} (H)

] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it [...] Read more.

We introduce the concrete category

{CRel}_{P} (H)

[resp.

{CRel}_{R} (H)

] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it is Cartesian closed over

Set

. Next, we introduce the subcategory

{CRel}_{P, R} (H)

[resp.

{CRel}_{R, R} (H)

] of

{CRel}_{P} (H)

[resp.

{CRel}_{R} (H)

] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in

{CRel}_{P, R} (H)

[resp.

{CRel}_{R, R} (H)

] quite different from those in

{CRel}_{P} (H)

[resp.

{CRel}_{R} (H)

]. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

14 pages, 306 KiB

Open AccessArticle

Refined Expected Value Decision Rules under Orthopair Fuzzy Environment

by Yige Xue and Yong Deng

Mathematics 2020, 8(3), 442; https://doi.org/10.3390/math8030442 - 18 Mar 2020

Cited by 15 | Viewed by 3006

Abstract

Refined expected value decision rules can refine the calculation of the expected value and make decisions by estimating the expected values of different alternatives, which use many theories, such as Choquet integral, PM function, measure and so on. However, the refined expected value [...] Read more.

Refined expected value decision rules can refine the calculation of the expected value and make decisions by estimating the expected values of different alternatives, which use many theories, such as Choquet integral, PM function, measure and so on. However, the refined expected value decision rules have not been applied to the orthopair fuzzy environment yet. To address this issue, in this paper we propose the refined expected value decision rules under the orthopair fuzzy environment, which can apply the refined expected value decision rules on the issues of decision making that is described in the orthopair fuzzy environment. Numerical examples were applied to verify the availability and flexibility of the new refined expected value decision rules model. The experimental results demonstrate that the proposed model can apply refined expected value decision rules in the orthopair fuzzy environment and solve the decision making issues with the orthopair fuzzy environment successfully. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

22 pages, 367 KiB

Open AccessArticle

Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

by Vilém Novák

Mathematics 2020, 8(3), 432; https://doi.org/10.3390/math8030432 - 16 Mar 2020

Cited by 5 | Viewed by 3804

Abstract

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic [...] Read more.

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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21 pages, 685 KiB

Open AccessArticle

Generalized Fuzzy Graph Connectivity Parameters with Application to Human Trafficking

by Arya Sebastian, John N Mordeson and Sunil Mathew

Mathematics 2020, 8(3), 424; https://doi.org/10.3390/math8030424 - 16 Mar 2020

Cited by 19 | Viewed by 5792

Abstract

Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been [...] Read more.

Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been studied theoretically on connectivity of normalized networks. Fuzzy graph theory answers to most of the problems in this area. Although the concept of connectivity in fuzzy graphs has been widely studied, one cannot find proper generalizations of connectivity parameters of unweighted graphs. Generalizations for some of the existing vertex and edge connectivity parameters in graphs are attempted in this article. New parameters are compared with the old ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. The existence of super fuzzy graphs with higher connectivity values are established for both old and new parameters. The new edge connectivity values for some wider classes of fuzzy graphs are also obtained. The generalizations bring substantial improvements in fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart from these, a new class of fuzzy graphs called generalized t-connected fuzzy graphs are studied. An algorithm for clustering the vertices of a fuzzy graph and an application related to human trafficking are also proposed. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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14 pages, 781 KiB

Open AccessArticle

The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

by Faisal Mehmood, Fu-Gui Shi, Khizar Hayat and Xiao-Peng Yang

Mathematics 2020, 8(3), 411; https://doi.org/10.3390/math8030411 - 13 Mar 2020

Cited by 16 | Viewed by 3251

Abstract

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we [...] Read more.

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

21 pages, 349 KiB

Open AccessArticle

Hesitant Fuzzy Topological Spaces

by Jeong-Gon Lee and Kul Hur

Mathematics 2020, 8(2), 188; https://doi.org/10.3390/math8020188 - 4 Feb 2020

Cited by 11 | Viewed by 3452

Abstract

In this study, we define a hesitant fuzzy topology and base, obtain some of their properties, respectively, and give some examples. Next, we introduce the concepts of a hesitant fuzzy neighborhood, Q-neighborhood, closure, and interior and obtain some of their properties, respectively. Furthermore, [...] Read more.

In this study, we define a hesitant fuzzy topology and base, obtain some of their properties, respectively, and give some examples. Next, we introduce the concepts of a hesitant fuzzy neighborhood, Q-neighborhood, closure, and interior and obtain some of their properties, respectively. Furthermore, we define a hesitant fuzzy continuous mapping and investigate some of its properties. Furthermore, we define a hesitant fuzzy subspace and obtain some of its properties. In particular, we obtain the Pasting lemma. We investigate the concept of hesitant fuzzy product space and study some of its properties. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

16 pages, 796 KiB

Open AccessArticle

Multipolar Intuitionistic Fuzzy Set with Finite Degree and Its Application in BCK/BCI-Algebras

by Kyung Tae Kang, Seok-Zun Song and Young Bae Jun

Mathematics 2020, 8(2), 177; https://doi.org/10.3390/math8020177 - 2 Feb 2020

Cited by 12 | Viewed by 2427

Abstract

When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and [...] Read more.

When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

14 pages, 782 KiB

Open AccessArticle

Multipolar Fuzzy p-Ideals of BCI-Algebras

by Mohammad Mohseni Takallo, Sun Shin Ahn, Rajab Ali Borzooei and Young Bae Jun

Mathematics 2019, 7(11), 1094; https://doi.org/10.3390/math7111094 - 12 Nov 2019

Cited by 15 | Viewed by 2602

Abstract

The notion of (normal) m-polar

(\in, \in)

-fuzzy p-ideals of BCI-algebras is introduced, and several properties are investigated. Relations between an m-polar

(\in, \in)

-fuzzy ideal and an m-polar [...] Read more.

The notion of (normal) m-polar

(\in, \in)

-fuzzy p-ideals of BCI-algebras is introduced, and several properties are investigated. Relations between an m-polar

(\in, \in)

-fuzzy ideal and an m-polar

(\in, \in)

-fuzzy p-ideal are displayed, and conditions for an m-polar

(\in, \in)

-fuzzy ideal to be an m-polar

(\in, \in)

-fuzzy p-ideal are provided. Characterization of m-polar

(\in, \in)

-fuzzy p-ideals are considered. Given an m-polar

(\in, \in)

-fuzzy ideal (resp., m-polar

(\in, \in)

-fuzzy p-ideal), a normal m-polar

(\in, \in)

-fuzzy ideal (resp., normal m-polar

(\in, \in)

-fuzzy p-ideal) is established. Using an m-polar

(\in, \in)

-fuzzy ideal, the quotient structure of BCI-algebras is constructed. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

14 pages, 790 KiB

Open AccessArticle

Approximation Properties in Felbin Fuzzy Normed Spaces

by Ju Myung Kim and Keun Young Lee

Mathematics 2019, 7(10), 1003; https://doi.org/10.3390/math7101003 - 22 Oct 2019

Cited by 5 | Viewed by 2588

Abstract

In this paper, approximation properties in Felbin fuzzy normed spaces are considered. These approximation properties are new concepts in Felbin fuzzy normed spaces. Definitions and examples of such properties are given and we make a comparative study among approximation properties in Bag and [...] Read more.

In this paper, approximation properties in Felbin fuzzy normed spaces are considered. These approximation properties are new concepts in Felbin fuzzy normed spaces. Definitions and examples of such properties are given and we make a comparative study among approximation properties in Bag and Samanta fuzzy normed spaces and Felbin fuzzy normed spaces. We develop the representation of finite rank bounded operators in our context. By using this representation, characterizations of approximation properties are established in Felbin fuzzy normed spaces. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

36 pages, 1428 KiB

Open AccessArticle

A Soft-Rough Set Based Approach for Handling Contextual Sparsity in Context-Aware Video Recommender Systems

by Syed Manzar Abbas, Khubaib Amjad Alam and Shahaboddin Shamshirband

Mathematics 2019, 7(8), 740; https://doi.org/10.3390/math7080740 - 12 Aug 2019

Cited by 17 | Viewed by 6235

Abstract

Context-aware video recommender systems (CAVRS) seek to improve recommendation performance by incorporating contextual features along with the conventional user-item ratings used by video recommender systems. In addition, the selection of influential and relevant contexts has a significant effect on the performance of CAVRS. [...] Read more.

Context-aware video recommender systems (CAVRS) seek to improve recommendation performance by incorporating contextual features along with the conventional user-item ratings used by video recommender systems. In addition, the selection of influential and relevant contexts has a significant effect on the performance of CAVRS. However, it is not guaranteed that, under the same contextual scenario, all the items are evaluated by users for providing dense contextual ratings. This problem cause contextual sparsity in CAVRS because the influence of each contextual factor in traditional CAVRS assumes the weights of contexts homogeneously for each of the recommendations. Hence, the selection of influencing contexts with minimal conflicts is identified as a potential research challenge. This study aims at resolving the contextual sparsity problem to leverage user interactions at varying contexts with an item in CAVRS. This problem may be investigated by considering a formal approximation of contextual attributes. For the purpose of improving the accuracy of recommendation process, we have proposed a novel contextual information selection process using Soft-Rough Sets. The proposed model will select a minimal set of influencing contexts using a weights assign process by Soft-Rough sets. Moreover, the proposed algorithm has been extensively evaluated using “LDOS-CoMoDa” dataset, and the outcome signifies the accuracy of our approach in handling contextual sparsity by exploiting relevant contextual factors. The proposed model outperforms existing solutions by identifying relevant contexts efficiently based on certainty, strength, and relevancy for effective recommendations. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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16 pages, 283 KiB

Open AccessFeature PaperArticle

A Portfolio Choice Problem in the Framework of Expected Utility Operators

by Irina Georgescu and Louis Aimé Fono

Mathematics 2019, 7(8), 669; https://doi.org/10.3390/math7080669 - 26 Jul 2019

Cited by 4 | Viewed by 2594

Abstract

Possibilistic risk theory starts from the hypothesis that risk is modeled by fuzzy numbers. In particular, in a possibilistic portfolio choice problem, the return of a risky asset will be a fuzzy number. The expected utility operators have been introduced in a previous [...] Read more.

Possibilistic risk theory starts from the hypothesis that risk is modeled by fuzzy numbers. In particular, in a possibilistic portfolio choice problem, the return of a risky asset will be a fuzzy number. The expected utility operators have been introduced in a previous paper to build an abstract theory of possibilistic risk aversion. To each expected utility operator, one can associate the notion of possibilistic expected utility. Using this notion, we will formulate in this very general context a possibilistic portfolio choice problem. The main results of the paper are two approximate calculation formulas for the corresponding optimization problem. The first formula approximates the optimal allocation with respect to risk aversion and investor’s prudence, as well as the first three possibilistic moments. Besides these parameters, in the second formula, the temperance index of the utility function and the fourth possibilistic moment appear. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

14 pages, 265 KiB

Open AccessFeature PaperArticle

Fuzzy Positive Implicative Filters of Hoops Based on Fuzzy Points

by Rajab Ali Borzooei, Mona Aaly Kologani, Mahdi Sabet Kish and Young Bae Jun

Mathematics 2019, 7(6), 566; https://doi.org/10.3390/math7060566 - 24 Jun 2019

Cited by 7 | Viewed by 2638

Abstract

In this paper, we introduce the notions of

(\in, \in)

-fuzzy positive implicative filters and

(\in, \in \lor q)

-fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, [...] Read more.

In this paper, we introduce the notions of

(\in, \in)

-fuzzy positive implicative filters and

(\in, \in \lor q)

-fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, and then we use the congruence relation on hoop defined in blue[Aaly Kologani, M.; Mohseni Takallo, M.; Kim, H.S. Fuzzy filters of hoops based on fuzzy points. Mathematics. 2019, 7, 430; doi:10.3390/math7050430] by using an

(\in, \in)

-fuzzy filter in hoop. We show that the quotient structure of this relation is a Brouwerian semilattice. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

25 pages, 585 KiB

Open AccessArticle

Granulation of Hypernetwork Models under the q-Rung Picture Fuzzy Environment

by Anam Luqman, Muhammad Akram and Ali N. A. Koam

Mathematics 2019, 7(6), 496; https://doi.org/10.3390/math7060496 - 1 Jun 2019

Cited by 21 | Viewed by 3006

Abstract

In this paper, we define q-rung picture fuzzy hypergraphs and illustrate the formation of granular structures using q-rung picture fuzzy hypergraphs and level hypergraphs. Further, we define the q-rung picture fuzzy equivalence relation and q-rung picture fuzzy hierarchical quotient [...] Read more.

In this paper, we define q-rung picture fuzzy hypergraphs and illustrate the formation of granular structures using q-rung picture fuzzy hypergraphs and level hypergraphs. Further, we define the q-rung picture fuzzy equivalence relation and q-rung picture fuzzy hierarchical quotient space structures. In particular, a q-rung picture fuzzy hypergraph and hypergraph combine a set of granules, and a hierarchical structure is formed corresponding to the series of hypergraphs. The mappings between the q-rung picture fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of the universal set is more efficient through q-rung picture fuzzy hypergraphs and the q-rung picture fuzzy equivalence relation. We also present an arithmetic example and comparison analysis to signify the superiority and validity of our proposed model. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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22 pages, 349 KiB

Open AccessArticle

Linguistic Spherical Fuzzy Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems

by Huanhuan Jin, Shahzaib Ashraf, Saleem Abdullah, Muhammad Qiyas, Mahwish Bano and Shouzhen Zeng

Mathematics 2019, 7(5), 413; https://doi.org/10.3390/math7050413 - 8 May 2019

Cited by 99 | Viewed by 4674

Abstract

The key objective of the proposed work in this paper is to introduce a generalized form of linguistic picture fuzzy set, so-called linguistic spherical fuzzy set (LSFS), combining the notion of linguistic fuzzy set and spherical fuzzy set. In LSFS we deal with [...] Read more.

The key objective of the proposed work in this paper is to introduce a generalized form of linguistic picture fuzzy set, so-called linguistic spherical fuzzy set (LSFS), combining the notion of linguistic fuzzy set and spherical fuzzy set. In LSFS we deal with the vague and defective information in decision making. LSFS is characterized by linguistic positive, linguistic neutral and linguistic negative membership degree which satisfies the conditions that the square sum of its linguistic membership degrees is less than or equal to 1. In this paper, we investigate the basic operations of linguistic spherical fuzzy sets and discuss some related results. We extend operational laws of aggregation operators and propose linguistic spherical fuzzy weighted averaging and geometric operators based on spherical fuzzy numbers. Further, the proposed aggregation operators of linguistic spherical fuzzy number are applied to multi-attribute group decision-making problems. To implement the proposed models, we provide some numerical applications of group decision-making problems. In addition, compared with the previous model, we conclude that the proposed technique is more effective and reliable. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

14 pages, 296 KiB

Open AccessArticle

Extended Fuzzy Metrics and Fixed Point Theorems

by Valentín Gregori, Juan-José Miñana and David Miravet

Mathematics 2019, 7(3), 303; https://doi.org/10.3390/math7030303 - 25 Mar 2019

Cited by 7 | Viewed by 3007

Abstract

In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that

⋀_{t > 0} M (x, y, t) > 0

. The continuous extension

M^{0}

of M to [...] Read more.

In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that

⋀_{t > 0} M (x, y, t) > 0

. The continuous extension

M^{0}

of M to

X^{2} \times [0, + \infty[

is called extended fuzzy metric. We prove that

M^{0}

generates a metrizable topology on X, which can be described in a similar way to a classical metric.

M^{0}

can be used for simplifying or improving questions concerning M; in particular, we expose the interest of this kind of fuzzy metrics to obtain generalizations of fixed point theorems given in fuzzy metric spaces. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

15 pages, 297 KiB

Open AccessArticle

Fixpointed Idempotent Uninorm (Based) Logics

by Eunsuk Yang

Mathematics 2019, 7(1), 107; https://doi.org/10.3390/math7010107 - 20 Jan 2019

Cited by 6 | Viewed by 3037

Abstract

Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard [...] Read more.

Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard completeness, i.e., completeness on real unit interval

[0, 1]

, was proved by Baldi and Ciabattoni. However, their proof is not algebraic and does not shed any light on the algebraic feature by which an idempotent uninorm is characterized, using operations defined by a fixpointed negation. To shed a light on this feature, this paper algebraically investigates logics based on fixpointed idempotent uninorms. First, several such logics are introduced as axiomatic extensions of uninorm mingle logic (UML). The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided. Next, standard completeness is established for the systems using an Esteva–Godo-style approach for proving standard completeness. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

10 pages, 1531 KiB

Open AccessArticle

Two Classes of Entropy Measures for Complex Fuzzy Sets

by Lvqing Bi, Zhiqiang Zeng, Bo Hu and Songsong Dai

Mathematics 2019, 7(1), 96; https://doi.org/10.3390/math7010096 - 17 Jan 2019

Cited by 42 | Viewed by 3167

Abstract

Complex fuzzy sets are characterized by complex-valued membership functions, whose range is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane. In this paper, we define two kinds of entropy measures for complex fuzzy sets, called [...] Read more.

Complex fuzzy sets are characterized by complex-valued membership functions, whose range is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane. In this paper, we define two kinds of entropy measures for complex fuzzy sets, called type-A and type-B entropy measures, and analyze their rotational invariance properties. Among them, two formulas of type-A entropy measures possess the attribute of rotational invariance, whereas the other two formulas of type-B entropy measures lack this characteristic. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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24 pages, 362 KiB

Open AccessArticle

Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets

by Hsien-Chung Wu

Mathematics 2019, 7(1), 11; https://doi.org/10.3390/math7010011 - 22 Dec 2018

Cited by 10 | Viewed by 4420

Abstract

The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual [...] Read more.

The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in

R

and establish some interesting results based on the upper and lower α-level sets. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

20 pages, 373 KiB

Open AccessArticle

Characterizations of Right Weakly Regular Semigroups in Terms of Generalized Cubic Soft Sets

by Muhammad Gulistan, Feng Feng, Madad Khan and Aslıhan Sezgin

Mathematics 2018, 6(12), 293; https://doi.org/10.3390/math6120293 - 30 Nov 2018

Cited by 5 | Viewed by 2746

Abstract

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of

[0, 1]

and a number from

[0, 1]

. Generalized cubic sets generalized the cubic [...] Read more.

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of

[0, 1]

and a number from

[0, 1]

. Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory. Full article

(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

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