Fuzzy Algebra Research Articles

Fuzzy Logic in Traffic Engineering: A Review on Signal Control

Since 1965 when the fuzzy logic and fuzzy algebra were introduced by Lotfi Zadeh, the fuzzy theory successfully found its applications in the wide range of subject fields. This is mainly due to its ability to process various data, including vague or uncertain data, and provide results that are suitable for the decision making. This paper aims to provide comprehensive overview of literature on fuzzy control systems used for the management of the road traffic flow at road junctions. Several theoretical approaches from basic fuzzy models from the late 1970s to most recent combinations of real-time data with fuzzy inference system and genetic algorithms are mentioned and discussed throughout the paper. In most cases, fuzzy logic controllers provide considerable improvements in the efficiency of traffic junctions’ management.

A fixed point approach to almost ternary homomorphisms and ternary derivations associated with the additive functional equation of n-Apollonius type in fuzzy ternary Banach algebras

A fixed point approach to almost ternary homomorphisms and ternary derivations associated with the additive functional equation of n-Apollonius type in fuzzy ternary Banach algebras

FUZZY ALGEBRA HOMOMORPHISMS AND FUZZY DERIVATIONS

In this paper, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach algebras and of derivations on fuzzy Banach algebras associated to the Cauchy-Jensen functional equation.

ON RIGHT f-DERIVATIONS OF INCLINE ALGEBRAS

Abstract. In this paper, we introduce the concept of a rightderivation in incline algebras and give some properties of inclinealgebras. Also, the concept of d-ideal is introduced in an inclinealgebra with respect to right derivation. 1. IntroductionZ. Q. Cao, K. H. Kim and F. W. Roush [4] introduced the notionof incline algebras in their book and later it was developed someauthors [1, 2, 3, 5]. Ahn et al [1] introduced the notion of quotientincline and obtained the structure of incline algebras. N. O. Alshehri [3]introduced the notion of derivation in incline algebra. Incline algebra is ageneralization of both Boolean and fuzzy algebra and it is a special typeof semiring. It has both a semiring structure and a poset structure. Itcan also be used to represent automata and other mathematical systems,to study inequalities for non-negative matrices of polynomials. In thispaper, we introduce the concept of a right derivation in incline algebrasand give some properties of incline algebras. Also, the concept of d-idealis introduced in an incline algebra with respect to right derivation.2. PreliminariesAn incline algebra is a set K with two binary operations denoted by + and satisfying the following axioms:(K1) x+ y = y + x;(K2) x+ (y + z) = (x+ y) + z;

Algebras with compatible quantale-enriched structures

Based on quantale-enriched category, we consider algebras with compatible quantale-enriched structures, which can be viewed as fuzzification of ordered algebraic structures. We mainly study groupoids and semigroups with compatible quantale-enriched structures from this viewpoint. Some basic concepts such as ideals, homomorphisms, residuated quantale-enriched groupoids are developed and some examples of them are given. Our approach gives a complement to the approach initiated by Rosenfeld to study fuzzy abstract algebra, and these two approaches are combined in the present paper to study fuzzy aspects of abstract algebra structures.

Fuzzy Acts over Fuzzy Semigroups and Sheaves

Although fuzzy set theory and sheaf theory have been developed and studied independently, Ulrich Hohle shows that a large part of fuzzy set theory is in fact a subeld of sheaf theory. Many authors have studied math- ematical structures, in particular, algebraic structures, in both categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra A, the set of fuzzy algebras over A and the set of subalgebras of the constant sheaf of algebras over A are order isomorphic. Then, among other things, we study the category of fuzzy acts over a fuzzy semigroup, so to say, with its universal algebraic as well as classic algebraic denitions.

The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets

In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of λ-cut sets of IFS and show that λ-cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of λ-cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup.

Robustness of fuzzy interval circulant-Hankel matrices

Fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a⊕b=max{a,b}, a⊗b=min{a,b}. A square matrix is circulant-Hankel if the input values in every row are the same as the values in the previous row, but they are cyclically shifted by one position to the left.The robustness and the X-robustness of fuzzy circulant-Hankel matrices are studied. We define the possible and universal robustness and X-robustness of interval matrices. The necessary and sufficient conditions for the possible and universal robustness and X-robustness of interval circulant-Hankel matrices are given.

ПРОЦЕДУРА ФОРМУВАННЯ МОДЕЛі ПРОГНОЗУВАННЯ ПАСАЖИРОПОТОКіВ НА ЗАЛіЗНИЧНИХ ЛіНіЯХ

Th e article deals with formation of a traffic assignment forecast model on railway lines while introducing high-speed trains. The procedure for adjusting a forecast model based on a genetic algorithm with coding has been proposed. The main aim of the research is to improve a traffic assignment forecast model on railway lines while introducing high speed trains based on evolution modelling. Methods of fuzzy algebra, genetic algorithms and mathematic programming have been implemented to solve the scientific problem under consideration. It enabled to develop a procedure for adjusting a mathematical model based on an objective function which minimizes an average relative error in forecast and actual data of the testing set. Apart from a search for a fuzzy relation in a relational equation, the article proposes a method to adjust membership functions of output linguistic terms for a variable models (within a genetic algorithm) to improve the accuracy of adjustment for a forecast model. The adjustment procedure proposed has improved the forecast model accuracy. A relative error in a forecast model for the testing set is less than 10,0144 %. The results of the research can be implemented on railways while designing automated programme complex for traffic assignment forecast between cities in strategic planning.

FUZZY EQUATIONAL CLASSES ARE FUZZY VARIETIES

In the framework of fuzzy algebras with fuzzy equalities and a complete lattice as a structure of membership values, we investigate fuzzy equational classes. They consist of special fuzzy algebras fullling the same fuzzy identities, dened with respect to fuzzy equalities. We introduce basic notions and the corresponding operators of universal algebra: construction of fuzzy subalgebras, homomorphisms and direct products. We prove that every fuzzy equational class is closed under these three operators, which means that such a class is a fuzzy variety.

EQ-algebras from the point of view of generalized algebras with fuzzy equalities

EQ-algebras introduced by Novák are algebras of truth values for a higher-order fuzzy logic (fuzzy type theory). In this paper, the compatibility of multiplication w.r.t. the fuzzy equality in an arbitrary EQ-algebra is examined. Particularly, an example indicates that the compatibility axiom does not always hold, and then a class of EQ-algebras satisfying the compatibility axiom is characterized by introducing a residuated integral-meet-semilattice-ordered-monoid-valued fuzzy algebra called a generalized algebra with a fuzzy equality.

The stability of fuzzy approximately Jordan mappings

In this paper we introduce the concept of fuzzy approximately Jordan mappings in fuzzy algebras, and study some of their basic properties. The main purpose of this paper is to study the stability of fuzzy approximately Jordan mappings in fuzzy algebras.

Fixed points and stability of functional equations in fuzzy ternary Banach algebras

Abstract By using Diaz and Margolis fixed point theorem, we establish the generalized Hyers-Ulam-Rassias stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras associated to the following ( m , n ) -Cauchy-Jensen additive functional equation: ∑ 1 ≤ i 1 < ⋯ < i m ≤ n 1 ≤ k l ≤ n k l ≠ i j , ∀ j ∈ { 1 , … , m } f ( ∑ j = 1 m x i j m + ∑ l = 1 n − m x k l ) = ( n − m + 1 ) n ( n m ) ∑ i = 1 n f ( x i ) . MSC:39B52, 46S40, 26E50.

Nearly n-homomorphisms and n-derivations in fuzzy ternary Banach algebras

Abstract Let n = 3 k + 2 for some k ∈ N . We investigate the generalized Hyers-Ulam stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras related to the generalized Cauchy-Jensen additive functional equation. MSC:39B52, 46S40, 26E50.

X-robustness of interval circulant matrices in fuzzy algebra

Fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a⊕b=max{a,b},a⊗b=min{a,b}.A matrix is X-robust if its eigenspace is achieved starting at each vector from a given interval vector X.The possible and universal X-robustness of interval matrices over fuzzy algebra are defined. Polynomial algorithms for checking the possible and universal X-robustness of interval circulant matrices are described.

On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equationf(xy)=xf(y)+f(x)yin an intuitionistic fuzzy normed algebra with unit.

On an algebra of linguistic truth-valued intuitionistic lattice-valued logic

In this paper, we propose a kind of linguistic truth-valued intuitionistic fuzzy lattice based on the point view of intuitionistic fuzzy set and linguistic truth-valued lattice implication algebra. As an algebra fundament of linguistic truth-valued intuitionistic fuzzy logic, some properties of linguistic truth-valued intuitionistic fuzzy algebra are discussed. The results show that linguistic truth-valued intuitionistic fuzzy lattice is a residual lattice, but it is not MTL-algebra, R0-algebra, BL-algebra, MV-algebra and quasi lattice implication algebra.

Development of a fuzzy logic-based inherent safety index for food industries

Inherent safety is a proactive approach for loss or elimination of hazard values and risk management. Expansion of safety relating to process and production in a supply chain in an uncertain environment has a critical effect on the competitive advantage of each food industrial organization. In this article, the fuzzy rating weights of each pair of the considered hazards and uncertain hazard values are described by linguistic expressions modelled by triangular fuzzy numbers. The fuzzy extent approach for the synthetic extent values of the pairwise comparison for handling fuzzy analytic hierarchical process is used to calculate the weight vector. The proposed fuzzy model is arranged in a hierarchical structure, so that the output of one interference system is used as an input for the second interference step. In the first of interference steps, the total hazard value for each identified process is calculated using the fuzzy algebra rules. In the second interference step, the inherent safety index for each process is determined on the basis of total hazard value, using the IF–THEN fuzzy logic rules. It is shown that the proposed fuzzy model is highly suitable as a decision making tool for making decisions about the inherent safety index of food industrial processes.

Matriks atas Aljabar Max-Plus Interval

This paper aims to discuss the matrix algebra over interval max-plus algebra (interval matrix) and a method tosimplify the computation of the operation of them. This matrix algebra is an extension of matrix algebra over max-plus algebra and can be used to discuss the matrix algebra over fuzzy number max-plus algebra via its alpha-cut.The finding shows that the set of all interval matrices together with the max-plus scalar multiplication operationand max-plus addition is a semimodule. The set of all square matrices over max-plus algebra together with aninterval of max-plus addition operation and max-plus multiplication operation is a semiring idempotent. As reasoningfor the interval matrix operations can be performed through the corresponding matrix interval, because thatsemimodule set of all interval matrices is isomorphic with semimodule the set of corresponding interval matrix,and the semiring set of all square interval matrices is isomorphic with semiring the set of the correspondingsquare interval matrix.

A model for “crisp reasoning” with fuzzy sets

In the setting of a general type of fuzzy algebras, this paper deals with a new theoretic view on the commonsense reasoning, consisting of a kind of Popper's search for conjectures and refutations. It is supposed that the reasoning is done in natural language, but only with nonambiguous precise and imprecise terms, respectively, represented by crisp and fuzzy sets. © 2012 Wiley Periodicals, Inc.