on fuzzy valued operations and fuzzy valued fuzzy sets this paper is devoted to investigate fuzzy valued operations and the lattice structures of the algebra of fuzzy values the algebra of fuzzy values is proven to be a complete completely distributive lattice and the list of all join irreducible elements is provided fuzzy valued t norms and t conorms are induced by arbitrary t norms and t conorms regardless of their continuity respectively moreover fuzzy valued fuzzy implications are constructed from fuzzy valued t norms induced by left continuous t norms and residuated lattices are constructed on the algebra of fuzzy values as a consequence fuzzy valued fuzzy rough sets and fuzzy valued fuzzy complete lattices are proposed and investigated in fuzzy valued fuzzy sets

a new approach to l t2 l urysohn and l completely hausdorff axioms in this paper new l t 2 l urysohn and l completely hausdorff axioms are introduced in l topological spaces relations between these axioms are established the l extended real line and its subspaces as well as pointwise metric spaces satisfy these axioms

convergence in topological molecular lattices abstract the theory of topological molecular lattices introduced by wang has general topology and fuzzy topology as special cases in this paper the author establishes the theory of ideal convergence in topological molecular lattices and gives characterizations of some topological notions e g closed elements hausdorffness continuous goh in topological molecular lattices etc in terms of ideals the author establishes the relation between ideal convergence and molecular net convergence introduced by wang

on the accumulation points of fuzzy sets abstract in this paper we first study the stratified structure of the derived sets and the distributional rules of the accumulation points of the f sets then we give some conditions that are equivalent to a b d a d b d for any a b 0 1 x finally we introduce the general f derived operator on x and by which we characterize the general fts x

on two weakers forms of continuous order homomorphisms the aim of this paper is to find further connections between two different current researches on fuzzy topology the one based on the concepts of neighborhood filter and zadeh function and the other based on the concepts of remote neighborhood ideal and order homomorphism on fuzzy lattices to approach this the paper studies two weaker forms of continuous order homomorphisms the semi continuous order homomorphism and the weakly semi continuous order homomorphism between l fuzzy topological spaces these order homomorphisms can be characterized in each of the notions remote neighborhood neighborhood ideal filter and molecular net in addition some relations connecting weakly semi continuous order homomorphisms semi hausdorff l fts and urysohn l fts have also been established

urysohn closedness on completely distributive lattices the purpose of this paper is to investigate the urysohn closedness of co topology on lattices which are not required to have an order reversing involution the concept of urysohn closedness on lattices is introduced and characterizations of urysohn closedness are given in terms of ideals and nets these characterizations are obtained mainly through the introduction of a kind of convergence for ideals and nets that we call l urysohn convergence

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

metrization theorems in l topological spaces abstract in this paper some characterizations of a pointwise uniformity are presented in l fuzzy set theory a new hausdorff separation axiom compatible with pointwise metric is introduced urysohn metrization theorem and alexandroff urysohn metrization theorem in general topology are generalized to l topology

reconstruction in a generalized model for translation invariant systems abstract we consider translation invariant ti operators on the set of maps from an abelian group g to called lg fuzzy sets where 0 is a complete lattice ordered group by defining minkowski and morphological operations on and considering order preserving operators we prove a reconstruction theorem this theorem which is called the strong reconstruction theorem srt is similar to the convolution theorem in the theory of linear and shift invariant systems and states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y the introduced framework is a general model for the theory of translation invariant systems and srt shows the consistency of it for the special cases when g r z srt implies the results of maragos and schafer

construct fuzzy lattices from a given symmetric complete lattice abstract fuzzy lattices are the basic objects which l fuzzy topology bases on we study relations of fuzzy lattices and symmetric complete lattices from category points of view a relevant fuzzy lattice l and a relevant symmetric complete heyting algebra l are constructed from a given symmetric complete lattice l we prove that the constructions and are two functors and they have reflective properties over relevant categories

some additive l fuzzy topological properties abstract the notion of sum for crisp topological spaces have extended to fuzzy topology by ghanim et al 1984 the concept of sum of l fuzzy topological spaces is introduced and its basic properties are discussed by the author 1993 in this paper the additivity of three kinds of l fuzzy topological properties strong separation properties countability and weakly n compactness are studied

conjugate topology of pointwise metric molecular lattices in this paper the pointwise pseudo metric for molecular lattices introduced by shi is studied a series of expressions concerning neighborhood balls and remote neighborhood balls are obtained some algebraic properties of the pointwise pseudo metric function of molecular lattices are presented and their topological structure are discussed in terms of the conjugate families of mappings these results show that there exist conjugate relations between neighborhoods and remote neighborhoods

a new fuzzy model of pattern recognition and hitch diagnoses of complex systems in this paper we give a new fuzzy model of pattern recognition and hitch diagnoses of complex systems which are based on afs structures ei algebras and eii algebras this model can be applied to the recognition and the diagnoses of complex systems whose relative sets are countable

short presburger arithmetic is hard we study the computational complexity of short sentences in presburger arithmetic short pa here by short we mean sentences with a bounded number of variables quantifiers inequalities and bo

a new definition of fuzzy compactness in this paper a new definition of fuzzy compactness is presented in l topological spaces when l is a complete demorgan algebra this definition does not rely on the structure of the basis lattice l and no distributivity in l is required the intersection of a fuzzy compact l set and a closed l set is fuzzy compact the continuous image of a fuzzy compact l set is fuzzy compact if the set of all prime elements of l is order generating then the alexander subbase theorem is true when l is a completely distributive demorgan algebra it is equivalent to lowen s fuzzy compactness and in this case its many characterizations are presented by means of open l sets and closed l sets

the uniform boundedness principle in l topological vector spaces in this paper we introduce and study the equicontinuity of a family of l fuzzy linear order homomorphisms we prove a generalization of the uniform boundedness principle in the classical topological vector spaces to l topological vector spaces

net theoretical convergence in l m fuzzy cotopological spaces in this paper we introduce the notion of topological l m fuzzy convergence structure and prove that for a given set there exists a bijection between the set of all topological l m fuzzy convergence structures and the set of all l m fuzzy cotopologies on this set this induces an isomorphism between the category of topological l m fuzzy convergence spaces and the category of l m fuzzy cotopological spaces

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

the inverse limit in the category of topological molecular lattices abstract in this paper we first introduce the definition of an inverse system in the category of topological molecular lattices and give the structure of the inverse limit of an inverse system then we establish the concept of a mapping between two inverse systems and induce the concept of a continuous mapping between the inverse limits of two inverse systems finally we discuss a series of properties of continuous mappings therefore the difficult problem in the theory of topological molecular lattices is solved

h λ completely hausdorff axiom on l topological spaces this paper defines the new concept of completely hausdorff axiom of an l topological space by means of l continuous mappings from an l topological space to the refined hutton s unit l interval by wang some characterizations of the completely hausdorff axiom defined in this paper are given and many nice properties of this kind of completely hausdorff axiom are proved for example it is hereditary and product invariant the refined hutton s unit l interval satisfy this kind of completely hausdorff axiom and when an l topological space satisfy this kind of completely hausdorff axiom every f convergent ideal does not have f limit points with different supports etc the relation between the completely hausdorff axiom defined in the paper and other separation axioms is discussed also

mixed integer quadratic programming is in np mixed integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral in this paper we prove that the decision version of mixed integer quadratic programming is in np thereby showing that it is np complete this is established by showing that if the decision version of mixed integer quadratic programming is feasible then there exists a solution of polynomial size this result generalizes and unifies classical results that quadratic programming is in np vavasis in inf process lett 36 2 73 77 17 and integer linear programming is in np borosh and treybig in proc am math soc 55 299 304 1 von zur gathen and sieveking in proc am math soc 72 155 158 18 kannan and monma in lecture notes in economics and mathematical systems vol 157 pp 161 172 springer 9 papadimitriou in j assoc comput mach 28 765 768 15

pointwise uniformities in fuzzy set theory a theory of pointwise quasi uniformities is based on fuzzy sets many theorems corresponding to the usual theorems in general topology are generalized on fuzzy sets in particular each fuzzy topological space is pointwise quasi uniformizable a pointwise uniformity is characterized in terms of a kind of complete regularity a uniformity on the fuzzy unit interval is constructed

subcategories of the category of l convex spaces abstract in this paper several types of l convex spaces are introduced including stratified l convex spaces convex generated l convex spaces weakly induced l convex spaces and induced l convex spaces their relations are discussed category theoretically firstly it is shown that there is a galois correspondence between the category sl cs of stratified l convex spaces resp the category wil cs of weakly induced l convex spaces and the category l cs of l convex spaces in particular sl cs and wil cs are both coreflective subcategories of l cs secondly it is proved that there is a galois correspondence between the category cs of convex spaces and the category sl cs resp wil cs specially cs can be embedded into sl cs and wil cs as a coreflective subcategory finally it is shown that the category cgl cs of convex generated l convex spaces the category il cs of induced l convex spaces and cs are isomorphic

urysohn separation property in topological molecular lattices abstract the theory of topological molecular lattices which is a generalization of the theories of point set topology fuzzy topology and l fuzzy topology is established by wang in this paper the concepts of some kinds of urysohn separation axioms in topological molecular lattices are introduced and systematically discussed several important results such as the u i i 1 0 1 2 3 4 urysohn regular and urysohn normal separation properties are both hereditary and topological invariant etc are given

applications of axiomatic fuzzy set clustering method on management strategic analysis abstract in this paper first the axiomatic fuzzy set afs clustering method x d liu w wang t y chai the fuzzy clustering analysis based on afs theory ieee transactions on systems man and cybernetics part b 35 5 2005 1013 1027 is investigated further by improving the algorithm to be more applicable then it is used to analyze the evaluation results of 30 companies which have been studied and analyzed by g s liang et al g s liang t y chou t c han cluster analysis based on fuzzy equivalence relation european journal of operational research 166 2005 160 171 compared with liang s algorithm the proposed method is more transparent understandable and interpretable this method can be applied to the management strategic analysis based on the datasets described by mixed features such as real numbers boolean logical values linguistic descriptions the illustrative examples show that the interpretations of the clustering results of the 30 companies are almost consistent with the expert s intuitions

hurewicz spaces and the selectively ω grouping property definitions are presented for two covering properties in an l topological space hurewicz property and selectively w grouping property the main theorem gives conditions for a space x n to be hurewicz by means of selectively w grouping property we prove that the fuzzy unit interval has the selectively w grouping property we also present l versions for some results about product of compact spaces

the fuzzy theory based on afs algebras and afs structure abstract in this paper we define a more general algebra class called afs algebras which includes ei algebra and eii algebra published by the author which are completely distributive molecular lattices we establish the fuzzy theory using afs algebras and afs structure published by the author and propose the concept of fuzzy variable and a simple controlling model based on afs theory

generalization of kolmogoroff s theorem to l topological vector spaces in this paper the concept of l fuzzy normed linear spaces is introduced and some of its properties are studied then we give a necessary and sufficient condition for l topological vector spaces to be l fuzzy normable this is a generalization of kolmogoroff s theorem in the classical topological vector spaces to l topological vector spaces

a note on urysohn separation property in topological molecular lattices as an extension of urysohn separation axioms in general topology several separation property were introduced by chen and wu urysohn separation property in topological molecular lattices fuzzy sets and systems 123 2001 177 184 in topological molecular lattices in this note it is proved that no 0 1 topological space satisfies chen wu separation axioms s

separation axioms in l fuzzy topological spaces i t0 and t1 abstract using the notion of remote neighborhood we define the separation axioms t 0 and t 1 in l fuzzy topological spaces l fts the relations between our definitions hutton and reilly s and wang s are discussed and the separations of hutton s fuzzy unit interval and gantner s fuzzy real line are examined characterizations of these l fts are proved and a series of properties of them are investigated moreover some results on minimal t 0 l fts and t 1 l fts are established which will be used immediately in our subsequent discussion

sr convergence theory in fuzzy lattices abstract convergence theory and order homomorphisms theory are two important objects of study in fuzzy lattices in this paper sr convergence theory for molecular nets and ideals is established and the concepts of r irresolute sr continuous and almost irresolute order homomorphisms in topological molecular lattices are introduced and systematically characterized some mutual relations between these concepts and others are obtained

existence results for translation nets ii abstract let d be a translation bruck net of order s and degree r with translation group g and let s q 1 q n be the prime power factorization of s we prove that r min q i 1 i 1 n moreover equality can be realized provided that g is the direct sum of elementary abelian groups similar results are obtained for translation s r nets and more generally for groups with large systems of large subgroups as a corollary we considerably strengthen a known result on symmetric translation nets by showing that any translation s s net has parameters of the form s p i p j for some prime p and also necessarily has elementary abelian translation group

categories isomorphic to the kubiak sostak extension of tml abstract in this paper we study wang guo jun s topological molecular lattices wang s tml in a kubiak sostak sense and construct a category the kubiak sostak extension of tml denoted by ftml then we prove that ftml is topological over cd op and tml is bireflective full subcategory of ftml finally we introduce several important categories to characterize ftml

on the order conditions of fuzzy convergence classes by means of the order structure of the related lattice the liminf condition of fuzzy convergence classes is proposed in this paper which reflects the essential difference between fuzzy convergence classes and ordinary convergence classes the relationship between the liminf condition and two related conditions proposed by liu and wang respectively are discussed the theory of fuzzy convergence classes based on liminf condition is established for topological molecular lattices l topological spaces in the sense of chang or lowen weakly induced spaces and induced spaces

new applications of l nets in l fuzzy multifunctions in this paper we will give new applications of l net in l fuzzy multifunctions using l net we will further study l fuzzy continuous almost continuous and r continuous multifunctions some interesting characterizations are obtained

syntopogenous structure on completely distributive lattice and its connectedness abstract in this paper the general theory of syntopogenous structures on completely distributive lattices is established the unified question of cotopology quasi uniformity and t structure is investigated the results of this paper complete the framework of the topological structure on completely distributive lattices and generalize the corresponding theory in general and fuzzy topology finally we examine the connectedness the following main results about connectedness are obtained 1 1 if f l1 s1 l2 s2 is an s1 s2 continuous goh function and d l1 is s1 connected element then f d is s2 connected element 2 2 let c l s be an s connected element and c d c then d is an s connected element 3 3 i li si is connected iff for any i i li si is connected

the category of pointwise s proximity spaces first it is proved that the category of pointwise quasi uniform spaces and pointwise quasi uniformly continuous morphisms is topological then the concept of pointwise s quasi proximity on lattice valued fuzzy set theory is introduced in such a way as to be compatible with pointwise quasi uniformity and pointwise p q metric each co topology can be induced by a pointwise s quasi proximity when a valued lattice is a completely distributive lattices equipped with an order reversing involution the concept of pointwise s proximity can be defined by means of pointwise s quasi proximity the relation between pointwise s quasi proximities and pointwise quasi uniformities is discussed it is proved that the category of pointwise s quasi proximity spaces is isomorphic to the category of totally bounded pointwise quasi uniform spaces the category of pointwise s proximity spaces is isomorphic to a subcategory of totally bounded pointwise uniform spaces hence they are topological

partial spreads over zq let p be a prime q pa and g z2nq we consider partial spreads in g i e collections of pairwise disjoint subgroups of order qn it is shown that the maximum size of such a collection is exactly pn 1 the method of proof consists in representing partial spreads in g by invertible n n matrices over zq and in finding a close relation to ordinary partial spreads over zp geometrically partial spreads over zq correspond to homogeneous translation nets

degrees of fuzzy compactness in l fuzzy topological spaces in this paper we introduce the notion of the degree of fuzzy compactness in l fuzzy topological spaces by means of an inequality proposed by shi in 2007 the notion is a generalization of the definition of fuzzy compactness in l topological spaces some properties of this new notion are researched the tychonoff theorem is generalized it is pointed out that our notion is not equivalent to other definition of degree of compactness

a new approach to the fuzzification of matroids in this paper the concept of closed fuzzy pre matroids is generalized to l fuzzy set theory when l is a complete lattice it is also called an l fuzzifying matroid in the definition of l fuzzifying matroids each subset can be regarded as an independent set to some degree when l is completely distributive an l fuzzifying matroid can be characterized by means of its l fuzzifying rank function an l fuzzifying matroid and its l fuzzifying rank function are one to one corresponding

minimal hausdorff l fuzzy topological spaces abstract in this paper the author introduced the concept of minimal hausdorff l fuzzy topological spaces investigated basic properties of it by means of fuzzy ideals and fuzzy nets and compared it with the concepts of compactness and h closedness

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

strong fuzzy compact sets and ultra fuzzy compact sets in l topological spaces abstract in this paper lowen s strong fuzzy compactness and ultra fuzzy compactness of i topological spaces are extended to l subsets for l a complete lattice the properties of strong fuzzy compact l subsets and ultra fuzzy compact l subsets are studied in detail here and comparisons between these notions as well as the notion of n compactness are shown

l m fuzzy matroids in this paper an extension of matroids is introduced in lattice valued fuzzy set theory such a generalized matroid is a pair made of a finite set and a fuzzy subfamily of its lattice valued fuzzy subsets satisfying three axioms it uses two lattices one for membership grades of elements to fuzzy sets and another for membership degrees of fuzzy sets in a fuzzy family of fuzzy sets it is a logical generalization of the notions of matroids and fuzzifying matroids in our definition of generalized fuzzy matroids each l fuzzy subset can be regarded to be independent of other ones to some degree the relevance of generalized fuzzy matroids is shown in the setting of fuzzy vector spaces and fuzzy graphs

l metric on the space of l fuzzy numbers abstract in this article a natural l topology is constructed on the set of l fuzzy numbers which is called the standard l topology it is proved that the standard l topology can be induced by an l metric the space of trapezoidal fuzzy numbers can be regarded as a subspace of the space of l fuzzy numbers an l metric on the set of trapezoidal fuzzy numbers is presented

approximation operators on complete completely distributive lattices rough set a tool for data mining deals with the vagueness and granularity in information systems in 2006 chen et al initiated the study of rough approximations on a complete completely distributive lattice ccd lattice for short and brought generalizations of rough sets into a unified framework in this paper we discuss the approximation operators on a ccd lattice based on the concept of neighborhood three kinds of upper approximation operators and a kind of lower approximation operator are constructed basic properties of lower and upper approximation operators are examined furthermore the relationships among these operators are analyzed

l proximities and totally bounded pointwise l uniformities in this paper we shall develop the study of connections between l quasi proximities in the sense of katsaras liu artico moresco and pointwise l quasi uniformities in the sense of shi we shall prove that there exists a one to one correspondence between l quasi proximity spaces and totally bounded pointwise l quasi uniform spaces

on afs algebra part ii this paper addresses the properties and structures of ei algebra and ein algebra in the framework of liu fuzzy sets and systems 95 1998 179 the main contribution of the paper is that the expressions of both irreducible elements or irreducible elements in molecular lattice em and ex1 xnm are developed and that the standard minimal family of elements in molecular lattice em and ex1 xnm is formulated it is proved that neither molecular lattice em nor molecular lattice ex1 xnm is a fuzzy lattice

groups with a large number of large disjoint subgroups it is known that if g is a group of order 4n2 and g contains n mutually disjoint subgroups of order 2n then the nonidentity elements of these subgroups form a difference set in g gluck recently discovered a nonabelian example with n 4 and showed it to be the only case with n 4 and g not elementary abelian we show here that the only examples with n 4 are elementary abelian 2 groups

connectivity in fuzzy topological molecular lattices abstract a new kind of the definitions for connectivity in a fuzzy topological molecular lattice called s connectivity is defined some properties for s connectivity are obtained and the relations between s connectivity and each connectivity given in 1 4 7 are discussed

q convergence of ideals in fuzzy lattices and its applications abstract in this paper we study q convergence and q convergence of ideals in fuzzy lattices by the concept of q remote neighborhood then we introduce the strong s irresoluteness s irresoluteness and s strong semicontinuity on fuzzy lattices we also study some properties of the notions above and strong semicontinuity 2 3 and s irresoluteness 3 4 with the aid of the q convergence q convergence of ideals

h l completely regular l fuzzy sets and their applications first we introduce the concept of h completely regular l fuzzy set in l fts and give some characterizations of it then we propose the concept of h complete regularization of l fts and investigate some properties of this complete regularization we also establish the relations between h completely regular l fts completely regular l fts and weakly induced l fts based on these results we obtain several characterizations of h completely regular l fts

ditopological texture spaces and fuzzy topology ii topological considerations this is the second of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of l topological spaces in the first paper in this series subtitled basic concepts the authors presented the notions of direlation and difunction between textures and introduced the category dftex the construct ftex and several related categories in this paper the category dfditop of ditopological texture spaces and bicontinuous difunctions is defined and the forgetful functor u dfditop dftex shown to be topological several properties are discussed including the existence of products and coproducts an equivalence with the category of classical hutton spaces is presented and the paper ends with a consideration of l valued sets and topologies from the viewpoint of ditopological texture spaces

regularity and normality of l m fuzzy topological spaces abstract in this paper regularity and normality axioms are introduced in l m fuzzy topological spaces where l and m are completely distributive lattices with order reversing involution each l m fuzzy topological space can be regarded to be regular and normal to some degree the relations among the four separation axioms are investigated they are equivalent to each other in an l m fuzzy metric space

on the existence of translation nets abstract the existence of a translation net of order s and degree r with translation group g is equivalent to the existence of r mutually disjoint subgroups of g of order s in this paper we consider p groups g of odd square order p2n and improve the known general upper bound on the number of mutually disjoint subgroups of order pn in g provided that g is not elementary abelian this solves problem 8 2 14 in d jungnickel latin squares their geometries and their groups a survey in coding theory and design theory ii d k ray chaudhuri ed pp 166 225 springer berlin new york 1990 we determine all groups of order p4 which are translation groups of translation nets with at least three parallel classes for all prime numbers p furthermore we construct p3 p2 1 translation nets with non abelian translation group of order p6 for all odd prime numbers p

on stratiform l fuzzy topologies and their application for a given l fuzzy topological space and a fixed prime element r of fuzzy lattice l the concept of l fuzzy r open sets is introduced and it is proved that all r open sets for l fuzzy topological space form a new l fuzzy topology which is called stratiform l fuzzy topology making use of the stratiform l fuzzy topology a series of new characterizations of l fuzzy continuous mapping and l fuzzy open mapping and three kinds of separateness hausdorff regular and normal are presented

approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory ii in this paper the representations of fuzzy concepts based on raw data have been investigated within the framework of afs axiomatic fuzzy set theory first a brief review of afs theory is presented and a completely distributive lattice the e i algebra is proposed secondly two kinds of e i algebra representations of fuzzy concepts are derived in detail in order to represent the membership functions of fuzzy concepts in the interval 0 1 the norm of afs algebra is defined and studied finally the relationships of various representations with their advantages and drawbacks are analyzed

connectedness in l fuzzy topological spaces based on the consideration to the layer structures of fuzzy lattices and the level topologies of l fuzzy topological spaces a connectedness is defined for an arbitrary l fuzzy set in this paper this definition reflects the degree of connectivity of an l fuzzy set it is shown that the connectedness of l fuzzy topological spaces is an l good extension multiplicative and preserved under continuous l valued zadeh functions and that the inverse limit of continuums is a continuum general l fuzzy intervals and h intervals are defined and connectedness of them are proved it is also shown that the connectedness of an l fuzzy topological space is equivalent to the connectedness of its induced i l fuzzy topological space

the topology of afs structure and afs algebras abstract in this paper we will study the topological structure of the afs structures published by the author discuss the topological molecular lattice structures on ei ei eii eii algebras published by the author and give the relations of these topological structures as applications we study the topological produced by a family of fuzzy concepts in the afs algebras and apply these to analyze relations among fuzzy concepts using these we believe that we can study the law of human thinking the most important fact is that all these can be operated by computers we are also sure afs theory will play an important role in information processing

pointwise pseudo metrics in l fuzzy set theory a theory of pointwise pseudo metrics is based on l fuzzy sets a pointwise pseudo metric is characterized by its remote neighborhood maps many theorems in general topology are generalized on l fuzzy sets the l fuzzy real line and the l fuzzy unit interval are pointwise pseudo metrizable a pseudo metrization theorem of pointwise uniformity is obtained relations between erceg s p metric pseudometric and pointwise p metric are discussed

extension of shi s quasi uniformities in a kubiak sostak sense abstract the aim of this paper is to study the extension of shi fu gui s quasi uniformities in a kubiak sostak sense the relationship between this extension of shi fu gui s quasi uniform molecular lattices shi s quml and the corresponding kubiak sostak extension of wang guo jun s topological molecular lattices wang s tml is discussed quml and tml denote the categories of shi s quml and wang s tml respectively cd is the category of completely distributive lattices with complete lattice morphisms as morphisms we prove that fquml the category of fuzzy quasi uniform molecular lattices the extension of shi s quml is a topological category over cd op and ftml the category of fuzzy topological molecular lattices the extension of wang s tml can be embedded in fquml we also prove that fquml is isomorphic to quml c m when m is a completely distributive lattice with multiplicative property where quml c m is the co tower extension of quml finally we study the kubiak sostak extension of hutton s quasi uniformities

lattice theoretic contexts and their concept lattices via galois ideals this paper introduces a concept of lattice theoretic contexts as well as their concept lattices a lattice theoretic context is a triple g m i with two complete lattices g m and their galois ideal i a lattice theoretic context and its concept lattice are a common generalization of classical fca pocs s formal fuzzy context one sided concept lattices generalized concept lattices and l fuzzy concept lattices with hedges when the lattices g m are completely distributive a reduction of the relation i in the lattice theoretic context g m i can be obtained related algorithms to construct concept lattices of l fuzzy contexts considered as lattice theoretic contexts are presented in the case of l being a completely distributive lattice we can reduce the number of elements objects or and attributes before computing the whole concept lattice then the related algorithm has lower complexity

the characterizations of semi continuous and irresolute order homomorphisms on fuzzes abstract the concepts of semi continuous and irresolute order homomorphisms on fuzzes were introduced in j jianghan petroleum institute 2 1988 96 97 in this paper we first present the theory with respect to s convergence s convergence of molecular nets and ideals on fuzzes secondly we introduce the notions of strong semicontinuity strong irresoluteness and s irresoluteness on fuzzes then we systematically study the characterizations of these notions with the aid of the s convergence s convergence of molecular nets and ideals and give the relations of them

minimizing cubic and homogeneous polynomials over integers in the plane we complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in a real vector space of dimension two can be done in polynomial time whereas optimizing a quartic polynomial in the same type of region is np hard we close the gap by showing that this problem can be solved in polynomial time for cubic polynomials furthermore we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable in polynomial time we show that this holds for polynomials that can be translated into homogeneous polynomials even when the translation vector is unknown we demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input thus req

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

on compactness of induced i l fuzzy topological spaces abstract the induced i l fuzzy topological spaces for l fuzzy topological spaces introduced by wang geping is a kind of important fuzzy topological space in this paper the author studies the fuzzy compactness of induced i l fuzzy topological spaces some available relations between molecular nets of an l fuzzy topological space and that of its induced i l fuzzy topological space are firstly presented by using these relations the author obtains that the induced i l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact if and only if the l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact the author also proves that the induced i l fuzzy topological space is paracompact if and only if the l fuzzy topological space is paracompact

on some gross misunderstandings about the theory of topological molecular lattices abstract some gross misunderstandings have arisen since the theory of topological molecular lattices briefly tmls was presented wang 1989 1992 certain scholars think that there is a correspondence between tmls and classical spaces and they could obtain the results in the theory of tmls hence fuzzy topology by simply translating the corresponding results in classical topology the aim of this paper is to clear out such misunderstandings

integer polynomial optimization in fixed dimension we classify according to their computational complexity integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed for the optimization of an integer polynomial over the lattice points of a convex polytope we show an algorithm to compute lower and upper bounds for the optimal value for polynomials that are nonnegative over the polytope these sequences of bounds lead to a fully polynomial time approximation scheme for the optimization problem

limit structures over completely distributive lattices the notion of a limit molecular lattice a completely distributive lattice with some kinds of convergence of ideals on it is introduced it is proved that the category of limit molecular lattices is a cartesian closed category containing the category of topological molecular lattices as a bireflective full subcategory and the category of limit spaces as a reflective and coreflective subcategory respectively noteworthy is the distinctive order property of the convergence of ideals on the limit molecular lattice compared to that of limit spaces the l fuzzy version of fuzzy limit space is introduced and it is proved that the category of l fuzzy limit spaces is a nonfull coreflective subcategory of the category of limit molecular lattices