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In this article the relationship between fuzzy consequence and fuzzy interior operators with fuzzy relations is analyzed. While the situation has been studied in depth if the fuzzy relation is a ⁎-preorder it is not the case for arbitrary fuzzy relations. If R is an arbitrary fuzzy relation, a fuzzy consequence γ R and a fuzzy interior δ R operators can be derived from R. Using the Representation Theorem for ⁎-preorders, from the columns and rows of R two ⁎-preorders, R c o l and R r o w , can be generated that coincide with the ⁎-preorders generated by γ R and δ R respectively. A duality result between them and the ⁎-coherence of γ R are also shown. [ABSTRACT FROM AUTHOR]

The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled. [ABSTRACT FROM AUTHOR]

A covering is a family of sets. This paper studies the generalization of rough set models via a generalization of the concept of neighborhoods when considering a general family of sets instead of a partition of a set. Given the notion of neighborhood, we propose four different rough set models based on a family of sets and study the reduction problem for such sets. The reduction relationship between rough sets and formal contexts is established. Three reduction algorithms are obtained and applied to identify all reducts. We extend these results to fuzzy sets. Using a family of fuzzy sets to replace a fuzzy covering or a fuzzy β covering, we define set-based fuzzy rough sets and consider three types of reduction. Finally, we give reduction algorithms that are required to identify all reducts. [ABSTRACT FROM AUTHOR]

Considering a family { T c } c ∈ [ 0 , 1 ] generated from a t-norm T , the degree of T -transitivity of a fuzzy relation R is revisited and proved to coincide with the greatest c for which R is T c -transitive. This fact gives rise to the study of new families of t-norms to generate different degrees of transitivity with respect to them. The mappings transforming fuzzy relations into transitive fuzzy relations smaller than or equal to the given ones are studied. [ABSTRACT FROM AUTHOR]

The data readability, complexity reduction of learning algorithms, and enhancing predictability are the most important reasons for using feature selection methods, especially when there exist lots of features. In recent years, unsupervised feature selection techniques are well explored. Among the various methods of feature selection, algorithms based on information theory are effective in selecting useful attributes and eliminating useless ones. Mutual information and symmetric uncertainty based methods can effectively estimate feature relevancy, but the use of bivariate measures ignores possible dependencies among more than two features. To address this limitation, this research introduces a novel unsupervised feature selection method, called Fuzzy Multivariate Symmetric Uncertainty-Feature Selection (FMSU-FS). The proposed method also overcomes the need for discretization for continuous features that causes information loss. To evaluate the effectiveness of the proposed method, FMSU is compared with conventional methods based on information theory. The experimental results on benchmark datasets show improvement of our approach in measuring clustering performance. [ABSTRACT FROM AUTHOR]

Social Networking Sites (SNS) have entailed a revolution for society. They have given a say to everyone regardless of their status and this has been translated into loads of data. The task of profiling users constitutes a way to learn from this data in order to show users only the content that is relevant to them. Several recommendation system techniques have been used to address this problem, being mainly based on what the user explicitly says about themselves, on what the user publishes in the SNS and on the similarities between users. However, in social media context, it is also possible to use relations between users. Considering basic relations like follower or followee to extract information from them may result in noise, since they do not imply that users share interest or even ideas. In this work, we present a fuzzy framework to enrich user profiles with complex properties in order to have an even better representation of them. We use basic relations defined by SNSs to complete the information available in user profiles with topics of interest and ideas towards them and to define deep relations that will enable new ways of analysis. We use these deep relations to create clusters of similar users that, ultimately, will allow the expansion of properties from known users to the rest of the cluster. We tested our proposal with a dataset of Tweets in Spanish related to a political event. Our experiments prove the potential that this approach has for a lot of applications in microblogging context. [ABSTRACT FROM AUTHOR]

In this paper we study the class of all fuzzy subgroups defined with respect to a given t-norm on a fixed group. We find necessary and sufficient conditions over two t-norms to guarantee that the class of all fuzzy subgroups induced by the first t-norm is contained in the class of all fuzzy subgroups with respect to by the second one. We characterize the functions that transform fuzzy subgroups into fuzzy subgroups. The close relationship between indistinguishability operators and fuzzy subgroups allows us to obtain similar results for some classes of indistinguishability operators. [ABSTRACT FROM AUTHOR]

Abstract In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets. [ABSTRACT FROM AUTHOR]

In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction. [ABSTRACT FROM AUTHOR]

Fuzzy relations and reciprocal relations are two popular tools for representing degrees of preference. It is important to note that they carry a different semantics and cannot be equated directly. We propose a simple transformation based on implication operators that allows to establish a one-to-one correspondence between both formalisms. It sets the basis for a common framework in which properties such as transitivity can be studied and definitions belonging to different formalisms can be compared. As a byproduct, we propose a new family of upper bound functions for cycle-transitivity. Finally, we unveil some interesting equivalences between types of transitivity that were left uncompared till now. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a fuzzification of Tarski's fixed point theorem without the assumption of transitivity. For this purpose a new structure – the so called L -complete propelattice, which generalizes complete lattices and completely lattice L -ordered sets, is introduced. Our results show that for L -fuzzy isotone maps on L -complete propelattices a variant of Tarski's fixed point theorem holds. Especially, the set of fixed points is nonempty and of a certain structure. [ABSTRACT FROM AUTHOR]

In this paper, topological characterizations of generalized fuzzy rough sets are investigated in the context of basic rough equalities. Various fuzzy topologies induced by different fuzzy relations are studied with respect to the lower fuzzy rough approximation operator determined by a fuzzy implicator, which is left-continuous in the first argument and right-continuous in the second argument. The relationships among fuzzy topologies induced by fuzzy relations are discussed. The algebraic structures of T -similarity set of fuzzy relations are investigated with respect to different fuzzy implicators, where T is a left-continuous t-norm. Moreover, the T -transitive subset of T -similarity set is proven to be a complete distributive lattice. [ABSTRACT FROM AUTHOR]

In 1978, a fuzzy eigenequation with max ⁡ − min product was introduced and studied by Sanchez. He provided us with its maximal solution. In this study, we examine the fuzzy eigenequation A ∘ x = λ ⁎ x , where ⁎ is an increasing operation satisfying some additional assumptions. The assumptions for which the greatest solution exists in the eigenvector family are examined. The notion of max ⁡ − ⁎ subspace in [ 0 , 1 ] is introduced and the properties of this are studied. In particular, the relationships between families of eigenvectors for certain matrices and the results of operations on these matrices are studied. [ABSTRACT FROM AUTHOR]

The theory of operators generated by binary fuzzy relations is quickly growing due its fundamental nature and potential applicability. The main goal of the paper is to present several representation theorems for operators induced by fuzzy relations on Pavelka's algebras (for example closure operators used in formal concept analysis, monadic operators or tense operators). Consequently we establish algebraic models with their semantics which are usable in the non-classical logic research and in the computer science research. [ABSTRACT FROM AUTHOR]

We show that although there exists no non-trivial (fuzzy) tolerance relation a partial order relation is compatible with (in the sense of Bělohlávek), the situation is quite different when considering its strict part. More specifically, we provide a representation of all fuzzy tolerance (and, in particular, all fuzzy equivalence) relations a strict order relation is compatible with. To that end, we introduce the notion of clone relation associated with a partially ordered set and discuss its basic properties. The mentioned representation is intimately connected with this clone relation. [ABSTRACT FROM AUTHOR]

A similarity-based approximate reasoning methodology that requires the construction of fuzzy relations was proposed by Raha et al. [26] . This paper investigates properties of four inferred conclusions calculated by Raha's similarity-based approximate reasoning method. The relationships among the four inferred conclusions are examined. The monotonicity and the approximation property for similarity-based approximate reasoning methods are defined. Then the monotonicity and the approximation property of Raha's similarity-based approximate reasoning method are studied. [ABSTRACT FROM AUTHOR]

The problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations and their applications. Several variations of the acclaimed Szpilrajn's theorem have been provided, inclusive of the case when some order conditions between elements are imposed on the extension. We extend the analysis of that topic by Alcantud to the fuzzy case. By appealing to generators to decompose (fuzzy) preference relations into strict preference and indifference relations, we give general extension results for the corresponding concept of compatible extension of a fuzzy reflexive relation. Then we investigate the conditions under which compatible order extensions exist such that certain elements are connected by the asymmetric part, resp., and certain other elements by the symmetric part, to respective elements with degree 1. [ABSTRACT FROM AUTHOR]

In this paper, we show that fuzzy Alexandrov topologies can be induced by fuzzy relations. The concept of T -similarity of fuzzy relations is introduced. Variations of the same fuzzy relation are investigated. The fact that every fuzzy relation is solely T -similar to some T -preorder fuzzy relation is proved. A characteristic condition for fuzzy relations to be T -transitive is established. These results illustrate that fuzzy relations can be studied by means of topology. Moreover, algebraic structures based on T -similarity of fuzzy relations are obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the “powering” of binary fuzzy relations (i.e. lifting a fuzzy relation R defined on a set X to the relation R + defined on the set F ( X ) of all fuzzy subsets of X ). We prove that for any complete residuated lattice L , the composition of the powers of two L -relations is always a subset of the power of their composition. Answering to a question posed by Georgescu, we prove that the converse is not always true. We prove that the composition of the powers of two L -relations is equal to the power of their composition if and only if L is a Heyting algebra. [ABSTRACT FROM AUTHOR]

Graded properties of binary and unary fuzzy connectives (valued in MTL Δ -algebras) are studied, including graded monotony, a generalized Lipschitz property, commutativity, associativity, unit and null elements, and the dominance relation between fuzzy connectives. The apparatus of Fuzzy Class Theory (or higher-order fuzzy logic) is employed as a tool for easy derivation of graded theorems on the connectives. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the notion of universal fuzzy automaton with membership values in a complete residuated lattice, whose states are the factorizations of this fuzzy language and transition function is defined using the inclusion degree of related fuzzy languages. Next, we define the homomorphism of fuzzy automata, prove that every automaton accepting a fuzzy language can canonically map into the universal fuzzy automaton of this language, which is called the universal property. For a fuzzy language, the connections between the universal fuzzy automaton and fuzzy minimal automata of the given fuzzy language are exploited. Finally, we give a method to construct the universal fuzzy automaton by a deterministic fuzzy automaton accepting the given fuzzy language, which is effective in the case that this deterministic fuzzy automaton is finite. [Copyright &y& Elsevier]

Abstract: It is well known that the solution set of a fuzzy relational equation with sup–inf composition is a join semilattice, in general, not a meet semilattice. This paper investigates the conditions under which the solution sets of fuzzy relational equations with sup-inf composition over complete Brouwerian lattices form lattices. We first give some properties of the decompositions of elements in complete lattices, then present a necessary and sufficient condition that the meet of a solution with any other solution is again a solution. Finally, we show some necessary and sufficient conditions that the solution sets are lattices. [Copyright &y& Elsevier]

Abstract: Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward–backward, and backward–forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an efficient algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed in Ignjatović et al. [On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets Syst. 161 (2010) 3081–3113], which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations. [Copyright &y& Elsevier]

Abstract: This paper focuses on pretopological and diagonal axioms for stratified L-convergence spaces recently defined by the authors in 2011. This notion is slightly stronger than Jäger''s stratified L-generalized convergence spaces. In particular, one pretopological axiom is stated without resorting to the notion of neighborhood L-filters; some generalizations of Kowalsky''s and Fischer''s diagonal conditions are presented. The relationship between these axioms are investigated. [Copyright &y& Elsevier]

Abstract: For a GL-monoid , Höhle introduced sets to formalize the mathematical theory of identity and existence in monoidal logic. For the purpose of analyzing deadlock situations in data flow networks, the notion of partial metric space was proposed by Matthews. Later on a lattice-theoretic generalization of (pseudo) partial metric spaces, extending the range of (pseudo) partial metrics to a value lattice , were studied under the name of pmetric spaces. Referring to a GL-monoid and a dual GL-monoid , we show that sets are order-theoretically dual to spaces, and apply this duality to the representation of spaces and the determination of their various categorical properties. In addition to this, the present paper provides not only some new results about the representation of sets and their categories, but also some non-trivial categorical connections between sets and spaces. [Copyright &y& Elsevier]

Abstract: We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints. [Copyright &y& Elsevier]

Abstract: Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables. [Copyright &y& Elsevier]

Abstract: For crisp relations the concept of a semi-order can be stated in a number of equivalent ways. When trying to extend this concept to the fuzzy setting, we observe that the (generalized) definitions fail to be equivalent. In this contribution, we discuss which is the most natural definition of a fuzzy semi-order, and study the hierarchy among the alternative definitions, in particular when using a t-norm without zero divisors. [Copyright &y& Elsevier]

Abstract: Crisp and ambiguous representations of closed subsets of one space by closed subsets of another space are introduced. It is shown that, for each pair of compact Hausdorff spaces, the set of (crisp or ) ambiguous representations is a lattice and a compact Hausdorff Lawson upper semilattice. The categories of ambiguous and representations are defined and investigated. [Copyright &y& Elsevier]

Abstract: In this paper some basic aspects of the theory of graded consequence are reconsidered, based on many results that appeared since the inception of graded consequences in 1986. Relationships of the notion of graded consequence with the notions of consequence in many-valued and fuzzy logics are discussed. Following this, a detailed study is carried out on the necessary conditions that follow from axioms generalizing the rules “ex falso quodlibet” and “reasoning by cases” added to the basic three axioms of graded consequence of reflexivity, monotonicity and cut inspired from Tarski and Gentzen. Towards the converse direction, in the process of finding non-trivial examples satisfying the five graded consequence axioms some sufficient conditions on the algebraic structures and the set of valuations are obtained. The significance of these conditions holding simultaneously is investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new relationship between fuzzy concept lattice and fuzzy graph. Given a fuzzy binary relation, we define an underlying fuzzy graph, and establish a new relation between the set of elements of the fuzzy concept lattice built from the fuzzy relation and the set of minimal disconnections of the associated fuzzy graph. This link is important from the point of view of application and computation. [Copyright &y& Elsevier]

Abstract: We show that there are natural ways to make a fuzzy set into an enriched category, and that in doing so, some basic results in fuzzy set theory are applications of enriched category theory. Thus we argue that the theory of enriched categories is a useful tool for fuzzy set theorists. [Copyright &y& Elsevier]

Abstract: In this paper we describe a segmentation method applied to images which are compressed by using Fuzzy Transforms. The segmentation of the images is realized via the FGFCM (Fast Generalized Fuzzy C-Means) clustering algorithm, which is robust to noise and outliers. The optimal number of clusters is determined via the PCAES (Partition Coefficient And Exponential Separation) validity index. We use a similarity measure defined via Lukasiewicz t-norm for comparison between the original image and the reconstructed images. The best results are obtained if this similarity measure overcomes a threshold value, experimentally determined from the analysis of the trend of it with respect to the PSNR (Peak Signal to Noise Ratio). [Copyright &y& Elsevier]

Abstract: Goguen categories were introduced as a suitable categorical description of -fuzzy relations, i.e., of relations taking values from an arbitrary complete Brouwerian lattice instead of the unit interval of the real numbers. In this paper we want to study the algebraic structures derived from Goguen categories by replacing its second-order axiom by some weaker versions. [Copyright &y& Elsevier]

Abstract: Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation errors. In some way, the more number of elicitations, the more probable inconsistency is. This is mostly the case within a fuzzy framework, even when the number of alternatives or object to be classified is relatively small. In this paper we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy relation, being T any left-continuous t-norm. The computed approximation we propose will take time complexity, where n is the number of elements under consideration, and is expected to produce a T-transitive opening. To the authors’ knowledge, there are no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties. [Copyright &y& Elsevier]

Abstract: In this paper we introduce and study the concepts of a uniform fuzzy relation and a (partially) uniform F-function. We give various characterizations and constructions of uniform fuzzy relations and uniform F-functions, we show that the usual composition of fuzzy relations is not convenient for F-functions, so we introduce another kind of composition, and we establish a mutual correspondence between uniform F-functions and fuzzy equivalences. We also give some applications of uniform fuzzy relations in approximate reasoning, especially in fuzzy control, and we show that uniform fuzzy relations are closely related to the defuzzification problem. [Copyright &y& Elsevier]

Abstract: We present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes. The method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations, which allows transferring basic properties of sup-T and inf-R compositions to a family of more than 30 composition-related operations, including sup-T and inf-R images, pre-images, Cartesian products, domains, ranges, resizes, inclusion, height, plinth, etc. Besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles, the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them, thus trivializing a large part of the theory of fuzzy relations. [Copyright &y& Elsevier]

Abstract: This paper studies fuzzy relations in the graded framework of Fuzzy Class Theory (FCT). This includes (i) rephrasing existing work on graded properties of binary fuzzy relations in the framework of Fuzzy Class Theory and (ii) generalizing existing crisp results on fuzzy relations to the graded framework. Our particular aim is to demonstrate that Fuzzy Class Theory is a powerful and easy-to-use instrument for handling fuzzified properties of fuzzy relations. This paper does not rephrase the whole theory of (fuzzy) relations; instead, it provides an illustrative introduction showing some representative results, with a strong emphasis on fuzzy preorders and fuzzy equivalence relations. [Copyright &y& Elsevier]

This paper is devoted to the discussion of the relationship between fuzzifying topologies and fuzzy relations. A saturated fuzzifying topology can be generated by a reflexive fuzzy relation; and conversely, a fuzzy preorder can be generated by a fuzzifying topology. In fact, there exists a one-to-one correspondence between the set of all saturated fuzzifying topologies and that of all fuzzy preorders. [Copyright &y& Elsevier]

Abstract: The present paper gives a state-of-the-art overview of representation and construction results for fuzzy weak orders. We do not assume that the underlying domain is finite. Instead, we concentrate on results that hold in the most general case that the underlying domain is possibly infinite. This paper presents three fundamental representation results, each of which also provides a construction method: (i) score function-based representations, (ii) inclusion-based representations, (iii) representations by decomposition into crisp linear orders and fuzzy equivalence relations, which also facilitates a pseudo-metric-based construction. [Copyright &y& Elsevier]

Abstract: In classical linguistic modelling for imprecise concepts and uncertain reasoning, a basic idea is to use fuzzy sets to describe the semantics of imprecise concepts, the theoretic foundation of this modelling is fuzzy logic. But in some practical applications, the determination of membership functions associated with vague concepts is difficult or impossible. In this paper, we present a new linguistic modelling technique based on the semantic similarity relation among linguistic labels. In this new linguistic model, a fuzzy relation is utilized to represent the semantic relation among linguistic labels. The elements in the fuzzy relation are interpreted as the degrees of similarities or the degrees of semantic overlapping between the corresponding linguistic labels. We develop an inference method for computing the degree of similarity between any two linguistic expressions which are the logic formulas generated by applying the logic connectives to the linguistic labels. This inference method utilizes the consonant mass assignment functions induced from the fuzzy similarity relation on linguistic labels. We show that the linguistic model presented in this paper satisfies some perfect properties, e.g. the excluded middle law, law of non-contradiction. This new linguistic modelling is illustrated finally in a linguistic decision-making example. [Copyright &y& Elsevier]

Abstract: In this paper, we recall a class of approximating formulas. Since they arose as a generalization of classical normal forms we denote them by the same term. Moreover, we will show that the fuzzy transform (F-transform) introduced in [Perfilieva, I., Fuzzy approach to solution of differential equations with imprecise data: application to reef growth problem, in: R.V. Demicco, G.J. Klir (Eds.), Fuzzy Logic in Geology, Academic Press, Amsterdam, 2003, pp. 275–300 (Chapter 9), Perfilieva, I., Fuzzy transforms, in: J.F. Peters, A. Skowron (Eds.), Transactions on Rough Sets II. Rough Sets and Fuzzy Sets, Lecture Notes in Computer Science, vol. 3135, 2004, pp. 63–81] as an approximation method for continuous functions can be viewed as a particular case of normal form. Therefore, the results valid for normal forms may be applied to F-transforms as well. [Copyright &y& Elsevier]

Abstract: This paper proposes a new heuristic search technique for obstacle avoidance of autonomous underwater vehicles (AUVs) that are equipped with a looking-ahead obstacle-avoidance sonar. The fuzzy relation between the sonar sections and the properties of a real-time environment is used as a core concept. The Lukasiewicz fuzzy implication operator is used in Bandler and Kohout''s Triangle Subproduct to calculate the relationship between the fuzzy relation and its transposed relation. This product relation the reveals the characteristics and interrelationships of the sonar sections. A direction of the sonar section that has good characteristics is selected as the successive heading for obstacle avoidance of AUVs. The simulation results clearly demonstrate that the heuristic search technique, which uses the BK-product of a fuzzy relation with a sonar partitioned into seven sections, enables AUVs to navigate safely through the obstacle to the goal with the optimal path. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. The (TC) axiom for fuzzy topology is proposed. It is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies (TC) axiom; and conversely, a fuzzy topology which obey (TC) axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation. That is to say, there exists a one-to-one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy (TC) axiom. [Copyright &y& Elsevier]

In this work, we investigate a relationship between extensionality and continuity of fuzzy relations. We found conditions when extensionality implies continuity and vice versa. [Copyright &y& Elsevier]

This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and one-to-one correspondence between linearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case. If we do not require a one-to-one correspondence between linearity and maximality, however, we obtain that an implication-based definition appears to constitute a sound compromise, in particular, if Lukasiewicz-type logics are considered. [Copyright &y& Elsevier]

Let I be a finite and J an infinite index sets, A = (aij)I×J and B = (bi)i∈I a given matrix and a column vector with elements from a complete Brouwerian lattice. In this paper, the problem of a minimal solution of a fuzzy relational equation A⊙X = B, where ⊙ is the sup–inf composite operation, is investigated. [Copyright &y& Elsevier]

We discuss fuzzy deductive reasoning achieved through the compositional rule of inference from the analogy point of view, merging it into the analogical scheme. Necessary and sufficient conditions on operators involved are investigated to fit the compositional rule of inference into the analogical scheme. Several examples including well known models are included. [Copyright &y& Elsevier]

Fuzzy relations are able to model vagueness, in the sense that they provide the degree to which two objects are related to each other. However, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees. Intuitionistic fuzzy sets, as defined by Atanassov (Instuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universes.One of the main concepts in relational calculus is the composition of two relations. Burillo and Bustince (Fuzzy Sets and Systems 78 (1996) 293; Soft Comput. 2 (1995) 5) have extended the sup-T composition of fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionistic fuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and the variants of these compositions given by De Baets and Kerre (Adv. Electron. Electron Phys. 89 (1994) 255). Some properties of these compositions are investigated: containment, convertibility, monotonicity, interaction with union and intersection. [Copyright &y& Elsevier]

The standard combinatoric model of belief functions can be generalized in two ways: (1) the crisp compatibility relation is replaced by a fuzzy relation and (2) the values of crisp compatibility relation are observed through a binary information channel with noise and, perhaps, subjected to a deformation, in other terms, the original compatibility relation is replaced by its randomized modification. Both the generalizations are investigated, separately, in more detail. [Copyright &y& Elsevier]