axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

a note on some types of filters in mtl algebras abstract borzooei et al recently introduced the notion of associative filters in mtl algebras and studied some of their properties in this paper we prove that the only proper filter of mtl algebra l that is associative according to their definition is l 0 in fact in any mtl algebra l if there exists a proper associative filter f in l then f l 0

on some types of fuzzy covering based rough sets fuzzy coverings are a natural extension of the coverings by replacing crisp sets with fuzzy sets recently an excellent introduction to the definition of a fuzzy covering is due to ma and two fuzzy covering based rough set models are presented in this paper by introducing some new definitions of fuzzy covering approximation spaces the properties of fuzzy covering approximation spaces and ma s fuzzy covering based rough set models are studied furthermore three new types of fuzzy covering based rough set models as generalizations of ma s models are first proposed in this paper first some properties of fuzzy covering and its fuzzy neighborhood family are proposed we present a necessary and sufficient condition for fuzzy neighborhood family induced by a fuzzy covering to be equal to the fuzzy covering itself then we study the characterizations of ma s fuzzy covering based rough set models and give a necessary and sufficient condition for two fuzzy coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation finally this paper proposes three new types of fuzzy covering based rough set models by introducing a new notion of a fuzzy complementary neighborhood

why are papers about filters on residuated structures usually trivial in this paper we introduce a notion of a t filter on residuated lattices which is a generalization of several special types of filters we provide some basic properties of t filters and show how particular results about special types of filters e g extension property triple of equivalent characteristics and quotient characteristics are uniformly covered by this simple general framework

l fuzzy covering relation we investigate lattice valued l fuzzy covering or fuzzy neighboring relation arising from a given lattice valued order we show that every l fuzzy covering is obtained by synthesis of crisp coverings arising from the corresponding cut orderings as a converse we introduce and investigate lattice valued relations which turn out to be fuzzy coverings of fuzzy orders

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets traditional rough set theory is mainly used to extract rules from and reduce attributes in databases in which attributes are characterized by partitions while the covering rough set theory a generalization of traditional rough set theory does the same yet characterizes attributes by covers in this paper we propose a way to reduce the attributes of covering decision systems which are databases characterized by covers first we define consistent and inconsistent covering decision systems and their attribute reductions then we state the sufficient and the necessary conditions for reduction finally we use a discernibility matrix to design algorithms that compute all the reducts of consistent and inconsistent covering decision systems numerical tests on four public data sets show that the proposed attribute reductions of covering decision systems accomplish better classification performance than those of traditional rough sets

adaptive boundary element methods with convergence rates this paper presents adaptive boundary element methods for positive negative as well as zero order operator equations together with proofs that they converge at certain rates the convergence rates are quasi optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods in particular no saturation type assumption is used the main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods and an inverse type inequality involving boundary integral operators on locally refined finite element spaces

a comparative study of fuzzy sets and rough sets this paper reviews and compares theories of fuzzy sets and rough sets two approaches for the formulation of fuzzy sets are reviewed one is based on many valued logic and the other is based on modal logic two views of rough sets are presented set oriented view and operator oriented view rough sets under set oriented view are closely related to fuzzy sets which leads to non truth functional fuzzy set operators both of them may be considered as deviations of classical set algebra in contrast rough sets under operator oriented view are different from fuzzy sets and may be regarded as an extension of classical set algebra

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

fuzzy rough sets based on generalized residuated lattices this paper is devoted to propose generalized l fuzzy rough sets as a further generalization of the notion of l fuzzy rough sets a quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non commutative generalized l fuzzy rough sets are characterized from both constructive and axiomatic approaches in the constructive approach various classes of generalized l fuzzy rough sets are investigated moreover the relationship between generalized l fuzzy rough sets and l topologies on an arbitrary universe is discussed with generalized lower and upper sets as an application of generalized l fuzzy rough sets fuzzy rough sets are proposed and studied on the unit interval 0 1 which are based on generalized residuated lattices induced by left continuous pseudo t norms

rough approximations on a complete completely distributive lattice with applications to generalized rough sets the generalizations of rough sets considered with respect to similarity relation covers and fuzzy relations are main research topics of rough set theory however these generalizations have shown less connection among each other and have not been brought into a unified framework which has limited the in depth research and application of rough set theory in this paper the complete completely distributive ccd lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed these definitions result from the concept of cover introduced on a ccd lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers when t similarity relation is considered the existing fuzzy rough sets are the special cases of our proposed approximations on a ccd lattice thus these generalizations of rough sets are brought into a unified framework and a wider mathematical foundation for rough set theory is established

granular variable precision fuzzy rough sets with general fuzzy relations the variable precision fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models however the variable precision fuzzy rough sets were only defined and investigated on fuzzy similarity relations in this paper the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations the equivalent expressions of the approximation operators are given with fuzzy co implications on arbitrary fuzzy relations which can calculate efficiently the approximation operators the granular variable precision fuzzy rough sets are characterized from the constructive approach which are investigated on different fuzzy relations the conclusions on the variable precision fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

communication between information systems with covering based rough sets communication between information systems is considered as an important issue in granular computing the concept of homomorphism is an effective mathematical tool to study information exchange between information systems this paper provides a study on some basic properties of covering information systems and decision systems under homomorphisms first we define consistent functions related to coverings and covering mappings between two universes and study their properties then we introduce the notions of homomorphisms of covering information systems and point out that a homomorphism is a special covering mapping between information systems furthermore we investigate some important properties of homomorphisms in covering information systems and decision systems it is proved that some basic properties of original systems such as set approximations attribute reductions can be reserved under the condition of homomorphisms in both covering information systems and covering decision systems

an axiomatic approach of fuzzy rough sets based on residuated lattices rough set theory was developed by pawlak as a formal tool for approximate reasoning about data various fuzzy generalizations of rough approximations have been proposed in the literature as a further generalization of the notion of rough sets l fuzzy rough sets were proposed by radzikowska and kerre in this paper we present an operator oriented characterization of l fuzzy rough sets that is l fuzzy approximation operators are defined by axioms the methods of axiomatization of l fuzzy upper and l fuzzy lower set theoretic operators guarantee the existence of corresponding l fuzzy relations which produce the operators moreover the relationship between l fuzzy rough sets and l topological spaces is obtained the sufficient and necessary condition for the conjecture that an l fuzzy interior closure operator derived from an l fuzzy topological space can associate with an l fuzzy reflexive and transitive relation such that the corresponding l fuzzy lower upper approximation operator is the l fuzzy interior closure operator is examined

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

a short note on t filters i filters and extended filters on residuated lattices the notion of a t filter on residuated lattices was introduced in order to generalize a big amount of particular results about special types of filters e g implicative positive implicative fantastic etc in this short note we are going to point out the relationship between t filters and i filters and also between t filters and extended filters on residuated lattices

two fuzzy covering rough set models and their generalizations over fuzzy lattices by introducing the new concepts of fuzzy covering and fuzzy neighborhood we define two new types of fuzzy covering rough set models which can be regarded as bridges linking covering rough set theory and fuzzy rough set theory we show the properties of the two models and reveal the relationships between the two models and some others moreover we present the matrix representations of the newly defined lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices finally we generalize the models and their matrix representations to l fuzzy covering rough sets which are defined over fuzzy lattices

fuzzy t filters and their properties abstract fuzzification of special types of filters on several different algebras of many valued logics has been very popular in recent years the main aim of this paper is to point out some general principles concerning particular results about fuzzification of special types of filters we introduce the notion of a fuzzy t filter which generalizes most types of special fuzzy filters e g fuzzy implicative fuzzy boolean fuzzy fantastic etc and prove some basic properties of fuzzy t filters

some types of filters in mtl algebras the aim of this work is to introduce the concepts of imtl filters and strong mtl filters in mtl algebras and show that these filters are related to imtl algebras and strong mtl algebras respectively we also introduce the concepts of eimtl and associative filters in mtl algebras the relationships between these filters and quotient algebras that are constructed via these filters are described we clarify that eimtl imtl and fantastic filters coincide in bl algebras whereas they have different properties in mtl algebras moreover the relations between them are delineated we prove that strong filters include some current filters such as implicative positive implicative and fantastic filters

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems attribute reduction is viewed as an important issue in data mining and knowledge representation this paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory the definitions of several kinds of attribute reducts are introduced the relationships among these reducts are then investigated in a fuzzy decision system it is proved that the concepts of fuzzy positive region reduct lower approximation reduct and generalized fuzzy belief reduct are all equivalent the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set in a consistent fuzzy decision system an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct but in an inconsistent fuzzy decision system a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general

the algebraic structures of generalized rough set theory rough set theory is an important technique for knowledge discovery in databases and its algebraic structure is part of the foundation of rough set theory in this paper we present the structures of the lower and upper approximations based on arbitrary binary relations some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended based on the concepts of definable sets in rough set theory two important boolean subalgebras in the generalized rough sets are investigated an algorithm to compute atoms for these two boolean algebras is presented

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

neighborhood operator systems and approximations this paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k step neighborhood systems based on a binary relation on a finite universe six families of binary relations are obtained and the corresponding six classes of k step neighborhood systems are derived extensions of pawlak s rough set approximation operators based on such neighborhood systems are proposed properties of neighborhood operator systems and rough set approximation operators are investigated and their connections are examined

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

characterizations and new subclasses of i filters in residuated lattices abstract filters play an important role in studying logical systems and the related algebraic structures various filters have been proposed in the literature in this paper we aim to develop a unifying definition for some specific filters called i filters which provide us with a meaningful method to study these filters and corresponding logical algebras in particular trivial characterizations of i filters non trivial characterizations of classes of i filters such as implicative fantastic and boolean filters and characterizations of homologous logical algebras are obtained next three new types of i filters named divisible filters strong and n contractive filters in residuated lattices are introduced particularly it is verified that n fold implicative bl algebras and n contractive bl algebras coincide finally we investigate the relationships between these specific i filters it is shown that a filter is a fantastic filter if and only if it is both a divisible filter and a regular filter

relationship among basic concepts in covering based rough sets uncertainty and incompleteness of knowledge are widespread phenomena in information systems rough set theory is a tool for dealing with granularity and vagueness in data analysis rough set method has already been applied to various fields such as process control economics medical diagnosis biochemistry environmental science biology chemistry psychology and conflict analysis covering based rough set theory is an extension to classical rough sets in covering based rough sets there exist several basic concepts such as reducible elements of a covering minimal descriptions unary coverings and the property that the intersection of any two elements is the union of finite elements in this covering these concepts appeared in the literature of covering based rough sets separately in this paper we study the relationships between them in particular we establish the equivalence of the unary covering and the covering with the property that the intersection of any two elements is the union of finite elements in this covering we also investigate the relationship between the covering lower approximation operation and the interior operator a characterization of the interior operator by the covering lower approximation operation is presented in this paper correspondingly we study the relationship between the covering upper approximation operation and the closure operator in addition we explore the conditions under which the covering upper approximation operation is monotone the study of the relationships between these concepts will help us have a better understanding of covering based rough sets

a new approach for classification of filters in residuated lattices in the mathematical literature many types of filters in residuated lattices or in special classes of residuated lattices bl algebras mtl algebras etc have been studied consequently many names have been proposed for these types of filters boolean filters heyting filters implicative filters positive implicative filters fantastic filters easy filters etc this multiplicity of names for the filters makes it very difficult to study them and the connections between them to avoid this inconvenience in this paper we propose a new approach for the study of these filters in residuated lattices in order to be easily accessible to the readers we also present a unified treatment of some particular results about filters in residuated lattices theorem 16 corollary 17

radicals in mtl algebras we study some special sets in mtl algebras the notion of the radical of a filter in mtl algebras is defined and several characterizations of the radical of a filter are given the sets of dense elements and double complemented elements in mtl algebras are defined and we state and prove some theorems on the properties of these sets and filters of mtl algebras we also prove that the radical of a positive implicative filter of an mtl algebra is a subset of double complemented elements

covering based rough set approximations we propose a framework for the study of covering based rough set approximations three equivalent formulations of the classical rough sets are examined by using equivalence relations partitions and s algebras respectively they suggest the element based the granule based and the subsystem based definitions of approximation operators covering based rough sets are systematically investigated by generalizing these formulations and definitions a covering of universe of objects is used to generate different neighborhood operators neighborhood systems coverings and subsystems of the power set of the universe they are in turn used to define different types of generalized approximation operators within the proposed framework we review and discuss covering based approximation operators according to the element granule and subsystem based definitions

semi divisible t norms on discrete scales semi divisibility of left continuous triangular norms is a weakened version of the divisibility axiom the t norms on l 0 1 n 2 n 1 which fulfil the semi divisibility axiom are studied the only semi divisible t norm with ran n t l is shown to correspond to the lukasiewicz t norm several examples of semi divisible discrete t norms are shown

approximations and uncertainty measures in incomplete information systems there are mainly two methodologies dealing with uncertainty measurement issue in rough set theory pure rough set approach and information theory approach pure rough set approach is based on the concepts of accuracy roughness and approximation accuracy proposed by pawlak information theory approach is based on shannon s entropy or its variants several authors have extended the information theory approach into incomplete information systems however there are few studies on extending the pure rough set approach to incomplete information systems this paper focuses on constructing uncertainty measures in incomplete information systems by pure rough set approach three types of definitions of lower and upper approximations and corresponding uncertainty measurement concepts including accuracy roughness and approximation accuracy are investigated theoretical analysis indicates that two of the three types can be used to evaluate the uncertainty in incomplete information systems experiments on incomplete real life data sets have been conducted to test the two selected types the first type and the third type of uncertainty measures results show that the two types of uncertainty measures are effective

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

on para quasi topological mv algebras abstract in this paper the notions of para quasi semi topological mv algebras are defined and their related properties are studied also topologies with which an mv algebra can be a para semi topological mv algebra are obtained clearly a topological mv algebra is a para quasi semi topological mv algebra but the converse is not true as shown by an example in addition we study ideals and filters in para quasi topological mv algebras and we show that a quasitopological mv algebra is a topological mv algebra if the ideal 0 or equivalently the filter 1 is open

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension