TH-Interval-Valued Fuzzy Ideal

The document outlined the definition of F-pseudo Z-b-subalgebra in addition to introducing fundamental properties. Multiple theorems have been established throughout the discussion using appropriate examples. We introduced a new concept after presenting the material.: (fuzzy set μdz fj) of pseudo Z-algebra. This research examines both the homomorphic images and inverse images from this study. We denote b-sg to represent b-subalgebra in this article.

In this article, we apply the concept of a t -intuitionistic fuzzy set to PMS-ideals in PMS-algebras. The notion of the t -intuitionistic fuzzy PMS-ideal of PMS-algebra is introduced, and several related properties are studied. The relationships between a t -intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-ideal are discussed in detail. A condition for an intuitionistic fuzzy set to be a t -intuitionistic fuzzy PMS-ideal is provided. The t -intuitionistic fuzzy PMS-ideals of PMS-algebra are described using their α , β level cuts. The homomorphism of a t -intuitionistic fuzzy PMS-ideal of a PMS-algebra is studied, and its homomorphic image and inverse image are explored. The Cartesian product of any two t -intuitionistic fuzzy PMS-ideals is discussed, and some related results are derived. The Cartesian product of the t -intuitionistic fuzzy PMS-ideals is also characterized using its α , β level cuts. The strongest t -intuitionistic fuzzy PMS-relation in a PMS-algebra is defined. Finally, the relationships between the strongest t -intuitionistic fuzzy PMS-relation and t -intuitionistic fuzzy PMS-ideal are studied.

In this paper the notion of fuzzy BE-algebras are introduced. We state and prove some theorems in fuzzy BE-algebras and level subalgebras. Some characterizations of fuzzy subalgebras of BE-algebras are given. Using fuzzy subalgebras, characterizations of Artinian and Noetherian BE-algebras are also established. Finally the notion of fuzzy topological BE-algebras is introduced. We state and prove the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.

Composition, cartesian and direct products of fuzzy multivalued mappings

In this paper an exclusive study of some standard (lattice) algebraic properties for Intuitionistic fuzzy (inverse) images of Intuitionistic fuzzy subsets is done. Further as in crisp setup, characterizations for injectivity and surjectivity of maps in terms of some (lattice) algebraic properties of Intuitionistic fuzzy images and Intuitionistic fuzzy inverse images are performed.

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.

Modeling prior information as a fuzzy set and using Zadeh’s extension principle, a general approach is presented how to rate linear affine estimators in linear regression. This general approach is applied to fuzzy prior information sets given by ellipsoidal α-cuts. Here, in an important and meaningful subclass, a uniformly best linear affine estimator can be determined explicitly. Surprisingly, such a uniformly best linear affine estimator is optimal with respect to a corresponding relative squared error approach. Two illustrative special cases are discussed, where a generalized least squares estimator on the one hand and a general ridge or Kuks–Olman estimator on the other hand turn out to be uniformly best.

There are many results on the extended operations of two fuzzy numbers based on the Zadeh's extension principle. For the calculation, we have to use existing operations between two <TEX>${\alpha}$</TEX>-cuts. In this paper, we define parametric operations between two <TEX>${\alpha}$</TEX>-cuts which are different from the existing operations. But we have the same results as the extended operations of Zadeh's principle.

This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies.

Enhanced linguistic computational models and their similarity with Yager’s computing with words

An often raised objection to fuzzy logic can be articulated as follows. Fuzzy sets are intended to represent vague collections of objects. On the one hand, boundaries of such collections are inherently imprecise. On the other hand, fuzzy sets are defined by membership degrees which are precise. There is, therefore, an obvious discrepancy between the imprecision of a collection and the precision of a fuzzy set representing the collection. The objection leads to a question posed in the title of this paper: do exact shapes of fuzzy sets matter? That is, do the exact values of membership degrees of fuzzy sets matter? This is a fundamental question, particularly when fuzzy sets are supplied as an input to further processing. This paper presents a particular answer to this question. We focus on the case when the fuzzy sets in question are part of the input I from which the output O is produced in a way which can be described by logical formulas. This case covers several widely used fuzzy logical models including extension principle, products of fuzzy relations, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. We present formulas which say how similar is the original output O to a new output O′ when the original input I is replaced by a new input I′. The presented formulas provide us with an analytical tool which enables us to answer the question of whether (and to what extent) the exact shapes of fuzzy sets in a particular fuzzy logical model matter. We present application of our formulas to selected topics including the above-mentioned extension principle, fuzzy relational products, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. Based on the analysis, we argue that in several applications of fuzzy logic, the exact shapes of fuzzy sets do not really matter.

The extension principle is one of the most fundamental principles in the fuzzy set theory. It provides a powerful technique in order to extend the domain of a classical function from real numbers to fuzzy sets. In the present paper, we propose a new program in R software to graphically show the behavior of the simplest type of the extension principle, where the function is one-to-one. Numerical examples are given to show the results of the proposed program by import-export figures and animations.

In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, a-cuts, interval arithmetic, possibility theory and fuzzy functions may go on to the rest of the book. A good general reference for fuzzy sets and fuzzy logic is [1], [6].

In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30.

On fuzzifications of discrete dynamical systems