Fuzzy Matrices and Relational Equations

In fuzzy clustering, we always use fuzzy equivalence matrix to find the clustering results. Although fuzzy equivalence matrix is one of the most important tools in fuzzy clustering, it’s hard to judge whether a fuzzy matrix is a fuzzy equivalence matrix directly. In this paper, some simple and useful properties of the fuzzy equivalence matrices have been proposed based on the basic properties proposed in the classi cal articles. Using these properties, the judgment whether a fuzzy matrix is a fuzzy equivalence matrix may become an easy work. Keywords: Fuzzy clustering, fuzzy equivalence matrix 1. INTRODUCTION Given a finite set of objects X, and a measure of similarity for an arbitrary pair of objects, the problem of clustering is to find a partition of X such that the degree of similarity is high for objects within each block (cluster) and low for objects in different blocks. The purpose of clustering is to find a natural classification of given objects on the basis of a conceived concept of similarity. The classification may help to uncover a structure in given data, improve our understanding of a complex system, or suggest a meaningful simplification of a given system [1]. The requirement that given objects be classified into pairwise disjoint subsets (crisp clusters) is overly strong in many practical applications. Fuzzy set theory allows us to weaken it and, hence, make clustering more realistic. So, fuzzy clustering becomes one of the hottest research branches of data mining. One of the approaches to fuzzy clustering is based on fuzzy equivalence relations defined on X. Every fuzzy equivalence relation induces a crisp partition in each of its .-cuts. Using this approach, the fuzzy clustering problem becomes the problem of identifying a fuzzy equivalence relation on X in terms of a conceived concept of similarity. Although this may not be possible to do directly, it is usually easy to determine a fuzzy compatibility relation in terms of an appropriate distance function defined on X. A distance function is always reflexive and symmetric, but it is not transitive. However, once we have a compatibility relation, we can readily obtain a meaningful fuzzy equivalence relation by calculating its transitive closure. The resulting fuzzy equivalence relation offers the user a sequence of crisp partitions on X, which are linearly ordered by refinement ordering and graded by values of . [0, 1]. The user is given a flexibility to choose one of these partitions on the basis of additional criteria [2-4]. But, to calculate the transitive closure of a N×N fuzzy matrix , we need to calculate the 2kth power of the matrix, where k = 1, 2, , N. In the processing, after we calculate the 2k+1th power of the matrix and find that the 2k+1th power of the matrix is equal to the 2kth power of the matrix, the 2kth power of the matrix is the transitive closure of the matrix. In fact, if we can find the 2kth power of the matrix is a fuzzy equivalence matrix directly, we do not need to calculate the 2k+1th power of the matrix. If we can judge a fuzzy matrix is a fuzzy equivalence matrix by means of some simple properties of the equivalence matrix, it may be helpful. In this paper, some properties of the fuzzy equivalence matrix have been proposed and the judgment whether a fuzzy *white70121@163.com

An m<TEX>${\times}$</TEX>n Boolean matrix A is called regular if there exists an n<TEX>${\times}$</TEX>m Boolean matrix X such that AXA = A. We have characterizations of Boolean regular matrices. We also determine the linear operators that strongly preserve Boolean regular matrices.

An efficient algorithm for finding the Kleene closure of regular expressions matrices or fuzzy regular expression matrices is presented and illustrated by examples. Properties of Kleene closure are also investigated. The results may have useful applications in automata theory, pattern recognition, and pictorial information systems.

A fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a fuzzy/crist set in a fuzzy approximation space. A fuzzy rough set algebra is a fuzzy set algebra with added dual pair of fuzzy rough approximation operators. In this paper, we study the mathematical structures of fuzzy rough set algebras in infinite universes of discourse. We first define the concept of fuzzy rough set algebras by the axiomatic approach. We then examine the properties of fuzzy rough approximation operators in different types of fuzzy rough set algebras. We also prove that if a system \(({\cal F}(U), \cap, \cup, \sim, L, H)\) is a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra then the derived system \(({\cal F}(U), \cap, \cup, \sim, LL, HH)\) is also a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra.KeywordsApproximation operatorsFuzzy rough setsFuzzy setsFuzzy rough set algebrasRough sets

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Emerging trends in soft set theory and related topics.

Hesitant fuzzy set is a generalization of the classical fuzzy set by returning a family of the membership degrees for each object in the universe. Since how to use the rough set model to solve fuzzy problems plays a crucial role in the development of the rough set theory, the fusion of hesitant fuzzy set and rough set is then firstly explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, the model of the hesitant fuzzy rough set is presented to approximate a hesitant fuzzy target through a hesitant fuzzy relation. In axiomatic approach, an operators-oriented characterization of the hesitant fuzzy rough set is presented, that is, hesitant fuzzy rough approximation operators are defined by axioms and then, different axiom sets of lower and upper hesitant fuzzy set-theoretic operators guarantee the existence of different types of hesitant fuzzy relations producing the same operators.

On the lattice and the algebra of fuzzy subsets of a universal algebra

We give necessary and sufficient conditions for cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

In this paper, we present a system called KAFA (Kaist Fuzzy Accelerator) which provides various fuzzy inference methods and fuzzy set operations. The basic idea of this study is to develop a more general purpose hardware system. The architecture has SIMD structure, which consists of two parts; a system control unit (main controller), and an arithmetic unit (fuzzy processing element (FPE)). Microinstruction codes are defined and any fuzzy operation can be programmed by using these microinstructions. Each FPE has the maximum speed of 10 M FLOPS. As the KAFA contains 128 FPE's, if a fuzzy set consists of 128 elements, we achieve the peak performance of 10 M FSOPS (fuzzy set operation per second) under 10 MHz clock frequency. This system also includes the parallel algorithms for defuzzification on the SIMD mode architecture using KAFA network. The prototype of the proposed architecture was developed with the FPGA chips. The speed of the KAFA holds promise for the development of the new fuzzy application system such as automatic control, fuzzy expert systems, real time systems and fuzzy databases.

This paper presents a general framework for the study of covering-based fuzzy approximation operators in which a fuzzy set can be approximated by some elements in a crisp or a fuzzy covering of the universe of discourse. Two types of approximation operators, crisp-covering-based rough fuzzy approximation operators and fuzzy-covering-based fuzzy rough approximation operators, are defined, their properties are examined in detail. Finally, the comparison of these new approximation operators is done, a sufficient and necessary condition is given under which some operators are equivalent, and approximation operator characterization of fuzzy partitions of the universe is obtained.KeywordsCrisp coveringsfuzzy coveringsfuzzy-covering-based fuzzy rough approximation operatorsfuzzy partitionsfuzzy sets

On nilpotent incline matrices

This paper describes the construction of three Artificial Neural Networks with fuzzy input and output, imitating the performance of fuzzy algebra conjunction operators. More specifically, it is applied over the results of a previous research effort that used T‐Norms in order to produce a characteristic torrential risk index that unified the partial risk indices for the area of Xanthi. Each one of the three networks substitutes a T‐Norm and consequently they can be used as equivalent operators. This means that ANN performing Fuzzy Algebra operations can be designed and developed.

In this paper, we shall embark on the study of the algebraic object known as a fuzzy group which serves as one of the fundamental building blocks for the subject which is called fuzzy abstract algebra. In our opinion, the fuzzy algebraic systems are usually sets on whose elements we can operate algebraically by this we mean that we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set and, in addition, we assume that these fuzzy algebraic operations are subject to certain rules, which are explicitly spelled out in what we call the axioms or postulates defining the system. In this abstract setting we then attempt to prove theorems about these very general structures. We should like to stress that these fuzzy algebraic systems and their axioms, must come from the experience of looking at many examples. Namely, they should be rich in meaningful results. Hence, the acceptable definition of fuzzy group and subgroup are presented with binary operations and on the basis of the specified parameter, called ambiguity rank, which fulfils the basic requirements. The properties of these fuzzy groups and their fundamental qualities are discussed and then, the several illustrative examples were given. The future prospect of this paper is a new attitude to fuzzy basic mathematics, which will be referred to in the end.

When the max-min operations on the unit real interval are considered as a particular case of fuzzy logic operations (Godel operations), then the max-min algebra can be viewed as a specific case of more general fuzzy algebra with operations max and T, where T is a triangular norm (in short: t-norm). Such max-T algebras are useful in various applications of the fuzzy set theory. In this chapter we investigate the structure of the eigenspace of a given fuzzy matrix in two specific max-T algebras: the so-called max-drast algebra , in which the least t-norm T (often called the drastic norm) is used, and max-Lukasiewicz algebra with Łukasiewicz t-norm L. For both of these max-T algebras the necessary and sufficient conditions are presented under which the monotone eigenspace (the set of all non-decreasing eigenvectors) of a given matrix is non-empty and, in the positive case, the structure of the monotone eigenspace is described. Using permutations of matrix rows and columns, the results are extended to the whole eigenspace.