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In this paper we present a way of perturbing reflexive, symmetric, antisymmetric, perfect antisymmetric, transitive and partially included intuitionistic fuzzy relations afterward obtaining the perturbation of another reflexive, symmetric, antisymmetric, perfect antisymmetric, transitive and partially included intuitionistic fuzzy relation. To do so we study the main properties of an operator that allows us to go from an intuitionistic fuzzy set to another also intuitionistic fuzzy set, we then apply this operator to intuitionistic fuzzy relations with different properties and we study the conditions there must be for the new intuitionistic fuzzy relation to maintain the original properties.

This paper focuses on the decomposition problem of fuzzy relations using the concepts of multiuniverse fuzzy propositional logic. Given two fuzzy propositions in different universes, it is always possible to construct a fuzzy relation in the common universe through a prescribed combination. However, the converse is not so obvious, if possible at all. In other words, given a fuzzy relation, how would we know if it really represents a certain relationship between some fuzzy propositions? It is important to recognize whether the given fuzzy relation is a meaningful representation of information according to certain criteria applicable to some fuzzy propositions that constitute the fuzzy relation itself. Two basic structures of decomposition are investigated. Necessary and sufficient conditions for decomposition of multiuniverse fuzzy truth functions in terms of one-universe truth functions are presented. An algorithm for decomposition is proposed.

We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup –* composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

A special kinds of regular fuzzy relations are discussed. Some facts about sup – min composition of these relations and inverse relations are proved.

In recent years, the innovation and improvement of forecasting techniques have caught more and more attention. Especially, in the fields of financial economics, management planning and control, forecasting provides indispensable information in decision-making process. If we merely use the time series with the closing price array to build a forecasting model, a question that arises is: Can the model exhibit the real case honestly? Since, the daily closing price of a stock index is uncertain and indistinct. A decision for biased future trend may result in the danger of huge lost. Moreover, there are many factors that influence daily closing price, such as trading volume and exchange rate, and so on. In this research, we propose a new approach for a bivariate fuzzy time series analysis and forecasting through fuzzy relation equations. An empirical study on closing price and trading volume of a bivariate fuzzy time series model for Taiwan Weighted Stock Index is constructed. The performance of linguistic forecasting and the comparison with the bivariate ARMA model are also illustrated.

A new map (Λ_E) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of Λ_E is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φ_E and ψ_E) are also reviewed, in order to compare them with Λ_E. Interesting properties of the fixed points of φ_E and are studied. Among others, the fixed points of Λ_E (Fix(Λ_E)) are proved to be the maximal fuzzy points of (X, E) and the fixed points of coincide with the Image of Λ_E. An isometric embedding of X into Fix(Λ_E) is established and studied.

The paper presents a preliminary investigation of potential methods for extracting semantic views of text contents under the form of structured sets of words, which go beyond standard statistical indexing. The aim is to build kinds of fuzzily weighted structured images of semantic contents. A preliminary step consists in identifying the different types of relations (is-a, part-of, related-to, synonymy, domain, glossary relations) that exist between the words of a text, using some general ontology such as WordNet. Then taking advantage of these relations, different types of fuzzy clusters of words can be built. Moreover, apart from its frequency of occurrence, the importance of a word may be also evaluated through some estimate of its specificity. A degree of "centrality" is also computed for each word in a cluster. The size of the clusters, the frequency, the specificity and the centrality of their words are indications that enable us to build a fuzzy set of sets of words that progressively "emerge" from a text, as being representative of its contents. The ideas advocated in the paper and their potential usefulness are illustrated on a running example and on two experiments. It is expected that obtaining a better representation of the semantic contents of texts may help in particular to give indications of what the text is about to a potential reader.

In this paper, we focus on the characterizations of some properties of fuzzy relations by using the notions of traces. The investigation is two-fold. On one hand, we attempt to reformulate the results obtained by Fodor under weakened conditions. On the other hand, we present some new characterizations of properties, mainly T-asymmetry, T-S-semitransitivity and T-S-Ferrers property, in terms of traces.

Social networks have gained a lot attention. They are perceived as a vast source of information about their users. Variety of different methods and techniques has been proposed to analyze these networks in order to extract valuable information about the users – things they do and like/dislike. A lot of effort is put into improvement of analytical methods in order to grasp a more accurate and detailed image of users. Such information would have an impact on many aspects of everyday life of people – from politics, via professional life, to shopping and entertainment.

The theory of fuzzy sets and systems, introduced in 1965, has the ability to handle imprecise and ambiguous information, and to cope with linguistic terms. The theory has evolved into such areas like possibility theory and computing with words. It is very suitable for processing data in a human-like way, and providing the results in a human-oriented manner.

The paper presents a short survey of works that use fuzzy-based technologies for analysis of social networks. We pose an idea that fuzzy-based techniques allow for introduction of humancentric and human-like data analysis processes. We include here detailed descriptions of a few target areas of social network analysis that could benefit from applications of fuzzy sets and systems methods.

In this paper we propose clustering methods based on weighted quasiarithmetic means of T-transitive fuzzy relations. We first generate a T-transitive closure R^T from a proximity relation R based on a max-T composition and produce a T-transitive lower approximation or opening R_T from the proximity relation R through the residuation operator. We then aggregate a new T-indistinguishability fuzzy relation by using a weighted quasiarithmetic mean of R^T and R_T. A clustering algorithm based on the proposed T-indistinguishability is thus created. We compare clustering results from three critical t_i-indistinguishabilities: minimum (t₃), product (t₂), and Łukasiewicz (t₁). A weighted quasiarithmetic mean of a t₁-transitive closure $R^{t_1}$ and a t₁-transitive lower approximation or opening $R_{t_1}$ with the weight $p=0.5$, demonstrates the superiority and usefulness of clustering begun by using a proximity relation R based on the proposed clustering algorithm. The algorithm is then applied to the practical evaluation of the performance of higher education in Taiwan.

In this paper, we have studied representability of both fuzzy biorders and fuzzy weak orders. It has also been shown that union of a finite family of fuzzy weak orders with respect to a t-norm T is a fuzzy quasi-transitive relation with respect to T. In the last theorem, we have obtained a characterization for a T_L-representable fuzzy weak order.

This paper reports some novel approach on linguistic logic with our intention to realize CWW, Computing With Words, via a simple example which consists of only five words. As a by product, this simple example of the linguistic logical system may serve as a mathematical model, modeling the degree of truthfulness in daily usage. The five words set of a linguistic variable modeling the degree of truthfulness are; true, nearly true, undecided, nearly false and false. We subjectively choose trapezoidal fuzzy numbers as our linguistic truth values in order to model our linguistic logic system. Firstly, some natural operations and linguistic logic operators are defined to suit our objective of developing a closed linguistic variable set. Then the computation of linguistic truth values for this linguistic logical system is developed in order to facilitate us to perform the linguistic inferences. Properties of these natural operations can be derived accordingly. It is perhaps quite rewarding to see numerous linguistic truth relations defined on a single linguistic truth set and linguistic implications ended up with numerous linguistic truth tables. In addition, the linguistic inferences of generalized modus ponens and generalized tollens determined by linguistic compositional rules based on the linguistic truth relation and some natural operations are introduced. The simple examples of the linguistic inferences of the various generalized tautologies are illustrated. Finally, we have proved via a simple dictionary that a closed and self consistent linguistic logical system indeed can be constructed and it is possible to move a chunk of information as modeled by a fuzzy set to a higher level according to the theory of semiotics. These results have shown some promise in realizing the appealing theory of CWW.

Fuzzy relations are fundamental in applications of fuzzy set theory and fuzzy logic. The entire literature on fuzzy relations as applied to fuzzy graph theory are based on Rosenfeld’s relations. Rosenfeld used minimum and maximum as the norm and conorm in his study of compositions of fuzzy relations. In this paper, we generalize fuzzy relations using arbitrary $t$-norms and $t$-conorms. Many of the results do not hold when minimum and maximum are replaced by an arbitrary norm and an arbitrary conorm. Reflexive, symmetric and transitive generalized fuzzy relations are also discussed and an application to human trafficking and illegal immigration is presented.

The axiomatic definition of fuzzy set, which is based on the axioms of membership degrees, has been introduced by Xiaodong Pan and Yang Xu. On this basis, this paper focuses on establishing the axiomatical foundation of membership degree for fuzzy relations (binary fuzzy relation). The concept of two-dimensional vague partition is introduced, on this basis, the concept of fuzzy relation in Zadeh’s sense is redefined based on two-dimensional vague partition from the perspective of axiomatization. The results obtained in this paper can be easily extended to multiple fuzzy relations.

In this contribution, we investigate a boundary value problem for second-order ordinary differential equations with fuzzy boundary conditions. We introduce a novel approach where a solution of the fuzzy boundary value problem is defined as a fuzzy relation.We propose to use a system of fuzzy relation equations based on the technique of F-transform to obtain an approximate solution to the given fuzzy boundary value problem.