Soft ideals of BCK/BCI-algebras based on fuzzy set theory

Characterizations of a (?, ? ∨ q?)-fuzzy subalgebra (ideal) are considered. Given an ∈-soft set, an (?, ? ∨ q?)-fuzzy subalgebra is established. Using the notion of (t, s)-fuzzy subalgebras, characterizations for an∈-soft set to be a (idealistic) soft BCK/BCI-algebra are provided. Using the notion of fuzzy p-ideals, a characterization of an∈-soft set to be a p-idealistic soft BCI-algebra is constructed. An equivalent condition for a q-soft set to be a p-ideal is given. Characterizations of a (∈,∈ ∨ q)-fuzzy p-ideal are initiated. Conditions for a (∈,∈∨ q)-fuzzy ideal to be a (∈,∈∨ q)-fuzzy p-ideal are stated.     

Soft set theory

In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.     

Fuzzy -ary polygroups related to fuzzy points

Recently, fuzzy n-ary sub-polygroups were introduced and studied by Davvaz, Corsini and Leoreanu-Fotea [B. Davvaz, P. Corsini, V. Leoreanu-Fotea, Fuzzy n-ary sub-polygroups, Comput. Math. Appl. 57 (2008) 141–152]. Now, in this paper, the concept of (,q)-fuzzy n-ary sub-polygroups, -fuzzy n-ary sub-polygroups and fuzzy n-ary sub-polygroup with thresholds of an n-ary polygroup are introduced and some characterizations are described. Also, we give the definition of implication-based fuzzy n-ary sub-polygroups in an n-ary polygroup, in particular, the implication operators in Łukasiewicz system of continuous-valued logic are discussed.     

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

Cubic structures applied to ideals of BCI-algebras

The notions of cubic a-ideals and cubic p-ideals are introduced, and several related properties are investigated. Characterizations of a cubic a-ideal are established. Relations between cubic p-ideals, cubic a-ideals and cubic q-ideals are discussed. The cubic extension property of a cubic a-ideal is discussed.     

Soft -ideals of soft BCI-algebras

Molodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft p-ideals and p-idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) p-ideals in BCI-algebras. We provide relations between fuzzy p-ideals and p-idealistic soft BCI-algebras.     

On the capability of fuzzy set theory to represent concepts

The aim of this paper is to examine the conclusions drawn by Osherson and Smith ["On the adequacy of prototype theory as a theory of concepts", Cognition 9 (1981), pp. 35-58] concerning the inadequacy of the apparatus of fuzzy set theory to represent concepts. Since Osherson and Smith derive their conclusions from specific examples, we show for each of these examples that the respective conclusion they arrive at is not warranted. That is, we demonstrate that fuzzy set theory is sufficiently expressive to represent the various strong intuitions and experimental evidence regarding the relation between simple and combined concepts that are described by Osherson and Smith. To pursue our arguments, we introduce a few relevant notions of fuzzy set theory.     

An application of fuzzy set theory to inventory control models

A method for solving an inventory control problem, of which input data are described by triangular fuzzy numbers will be presented here. The continuous review model of the inventory control problem with fuzzy input data will be focused in, and a new solution method will be presented. For the reason that the result should be a fuzzy number because of fuzzy input data, and the certain number about order quantity is prefered in the real-world, it is necessary to transform the fuzzy result to crisp one. The interval mean value concept is used here to help to solve this problem. Under the condition of total cost minimum, the interval order quantity maximum can be obtained.     

Arithmetic operators in interval-valued fuzzy set theory

We introduce the addition, subtraction, multiplication and division on L^I, where L^I is the underlying lattice of both interval-valued fuzzy set theory [R. Sambuc, Fonctions Φ-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Université de Marseille, France, 1975] and intuitionistic fuzzy set theory [K.T. Atanassov, Intuitionistic fuzzy sets, 1983, VII ITKR’s Session, Sofia (deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the ?ukasiewicz t-norm and the product t-norm on the unit interval to L^I and some related operators can be written in a similar way as their counterparts on ([0,1],?).     

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338–353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682–687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80–98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69–81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313–334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169–186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741–747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.      

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

On characterizations of generalized fuzzy ideals of BCI-algebras

The aim of this study is to introduce the notions of (∈,∈∨ q)-FSI-ideals and (∈,∈∨ q)-FSC-ideals of BCI-algebras and to investigate some of their related properties. Some characterization theorems of these generalized fuzzy ideals are derived. The relationship among these generalized fuzzy ideals of BCI-algebras is considered. We show that a fuzzy set of a BCI-algebra is an (∈,∈∨ q)-FSI-ideal if and only if it is both an (∈,∈∨ q)-FSC-ideal and an (∈,∈∨ q)-FPI-ideal. Finally, we consider the concepts of implication-based FSI (resp., FSC)-ideals of BCI-algebras.     

A note on fuzzy set theory in scanning

Scanning of a natural picture handled by an image processing system always causes a loss of information, i.e., the micro picture elements become uncertain and vague. This is essential when the picture is to be used for measuring or evaluation of structural parameters. To make the induced uncertainty and vagueness transparent up to the computed results, specifications and procedures from fuzzy set theory are presented and suggested. In particular, distances of pixels are considered. For a demonstration of how this uncertainty can influence conclusions from the given picture, the computation of the approximate gradient of a curve is considered     

基于模糊论的数据挖掘研究综述

近年来,将模糊集理论应用到数据挖掘研究中成为数据挖掘领域的一个研究热点。为追踪其研究进展,探讨未来的研究方向,对模糊集理论在数据挖掘中的主要研究方向(聚类分析、关联挖掘、分类)进行了综述,主要阐述数据和模式的表示、模式相似性计算等关键问题。可以看出,充分利用模糊论强大的模糊数据建模功能,并且与其它智能化处理技术相结合,是当前这一领域研究的主流技术。指出了存在的若干问题,并对研究前景进行展望。     

Using fuzzy set theory and scale-free network properties to relate MEDLINE terms

Automated construction and annotation of biological networks is becoming increasingly important in bioinformatics as the amount of biological data increases. At the center of this are metrics required for relating biological entities such as genes, diseases, signaling molecules and chemical compounds. Co-occurrence of terms within abstracts is widely used to establish tentative relationships because it is easy to use, implement, understand, and is reasonably accurate. However, it is also very imprecise – the cutoffs for how many co-occurrences of terms are necessary to establish a relationship is arbitrary and the nature of the relationship is generic. Since the frequency of co-occurrence for terms usually follows a scale-free distribution, this property can be exploited to define degree of membership in fuzzy sets. Beginning with a set of co-occurrences for any biomedical term (or its synonyms), relations are defined by the overlap of sets, normalizing by the area under the curve that the two sets share. The ability of this method to rank the relative specificity of biological relationships is tested by comparing cumulative term co-occurrences within 7.5 million MEDLINE abstracts with focused summaries of gene function and disease association within LocusLink. On average, the fuzzy set ranking (FSR) was in the top 0.6% of all potential associations, showing a good correlation between domain overlap and the biological association between two terms.     

Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2-3) (1990) 191-209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.     

Two-dimensional shape decomposition using fuzzy subset theory applied to automated chromosome analysis

This paper describes a technique to transform a two-dimensional shape into a generalized fuzzy binary relation whose clusters represent the meaningful simple parts of the shape. The fuzzy binary relation is defined on the set of convex and concave boundary points, implying a piecewise linear approximation of the boundary, and describes the dissemblance of two vertices to a common cluster. Next some fuzzy subsets are defined over the points which determine the connection between the clusters.The decomposition method first determines nearly convex regions, which are subgraphs of the total graph, and then selects the greatest nearly convex region which satisfies best the defined fuzzy subsets and relations. Using this procedure on touching chromosomes defining the simple parts to be the separated chromosomes, the decomposition often corresponds well to the decomposition that a human might make.     

Fuzzy rough set theory for the interval-valued fuzzy information systems

The concept of the rough set was originally proposed by Pawlak as a formal tool for modelling and processing incomplete information in information systems, then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set theory by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory, i.e., the interval-valued fuzzy rough set model is presented based on the interval-valued fuzzy information systems which is defined in this paper by a binary interval-valued fuzzy relations RF⁽ⁱ⁾(U×U) on the universe U. Several properties of the rough set model are given, and the relationships of this model and the others rough set models are also examined. Furthermore, we also discuss the knowledge reduction of the classical Pawlak information systems and the interval-valued fuzzy information systems respectively. Finally, the knowledge reduction theorems of the interval-valued fuzzy information systems are built.     

Fuzzy time series forecasting based on axiomatic fuzzy set theory

Neural Computing and Applications - In fuzzy time series, a way of representing their original numeric data through a collection of fuzzy sets plays a pivotal role and impacts the prediction...