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The main aim of this paper is to connect R-fuzzy sets and type-2 fuzzy sets, so as to provide a practical means to express complex uncertainty without the associated difficulty of a type-2 fuzzy set. The paper puts forward a significance measure, to provide a means for understanding the importance of the membership values contained within an R-fuzzy set. The pairing of an R-fuzzy set and the significance measure allows for an intermediary approach to that of a type-2 fuzzy set. By inspecting the returned significance degree of a particular membership value, one is able to ascertain its true significance in relation, relative to other encapsulated membership values. An R-fuzzy set coupled with the proposed significance measure allows for a type-2 fuzzy equivalence, an intermediary, all the while retaining the underlying sentiment of individual and general perspectives, and with the adage of a significantly reduced computational burden. Several human based perception examples are presented, wherein the significance degree is implemented, from which a higher level of detail can be garnered. The results demonstrate that the proposed research method combines the high capacity in uncertainty representation of type-2 fuzzy sets, together with the simplicity and objectiveness of type-1 fuzzy sets. This in turn provides a practical means for problem domains where a type-2 fuzzy set is preferred but difficult to construct due to the subjective type-2 fuzzy membership. [ABSTRACT FROM AUTHOR]

The main purpose of the article is to show feasibility and efficiency of combining different types of simulating a hypervelocity impact interaction of solids within a single system. It allows effective application of new mathematical apparatus for analysis and comparison of experimental results. The paper considers the features of an integrated system that combines the results of experiments reproducing the high-velocity impact interaction of solids. The experiments included different types of modeling. System structure and the tasks solved within individual functional modulus were discussed. Based on the theory of fuzzy sets, new algorithms were proposed for comparing edges of the images obtained in the physical and computational experiment. Mathematical models were verified by such procedure. Differences in the shape and size of a cloud of particles that appears after penetrating a single barrier were estimated in the framework of designed models, as well as the differences in the sizes of punched holes using fuzzy variables and related membership functions. When analyzing experimental results, fuzzy sets enable switching from data structures to knowledge structures that are necessary to create an intelligent system for analyzing and predicting the effects of the interaction between space debris and spacecraft structures. • Types of simulating a hypervelocity impact interaction of solids can be combined. • This combining is efficient and feasible. • New algorithms were proposed for comparing edges of the images. • Fuzzy sets enable switching from data structures to knowledge structures. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study t-norms on the lattice of closed subintervals of the unit interval. Unlike for t-norms on a product lattice for which there exists a straightforward characterization of t-norms which are join-morphisms, respectively meet-morphisms, the situation is more complicated for t-norms in interval-valued fuzzy set theory. In previous papers several characterizations were given of t-norms in interval-valued fuzzy set theory which are join-morphisms and which satisfy additional properties, but little attention has been paid to meet-morphisms. Therefore, in this paper, we focus on t-norms which are meet-morphisms. We consider a general class of t-norms and investigate under which conditions t-norms belonging to this class are meet-morphisms. We also characterize the t-norms which are both a join- and a meet-morphism and which satisfy an additional border condition. [Copyright &y& Elsevier]

Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In order to handle these types of problem parameters, some fuzzy extensions of soft set theory are presented, yielding fuzzy soft set theory. Fuzzy description logics (DLs) are a family of logics which allow the representation of and the reasoning within structured knowledge affected by vagueness. In this paper we extend fuzzy soft sets with fuzzy DLs, i.e., present an extended fuzzy soft set theory by using the concepts of fuzzy DLs to act as the parameters of fuzzy soft sets. We define some operations for the extended fuzzy soft sets. Moreover, we prove that certain De Morgan’s laws hold in the extended fuzzy soft set theory with respect to these operations. In fact, the extended fuzzy soft set theory based on fuzzy DLs presented in this paper is a fuzzy extension of the extended soft set theory based on DLs. [Copyright &y& Elsevier]

Abstract: In previous papers, we introduced a new reasoning method based on quantifying linguistic domains, established a new fuzzy control algorithm, called hedge-algebras-based controller (HAC), and applied it to solve some fuzzy control problems. The HAC does not require fuzzy sets to provide the semantics of the linguistic terms used in the fuzzy rule system rather the semantics is obtained through the semantically quantifying mappings (SQMs). It was shown that the new method is very effective, i.e. for these problems it is always able to efficiently control the process toward the stable state. In the algebraic approach, the design of an HAC leads to the determination of the parameters of SQMs, which are the fuzziness measure of primary terms and linguistic hedges occurring in the fuzzy model, and the weights of a weighted averaging operator. However, there exists another problem, namely, how one can determine the optimal parameters of the method. In the present paper, we improved the HAC''s design by adding an optimal step that aims to find the optimal parameters of HAC using a genetic algorithm (GA). To show the effectiveness of the proposed method, we apply it to solve again the inverted pendulum problem and the problem of holding an object on a “hill”. The results demonstrate the good performance of the designed HAC. [Copyright &y& Elsevier]

This paper makes the attempt to explain the foundations of fuzzy sets from the point of view of universal algebra. The free fuzzy set and the fuzzy power set are constructed. Moreover, fuzzy power sets give rise to a monad. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets. [ABSTRACT FROM AUTHOR]

This paper proposes a short overview of models and methods based on fuzzy sets for image processing and image understanding, from low level to higher level interpretation. Recent trends are highlighted. [ABSTRACT FROM AUTHOR]

The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. [ABSTRACT FROM AUTHOR]

The study of concept lattices and oriented concept lattices with fuzzy attributes provides complementary conceptual structures, which can be used to search, analyze and extract information from large datasets. In this paper, we put forward new types of similarity relations between objects or attributes in fuzzy attribute-oriented concept lattices. We analyze how these similarities are related using the definitions and considering the reducibility between antitone and isotone fuzzy concept lattices as well. A comparison is made between two relations which measure the similarity of fuzzy oriented concepts. These relations are employed in factorizing the fuzzy attribute-oriented concept lattice, in order to reduce its complexity. [ABSTRACT FROM AUTHOR]

Both rough and fuzzy set theories offer interesting tools for dealing with imperfect data: while the former allows us to work with uncertain and incomplete information, the latter provides a formal setting for vague concepts. The two theories are highly compatible, and since the late 1980s many researchers have studied their hybridization. In this paper, we critically evaluate most relevant fuzzy rough set models proposed in the literature. To this end, we establish a formally correct and unified mathematical framework for them. Both implicator–conjunctor-based definitions and noise-tolerant models are studied. We evaluate these models on two different fronts: firstly, we discuss which properties of the original rough set model can be maintained and secondly, we examine how robust they are against both class and attribute noise. By highlighting the benefits and drawbacks of the different fuzzy rough set models, this study appears a necessary first step to propose and develop new models in future research. [ABSTRACT FROM AUTHOR]

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory. Two functors, the forward globalization functor and the backward globalization functor, that map a fuzzy set with a fuzzy preorder to a crisp set with a fuzzy preorder are considered. It is shown that both of them preserve cocompleteness with respect to several classes of weights. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to consider some methods for constructing Q-algebras and to give the concrete form of a free (unital) Q-algebra over a set (a semigroup, an ordered semigroup). [Copyright &y& Elsevier]

Abstract: In this paper we introduce extended multi-polarity and discuss its relation to classification problems. We show a connection between extended multi-polarity and multi-polarity; and we define extended multi-polar aggregation operators. We also generalize the notion of bipolar-valued fuzzy sets into multi-polar-valued fuzzy sets and show their link to extended multi-polarity. Further we show that the same link is present between reduced form of multi-polar (bipolar)-valued fuzzy sets and multi-polar aggregation (bipolar aggregation). Several examples are also given including extended multi-polar t-norms. [Copyright &y& Elsevier]

Abstract: In the present paper, using novel generalizations of the Hukuhara difference for fuzzy sets, we introduce and study new generalized differentiability concepts for fuzzy valued functions. Several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them. Characterization and relatively simple expressions are provided for the new derivatives. [Copyright &y& Elsevier]

Abstract: Incomplete data in soft sets lead to uncertainty and inaccuracy in representing and handling information. This paper introduces notions of complete distance between two objects and relative dominance degree between two parameters. Based on both the notions, an object-parameter method is proposed to predict unknown data in incomplete fuzzy soft sets. The proposal makes full use of known data, including the information from the relationship between known values of all objects on a certain parameter and the information from the relationship between known values of an object on all parameters. The effectiveness of the proposal is verified by many examples under the compared investigation of classical predicted methods. [Copyright &y& Elsevier]

Abstract: This paper shows that observations made by different authors at different times regarding one-to-one correspondence between the family of fuzzy preorders on a nonempty set and the family of all fuzzy topologies on this set satisfying certain extra conditions are essentially equivalent. [Copyright &y& Elsevier]

Abstract: In this paper, the concept of an -fuzzy sublattice (ideal, filter) and -fuzzy sublattice (ideal, filter) in a lattice is introduced, which is a generalization of the concept as given by Davvaz and Kazancı(2011) [18]. Characterizations for an -fuzzy sublattice (ideal, filter) and -fuzzy sublattice (ideal, filter) are provided. Different classes of lattices are characterized by the properties of these fuzzy sublattices (ideals, filters). Using the notions of a fuzzy sublattice (ideal, filter) with thresholds, characterization of a fuzzy sublattice (ideal, filter), an -fuzzy sublattice (ideal, filter) and -fuzzy sublattice (ideal, filter) are discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we show that there exists a one to one correspondence between a certain class of fuzzy pseudometrics (in the sense of Kramosil and Michalek) and [0,1)-indexed families of ordinary pseudometrics satisfying a property of lower semicontinuity. The aforementioned bijection is proved to be independent of the t-norm and it provides a representation theorem for a large class of fuzzy pseudometric spaces. Further, the relations between the uniformities and topologies both generated by the fuzzy pseudometric and by the corresponding family of ordinary pseudometrics are also investigated. [Copyright &y& Elsevier]

Abstract: This paper discusses basic notions underlying fuzzy sets, especially gradualness, uncertainty, vagueness and bipolarity, in order to clarify the significance of using fuzzy sets in practice. Starting with the idea that a fuzzy set may represent either a precise gradual composite entity or an epistemic construction refereeing to an ill-known object, it is shown that each of this view suggests a different use of fuzzy sets. Then, it is argued that the usual phrase fuzzy number is ambiguous as it induces some confusion between gradual extensions of real numbers and gradual extensions of interval calculations. The distinction between degrees of truth that are compositional and degrees of belief that cannot be so is recalled. The truth-functional calculi of various extensions of fuzzy sets, motivated by the desire to handle ill-known membership grades, are shown to be of limited significance for handling this kind of uncertainty. Finally, the idea of a separate handling of membership and non-membership grades put forward by Atanassov is cast in the setting of reasoning about bipolar information. This intuition is different from the representation of ill-known membership functions and leads to combination rules differing from the ones proposed for handling uncertainty about membership grades. [Copyright &y& Elsevier]

Abstract: Molodtsov (1999) initiated the concept of soft sets in . Maji et al. (2003) defined some operations on soft sets in . Aktaş and Çağman (2007) generalized soft sets by defining the concept of soft groups in . After them, Sun et al. (2008) gave soft modules in . In this paper, the concept of an intuitionistic fuzzy soft module is introduced and some operations on intuitionistic fuzzy soft sets are given. Finally, some of its basic properties are studied. [Copyright &y& Elsevier]

Abstract: Classical fuzzy mathematical morphology is one of the extensions of original binary morphology to greyscale morphology. Recently, this theory was further extended to interval-valued fuzzy mathematical morphology by allowing uncertainty in the grey values of the image and the structuring element. In this paper, we investigate the construction of increasing interval-valued fuzzy operators from their binary counterparts and work this out in more detail for the morphological operators, which results in a nice theoretical link between binary and interval-valued fuzzy mathematical morphology. The investigation is done both in the general continuous and the practical discrete case. It will be seen that the characterization of the supremum in the discrete case leads to stronger relationships than in the continuous case. [Copyright &y& Elsevier]

Abstract: Following the work of Zadeh and Xue-hai Yuan on the definition of a convex fuzzy subset, three new kinds of definitions of the convex fuzzy sublattices are proposed in this paper. First, by using the relations between fuzzy points and fuzzy sets, the definition of a -convex fuzzy sublattice is introduced. The acceptable nontrivial concepts obtained in this manner are the -convex fuzzy sublattice and -convex fuzzy sublattice. Second, the -convex fuzzy sublattice is presented and it is also characterized by the level -convex sublattice of the given fuzzy set. Finally, -fuzzy ideal and -fuzzy sublattice are introduced and several theorems are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, fuzzy soft subspace topology are studied. [Copyright &y& Elsevier]

Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making. [ABSTRACT FROM AUTHOR]

Abstract: In this paper an idea of soft mappings is given and some of their properties are studied. Images and inverse images of crisp sets and soft sets under soft mappings are also studied. An application of soft mapping in medical diagnosis has been shown. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies connectedness of Goetschel–Voxman fuzzy matroids (briefly, G–V fuzzy matroids), an analog of connectedness of crisp finite matroids. Based on the results of fuzzy circuits given by Goetschel and Voxman, the transitivity theorem concerning fuzzy circuits of G–V fuzzy matroids is established, and thus the useful notion of refined G–V fuzzy matroid is introduced. The connectedness of refined G–V fuzzy matroids is then defined by using an equivalence relation on the set of all fuzzy points on the ground set, and some expected properties of these G–V fuzzy matroids are presented. Additionally, five kinds of fuzzy matroids are compared. [ABSTRACT FROM AUTHOR]

Abstract: This paper points out an error in M. Mareš (Fuzzy coalition structures, Fuzzy Sets and Systems 114 (2000) 23–33), and gives two modified definitions to improve the model of fuzzy cooperative game. [ABSTRACT FROM AUTHOR]

Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed. Our interval-valued intuitionistic fuzzy soft set theory is a combination of an interval-valued intuitionistic fuzzy set theory and a soft set theory. In other words, our interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. The complement, “and”, “or”, union, intersection, necessity and possibility operations are defined on the interval-valued intuitionistic fuzzy soft sets. The basic properties of the interval-valued intuitionistic fuzzy soft sets are also presented and discussed. [Copyright &y& Elsevier]

Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainty. However, it is difficult to be used to represent the vagueness of problem parameters. In this paper, we introduce the notion of vague soft set which is an extension to the soft set. The basic properties of vague soft sets are presented and discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes. [Copyright &y& Elsevier]

Abstract: Molodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft -ideals and -idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) -ideals in BCI-algebras. We provide relations between fuzzy -ideals and -idealistic soft BCI-algebras. [Copyright &y& Elsevier]

Abstract: In this paper we firstly define some new types of fuzzy starshapedness, discuss the relationships among the different types of fuzzy starshapedness, and then present some basic properties of them. Finally we develop several important results on the shadows of starshaped fuzzy sets. [Copyright &y& Elsevier]

Abstract: Formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems. The purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment. Properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined. Based on both lattice-theoretic and fuzzy set-theoretic operators, two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined. Finally, properties of the rough fuzzy set approximation operators are presented in detail. [Copyright &y& Elsevier]