Fuzzy Algebra Research Articles

On simultaneously nilpotent antiring matrices

Antirings are an important type of semirings, which generalize Boolean algebra, fuzzy algebra, distributive lattice and incline. In this paper, we study the issue of nilpotent antiring matrices, provide some properties and characterizations of the simultaneous nilpotence for a finite number of antiring matrices, present some methods for calculating the simultaneously nilpotent index of a finite number of antiring matrices. Partial results obtained in this paper generalize and develop the corresponding ones on nilpotent antiring matrices and on simultaneously nilpotent fuzzy matrices (lattice matrices).

Almost Jordan homomorphisms and Jordan derivations associated to the parametric-additive functional equation on fuzzy Banach algebras

Abstract In this article, we establish the generalized Hyers-Ulam (or Hyers-Ulam-Rassais) stability of Jordan homomorphisms and Jordan derivations of the following parametric additive functional equation: ∑ i = 1 m f ( x i ) = 1 2 m ∑ i = 1 m f m x i + ∑ j = 1 , j ≠ i m x j + f ∑ i = 1 m x i for a fixed positive integer m with m ≥ 2, on fuzzy Banach algebras. The concept of Ulam-Hyers-Rassias stability originated from Rassias stability theorem that appeared in his article. Mathematics Subject Classification: Primary, 46S40; Secondary, 39B52; 39B82; 26E50; 46S50; 46H25.

Higher Ring Derivation and Intuitionistic Fuzzy Stability

We take account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.

Interval eigenvectors of circulant matrices in fuzzy algebra

Interval eigenvectors of circulant matrices in fuzzy algebra

Weak stability of interval orbits of circulant matrices in fuzzy algebra

Weak stability of interval orbits of circulant matrices in fuzzy algebra

R −Generalized Fuzzy Convex Sets

By combining R − implication operators and generalized fuzzy convex sets, the concept of a new fuzzy algebra structure, such R − generalized fuzzy convex set, is introduced. Furthermore, some fundamental properties of such fuzzy convex set and the interrelation between R − generalized fuzzy convex sets and generalized fuzzy convex sets in certain conditions are discussed.

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.

SEMI MODUL POLINOMIAL FUZZY ATAS ALJABAR MAX-PLUS FUZZY

In this paper, we discuss the extension of fuzzy max-plus algebra on polynomial with coefficient in fuzzy number. We also proof that fuzzy polynomial is semi modul over fuzzy max-plus algebra.

POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E. Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of L. A. Zadeh. On the next step, we provide a direct lift of the backward powerset operator using the notion of categorical biproduct. The obtained framework is readily extended to the variable-basis case, justifying the powerset theories currently popular in the fuzzy community. At the end of the paper, our general variety-based setting postulates the requirements, under which a convenient variety-based powerset theory can be developed, suitable for employment in all areas of fuzzy mathematics dealing with fuzzy powersets, including fuzzy algebra, logic and topology.

Standard Bases of a Vector Space Over a Linearly Ordered Incline

Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.

Fuzzy algebras as a framework for fuzzy topology

The paper introduces a variety-based version of the notion of the ( L, M)-fuzzy topological space of Kubiak and Šostak and embeds the respective category into a suitable modification of the category of topological systems of Vickers. The new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature, paving the way for studying the problem of interweaving algebra and topology in mathematics, which was raised by Denniston, Melton and Rodabaugh in their recent research on variable-basis topological systems over the category of locales.

Some Properties of Intuitionistic Fuzzy Lie Algebras over a Fuzzy Field

The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the “necessity” and “possibility” operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of homomorphic images.

On Generalized Transitive Matrices

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considered. Some properties of compositions of incline matrices are also given, and a new transitive incline matrix is constructed from given incline matrices. Finally, the issue of the canonical form of a transitive incline matrix is discussed. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

Using fuzzy cellular automata to access and simulate urban growth

In this paper we present a methodological framework designed to access urban growth dynamics and simulate urban growth. To do so, it utilizes the descriptive power of Fuzzy Logic and Fuzzy Algebra to map the effects of various parameters to the urban growth phenomenon and express them in comprehensible terms. Sensitive Sum, a new fuzzy operator is proposed to employ a parallel connection between the effects of separate variables while taking into account the (statistical) correlation between them. As a result, the model implements a reducible/extensible form of Knowledge Base which can include both data-driven and empirical rules and does not require certain variables/data to run. In order to simulate urban growth Cellular Automata techniques are incorporated that are enhanced by pseudo-agent behavior. The proposed model is applied in the broader Mesogia area in east Attica (Athens, Greece) for the period 2000–2007 during which urban land cover grew by 66% and appears to capture the urban growth dynamics occurred in a satisfactory way.

AN INTELLIGENT INFORMATION SYSTEM FOR FUZZY ADDITIVE MODELLING (HYDROLOGICAL RISK APPLICATION)

In this paper we propose and construct Fuzzy Algebraic Additive Model, for the estimation of risk in various fields of human activities or nature’s behavior. Though the proposed model is useful in a wide range of scientific fields, it was designed for to torrential risk evaluation in the area of river Evros. Clearly the model’s performance improves when the number of parameters and the actual data increases. A Fuzzy Decision Support System was designed and implemented to incorporate the model’s risk estimation capacity and the risk estimation output of the system was compared with the output of other existing methods with very interesting results.

Mathematics of fuzziness--basic issues

Mathematics of Fuzziness: Basic Issues introduces a basic notion of 'fuzziness' and provides a conceptual mathematical framework to characterize such fuzzy phenomena in Studies in Fuzziness and Soft Computing. The book systematically presents a self-contained introduction to the essentials of mathematics of fuzziness ranging from fuzzy sets, fuzzy relations, fuzzy numbers, fuzzy algebra, fuzzy measures, fuzzy integrals, and fuzzy topology to fuzzy control in a strictly mathematical manner. It contains most of the authors' research results in the field of fuzzy set theory and has evolved from the authors' lecture notes to both undergraduate and graduate students over the last three decades. A lot of exercises in each chapter of the book are particularly suitable as a textbook for any undergraduate and graduate student in mathematics, computer science and engineering. The reading of the book will surely lay a solid foundation for further research on fuzzy set theory and its applications.

A fuzzy pert approach to evaluate plant construction project scheduling risk under uncertain resources capacity

A plant construction project always involves lots of activities. Precise information about the activities duration is unfortunately unavailable due to the uncertain resources capacity. The fuzzy program evaluation and review technique (PERT) has been widely applied to solve the fuzzy project scheduling problem. This paper presents an extended fuzzy PERT approach with four major improvement aspects to support the construction project scheduling management: 1) Evaluate operation fuzzy times based on available working volumes, resources quantity and fuzzy capacity of resources, 2) Adopting a maximal ilevelcut method to compare the fuzzy precedent activities times to determine the reasonable earliest starting times of each activity, 3) Using fuzzy algebra method instead of fuzzy subtraction method to compute the fuzzy latest starting times and 4) Developing a project scheduling risk index (PSRI) to assist the decision maker to evaluate the project scheduling risk. Simulations experiments are conducted and demonstrated satisfactory results.

Study and Analysis the Mathematical Operations of Fuzzy Logic

The last decade of this 20th century provides a wide spread of applications of one of the computer techniques, which is called Fuzzy Logic. This technique depends mainly on the fuzzy set theory, which is considered as a general domain with respect to the conventional set theory. This paper presents in initiative the fuzzy sets theory and fuzzy logic as a complete mathematics system. Here it was explained the concept of fuzzy set and defined the operations of fuzzy logic. It contains eleven operations beside the other operations which related to fuzzy algebra. Such search is considered as an enhancement for supporting the others waiting search activities in this field.

Study and Analysis the Mathematical Operations of Fuzzy Logic

The last decade of this 20th century provides a wide spread of applications of one of the computer techniques, which is called Fuzzy Logic. This technique depends mainly on the fuzzy set theory, which is considered as a general domain with respect to the conventional set theory. This paper presents in initiative the fuzzy sets theory and fuzzy logic as a complete mathematics system. Here it was explained the concept of fuzzy set and defined the operations of fuzzy logic. It contains eleven operations beside the other operations which related to fuzzy algebra. Such search is considered as an enhancement for supporting the others waiting search activities in this field.

Uncertainty in value stream mapping analysis

In this article, two alternative approaches to include variability analysis in value stream mapping (VSM) are presented. Notoriously, VSM is one of the best tools to map a process and to identify its main criticalities in order to enhance lean manufacturing. Unfortunately, in the standard approach, data are recorded on the map as deterministic, disregarding the real variability of the process. This is a strong limit because, especially within manufacturing processes, variability has a significant impact on costs and leads to several wastes. Therefore, it must be carefully analysed and reduced before lean manufacturing can be set into place. To overcome this weakness, this article proposes two alternative approaches based on statistics and fuzzy algebra, respectively. Both methods are designed to support practitioners and can be easily applied to several industrial applications. Their practicality is finally demonstrated through an industrial application of relevance. Obtained results are compared and respective advantages/disadvantages are presented and discussed.