multi adjoint t concept lattices the t concept lattice is introduced as a set of triples associated to graded tabular information interpreted in a non commutative fuzzy logic following the general techniques of formal concept analysis and based on the works by georgescu and popescu given a non commutative conjunctor it is possible to provide generalizations of the mappings for the intension and the extension in two different ways and this generates a pair of concept lattices in this paper we show that the information common to both concept lattices can be seen as a sublattice of the cartesian product of both concept lattices the multi adjoint framework can be applied to this general t concept lattice and its usefulness is illustrated by a working example

generalized one sided concept lattices with attribute preferences the main aim of this paper is to introduce the preference relations on generalized one sided concept lattices which represent a fuzzy generalization of fca with classical object clusters and fuzzy attributes in our case a preference relation is modeled by a linear well quasi order on the set of all attributes we describe concept forming operators based on a galois connection which is defined between the power set of objects and the fuzzy sets of attributes with lexicographic order induced by the preference relation the representation theorem for such kind of concept lattices is also presented

computing sets of graded attribute implications with witnessed non redundancy in this paper we extend our previous results on sets of graded attribute implications with witnessed non redundancy we assume finite residuated lattices as structures of truth degrees and use arbitrary idempotent truth stressing linguistic hedges as parameters which influence the semantics of graded attribute implications in this setting we introduce algorithm which transforms any set of graded attribute implications into an equivalent non redundant set of graded attribute implications with saturated consequents whose non redundancy is witnessed by antecedents of the formulas as a consequence we solve the open problem regarding the existence of general systems of pseudo intents which appear in formal concept analysis of object attribute data with graded attributes and linguistic hedges furthermore we show a polynomial time procedure for determining bases given by general systems of pseudo intents from sets of graded attribute implications which are complete in data

set approximations in fuzzy formal concept analysis formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems the purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined based on both lattice theoretic and fuzzy set theoretic operators two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined finally properties of the rough fuzzy set approximation operators are presented in detail

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

variable threshold concept lattices in this paper the definition of a variable threshold concept lattice is introduced based on a galois connection three kinds of variable threshold concept lattices in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold are defined the number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice the three kinds of variable threshold concept lattices are constructed between two crisp sets between a crisp set and a fuzzy set and between a fuzzy set and a crisp set their properties are analogous to that of the classical concept lattices and can be induced by the fuzzy concept lattice

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

a comparative study of adjoint triples adjoint triples are helpful as basic operators used in several domains for example adjoint triples play an important role in two important frameworks multi adjoint logic programming and multi adjoint concept lattices this paper shows that adjoint triples are an interesting generalization of t norms and their residuated implications since they preserve their main properties as well as they help to increase the flexibility of the operators used for computation in the considered framework furthermore these operators will be related to other important general triples specifically implication triples and the definition of these latter operators will be improved

dual multi adjoint concept lattices several papers relate different alternative approaches to classical concept lattices such as property oriented and object oriented concept lattices and the dual concept lattices whereas the usual approach to the latter is via a negation operator this paper presents a fuzzy generalization of the dual concept lattice the dual multi adjoint concept lattice in which the philosophy of the multi adjoint paradigm is applied and no negation on the lattices is needed

an efficient method to factorize fuzzy attribute oriented concept lattices abstract factorization by similarity is a mathematical technique used in formal concept analysis with fuzzy attributes for reducing the complexity of different types of fuzzy concept lattices in this paper we find the structure of the factor lattice of a fuzzy attribute oriented concept lattice namely the intervals representing the blocks of this lattice we provide a procedure for generating the infimum and the supremum concepts of these intervals as fixpoints of a fuzzy closure operator this theoretical result allows to develop a more efficient algorithm for building the factor lattice of the fuzzy attribute oriented concept lattice a comparison between our approach and the existing algorithms is presented

on the structure of rough prime primary ideals and rough fuzzy prime primary ideals in commutative rings this paper is a continuation of ideas presented by davvaz roughness in rings inform sci 164 2004 147 163 roughness based on fuzzy ideals inform sci 176 2006 2417 2437 we introduce the notions of rough prime primary ideals and rough fuzzy prime primary ideals in a ring and give some properties of such ideals also we discuss the relations between the upper and lower rough prime primary ideals and the upper and lower approximations of their homomorphism images

phase transition in the 1d random field ising model with long range interaction we study one dimensional ising spin systems with ferromagnetic long range interaction decaying as n 2 alpha in left frac 12 frac ln 3 ln2 1 right in the presence of external random fields we assume that the random fields are given by a collection of symmetric independent identically distributed real random variables gaussian or subgaussian we show for temperature and strength of the randomness variance small enough with ip 1 with respect to the random fields that there are at least two distinct extremal gibbs measures

associatively tied implications we say that an implication operator a on a complete lattice l is associatively tied if there is a binary operation t on l that ties a that is the identity a a gamma a t holds for all in l this property extends to multiple valued logic the following equivalence in classical logic x y z x y z we show that in this case there exists an associative binary operation ta that ties a hence the nomenclature we study properties of that ta when a is associatively tied we then seek a characterization for the validity of associative tiedness for an implication a phrased in terms of a and two adjoints of it

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

a multi level conceptual data reduction approach based on the lukasiewicz implication starting from fuzzy binary data represented as tables in the fuzzy relational database in this paper we use fuzzy formal concept analysis to reduce the tables size to only keep the minimal rows in each table without losing knowledge i e association rules extracted from reduced databases are identical at given precision level more specifically we develop a fuzzy extension of a previously proposed algorithm for crisp data reduction without loss of knowledge the fuzzy galois connection based on the lukasiewicz implication is mainly used in the definition of the closure operator according to a precision level which makes data reduction sensitive to the variation of this precision level

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

formal concept analysis via multi adjoint concept lattices several fuzzifications of formal concept analysis have been proposed to deal with uncertain information in this paper we focus on concept lattices under a multi adjoint paradigm which enriches the language providing greater flexibility to the user in that he she can choose from a number of different connectives multi adjoint concept lattices are shown to embed different fuzzy extensions of concept lattices found in the literature the main results of the paper being the representation theorem of this paradigm and the embedding of other well known approaches

absolutely minimal lipschitz extension of tree valued mappings we prove that every lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal lipschitz extension amle we relate these extensions to a stochastic game called politics a generalization of a game called tug of war that has been used in peres et al j am math soc 22 1 167 210 2009 to study real valued amles

extending conceptualisation modes for generalised formal concept analysis formal concept analysis fca is an exploratory data analysis technique for boolean relations based on lattice theory its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes pairs of dually isomorphic sets of objects and attributes called formal concepts form a concept lattice but actually model only a conjunctive mode of conceptualisation in this paper we augment this formalism in two ways first we extend fca to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal logic motivated way this creates the three new types of concepts and lattices of extended fca viz the lattice of neighbourhood of objects that of attributes and the lattice of unrelatedness second we consider incidences with values in idempotent semirings concretely the completed max plus or schedule algebra email protected m a x and focus on generalising fca to try and replicate the modes of conceptualisation mentioned above to provide a concrete example of the use of these techniques we analyse the performance of multi class classifiers by conceptually analysing their confusion matrices

on heterogeneous formal contexts we propose a new type of fuzzification for formal concept analysis that works with heterogeneous values in a context and illustrate this with an example we formulate and prove an appropriate counterpart to the so called basic theorem of a concept lattice we show that this is a generalization of the previous approaches it covers the so called generalized concept lattice and multi adjoint t concept lattices

a heuristic knowledge reduction method for decision formal contexts computing a minimal reduct of a decision formal context by boolean reasoning is an np hard problem thus it is essential to develop some heuristic methods to deal with the issue of knowledge reduction especially for large decision formal contexts in this study we first investigate the relationship between the concept lattice of a formal context and those of its subcontexts in preparation for deriving a heuristic knowledge reduction method then we construct a new framework of knowledge reduction in which the capacity of one concept lattice implying another is defined to measure the significance of the attributes in a consistent decision formal context based on this reduction framework we formulate an algorithm of searching for a minimal reduct of a consistent decision formal context it is proved that this algorithm is complete and its time complexity is polynomial some numerical experiments demonstrate that the algorithm can generally obtain a minimal reduct and is much more efficient than some boolean reasoning based methods

note on generating fuzzy concept lattices via galois connections the aim of this paper is to compare an approach of creating fuzzy concept lattices proposed by popescu with several other approaches particularly we show that this approach is in some way equivalent to the approach of krajci called generalized concept lattices we also give a straightforward generalization of popescu s approach to non homogeneous cases

using concept lattice theory to obtain the set of solutions of multi adjoint relation equations an important goal in the fuzzy relation equations framework is to obtain the whole set of solutions this paper introduces algebraic properties in the multi adjoint concept lattice setting moreover using the relationship between these concept lattices and multi adjoint relation equations the complete set of solutions of these general equations is characterized and computed avoiding the necessity of considering minimal solutions

description of sup and inf preserving aggregation functions via families of clusters in data tables abstract connection between the theory of aggregation functions and formal concept analysis is discussed and studied thus filling a gap in the literature by building a bridge between these two theories one of them living in the world of data fusion the second one in the area of data mining we show how galois connections can be used to describe an important class of aggregation functions preserving suprema and by duality to describe aggregation functions preserving infima our discovered method gives an elegant and complete description of these classes also possible applications of our results within certain biclustering fuzzy fca based methods are discussed

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

multi adjoint relation equations definition properties and solutions using concept lattices this paper generalizes fuzzy relation equations following the multi adjoint philosophy moreover the solutions of these general fuzzy relation equations and the concepts of a multi adjoint property oriented concept lattice are related and several results are obtained from the theory of concept lattices as a consequence of this relevant relation more properties about these general equations can be proven from the theory of concept lattices and the algorithms developed to compute concept lattices can be used to obtain solutions furthermore an interesting application to fuzzy logic programming has been introduced in which an important problem in this topic has been interpreted in terms of solving a multi adjoint relation equation

on multi adjoint concept lattices based on heterogeneous conjunctors sets of attributes and objects in fuzzy formal concept analysis are usually different and hence it might not make sense to evaluate them on the same carrier in this context the operators used to obtain the concept lattice could be defined by associating different lattices to attributes and objects several reasons exist for which we need to evaluate the sets of attributes and objects in the same carrier following this direction we introduce a new definition of a concept lattice where objects and attributes are evaluated on the same lattice l although operators evaluating objects and attributes in different carriers are used moreover we study the relationship between this new concept lattice and the alternative one which can be obtained directly by using different carriers for the sets of attributes and objects

rough set model based on formal concept analysis this paper proposes a rough set model based on formal concept analysis in this model a solution to an algebraic structure problem is first provided in an information system a lattice structure is inferred from the information system and corresponding nodes are called rough concepts how to deal with common problems in rough set theory based on rough concepts is then explored such as upper and lower approximation operators reducts and cores decision dependency has become a common form of knowledge representation owing to its properties of expressiveness and ease of understanding so it has been widely used in practice finally application of rough concepts to the extraction of decision dependencies from a decision table is studied a complete and non redundant set of decision dependencies can be obtained from a decision table examples demonstrate that application of the method presented in this paper is valid and practicable the results not only provide a better understanding of rough set theory from the perspective of formal concept analysis but also demonstrate a new way of combining rough set theory and formal concept analysis

on the dedekindmacneille completion and formal concept analysis based on multilattices the dedekindmacneille completion of a poset p can be seen as the least complete lattice containing p in this work we analyze some results concerning the use of this completion within the framework of formal concept analysis in terms of the poset of concepts associated with a galois connection between posets specifically we show an interesting property of the dedekindmacneille completion in that the completion of the concept poset of a galois connection between posets coincides with the concept lattice of the galois connection extended to the corresponding completions moreover we study the specific case when p has multilattice structure and state and prove the corresponding representation theorem

characterizing reducts in multi adjoint concept lattices abstract the construction of reducts that is minimal sets of attributes containing the main information of a database is a fundamental task in different frameworks such as in formal concept analysis fca and rough set theory rst this paper will be focused on a general fuzzy extension of fca called multi adjoint concept lattice and we present a study about the attributes generating meet irreducible elements and on the reducts in this framework from this study we introduce interesting results on the cardinality of reducts and the consequences in the classical case

quasi additive estimates on the hamiltonian for the one dimensional long range ising model in this work we study the problem of getting quasi additive bounds for the hamiltonian of the long range ising model when the two body interaction term decays proportionally to 1d2 0 1 we revisit the paper by cassandro et al j math phys 46 053305 2005 where they extend to the case 0 ln3ln2 1 the result of the existence of a phase transition by using a peierls argument given by frohlich and spencer commun math phys 84 87 101 1982 for 0 the main arguments of cassandro et al j math phys 46 053305 2005 are based in a quasi additive decomposition of the hamiltonian in terms of hierarchical structures called triangles and contours which are related to the original definition of contours introduced by frohlich and spencer commun math phys 84 87 101 1982 in this work we study the existence of a quasi additive decomposition of the hamiltonian in terms of the contours defined in the work of cassandro et al j math phys 46 053305 2005 the most relevant r

lipschitz quotients from metric trees and from banach spaces containing l1 abstract if x is any separable banach space containing l1 then there is a lipschitz quotient map from x onto any separable banach space y

constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the paper deals with the isomorphism between the constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the essential point of the former approach encompasses the full diversification of data structures within a formal context in particular we use a different complete lattice for diverse objects a different complete lattice for diverse attributes and a different poset for diverse matrix fields the latter framework with heterogeneous hedges results in a reduction in the size of the corresponding concept lattice we present the properties of constraint heterogeneous approach that is associated with the fixpoints of hedges and we add remarks to the related studies

a new algebraic structure for formal concept analysis formal concept analysis fca originally proposed by wille 39 is an important theory for data analysis and knowledge discovery concept lattice is the core of the mathematical theory of formal concept analysis to address the requirements of real word applications concept lattice has been extended to many other forms from the theoretical point of view and possible applications in this paper with the aim of deriving the mathematical properties of formal concepts from the point of algebra we propose a new algebra system for the formal context under the frame of the proposed system some interesting properties of formal concepts are explored which could be applied to explore concept hierarchy and ontology merging

formal query systems on contexts and a representation of algebraic lattices formal concept analysis fca has proven to be a feasible framework for information retrieval ir in this paper with the aim of exploring the mathematical properties of the fca based document retrieval paradigm we propose the notions of conjunctive and disjunctive query on the formal contexts and investigate the derived query systems from the order theoretical perspective we show that the conjunctive query system is isomorphic to the galois lattice of the underlying context while the disjunctive query system can be join densely generated from the galois lattice of the underlying context up to isomorphism next we introduce directed disjunctive query systems and study their connection with algebraic lattices as a result we show that the notion of directed disjunctive query provides an alternative approach to restructuring algebraic lattices

on equivalence of conceptual scaling and generalized one sided concept lattices the methods of conceptual scaling and generalized one sided concept lattices represent different possibilities on how to deal with many valued contexts we briefly describe these methods and prove that they are equivalent in particular we show that the application of these two approaches to a given many valued context yields the same closure system on the set of all objects based on this equivalence we propose a possible attribute reduction of one sided formal contexts

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

the use of linguistic variables and fuzzy propositions in the l fuzzy concept theory the use of linguistic variables and fuzzy propositions in the interval valued l fuzzy contexts can be an interesting tool to extract a more complete information from them in this paper we analyze three different situations first we obtain significant relations in order to study all the objects and attributes of the interval valued l fuzzy context by means of the interval valued l fuzzy concepts after that we show how to replace the erroneous values to be able to study in a suitable way the context finally we use the linguistic labels to obtain a subcontext that represents our interest of study we also show an experimental evaluation in the paper

certificateless threshold cryptosystem secure against chosen ciphertext attack this paper proposes the first certificateless threshold decryption scheme which avoids both the single point of failure in the distributed networks and the inherent key escrow problem in identity based cryptosystem the scheme is secure against threshold chosen ciphertext attack it tolerates the type i adversary that can replace public key and the type ii adversary that has access to the system s master key the formal proof of security is presented in the random oracle model assuming some underlying problems closely related to the bilinear diffie hellman problem are computationally hard

bipolar fuzzy graph representation of concept lattice formal concept analysis fca is a mathematical framework for knowledge processing tasks fca has been successfully incorporated into fuzzy setting and its extension interval valued fuzzy set for handling vagueness and impreciseness in data however the analysis in such settings is restricted to unipolar space recently some applications of bipolar information are shown in bipolar fuzzy graph lattice theory as well as in fca the adequate analysis of bipolar information using fca requires incorporation of bipolar fuzzy set and an appropriate lattice structure for this purpose we propose an algorithm for generating the bipolar fuzzy formal concepts a method for cut of bipolar fuzzy formal context and its implications with illustrative examples

set valued ordered information systems set valued ordered information systems can be classified into two categories disjunctive and conjunctive systems through introducing two new dominance relations to set valued information systems we first introduce the conjunctive disjunctive set valued ordered information systems and develop an approach to queuing problems for objects in presence of multiple attributes and criteria then we present a dominance based rough set approach for these two types of set valued ordered information systems which is mainly based on substitution of the indiscernibility relation by a dominance relation through the lower upper approximation of a decision some certain possible decision rules from a so called set valued ordered decision table can be extracted finally we present attribute reduction also called criteria reduction in ordered information systems approaches to these two types of ordered information systems and ordered decision tables which can be used to simplify a set valued ordered information system and find decision rules directly from a set valued ordered decision table these criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules

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

formal concept analysis based on the topology for attributes of a formal context formal concept analysis fca concerns the hierarchical structures induced by a binary relation between a pair of sets it is widely applied in data analysis information retrieval and knowledge discovery within the framework of fca computing all formal concepts is the main challenge due to its exponential complexity it has be proved that all formal concepts is a closure system on objects hence the closure operator on objects are general used to generate all formal concepts of a formal context in the lexicographic tree approach a base on objects is used to generate extensions of all formal concepts and construct the formal concept lattice inspired from the approach we concentrate a base on attributes and generate intensions of all formal concepts in this paper to this end we firstly analyze an example the lexicographic tree approach is used to generate all formal concepts from the time complexity point of view we present that it is not trivial to generate the base on attributes by a simple symmetrical way from the methods based on objects then we deduce a set valued mapping from attributes to the power set of attributes in a formal context and define a binary relation on attributes by the set valued mapping using the binary relation on attributes we construct an approximation space and a topology for attributes respectively and obtain a base for the topology we prove that intensions of all formal concepts are included in the topology for attributes this means that the base can be used to generate intensions of all formal concepts of the formal context and construct the formal concept lattice more general our results represent relationships and the hierarchical structures among attributes of the formal context we present some typical applications in which the topology for attributes and the base for the topology are applied for association rules discovery from a formal context and linguistic concept analysis

attribute reduction in generalized one sided formal contexts in this paper we present a new pair of adjoint mappings between a power set and the direct product of complete lattices the proposed pair of adjoint mappings form a galois connection and the corresponding concept lattice is constructed from a generalized one sided formal context we also propose a lattice keep based attribute reduction approach for generalized one sided formal contexts specifically we present concrete judgment theorems and an algorithm to calculate the attribute reducts in generalized one sided formal contexts furthermore according to the importance of attributes we discuss attributive characteristics for the proposed generalized one sided concept lattice

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

reducing concept lattices by means of a weaker notion of congruence abstract attribute and size reductions are key issues in formal concept analysis in this paper we consider a special kind of equivalence relation to reduce concept lattices which will be called local congruence this equivalence relation is based on the notion of congruence on lattices with the goal of losing as less information as possible and being suitable for the reduction of concept lattices we analyze how the equivalence classes obtained from a local congruence can be ordered moreover different properties related to the algebraic structure of the whole set of local congruences are also presented finally a procedure to reduce concept lattices by the new weaker notion of congruence is introduced this procedure can be applied to the classical and fuzzy formal concept analysis frameworks

on attribute reduction in concept lattices methods based on discernibility matrix are outperformed by basic clarification and reduction abstract several recent papers have studied the problem of attribute reduction in concept lattices by using a discernibility matrix in various extensions of formal concept analysis we recall the clarification and reduction method by ganter wille and show that it is strictly superior to the methods based on discernibility matrix consequently we show how to adapt the clarification and reduction method to the extensions of formal concept analysis considered in the recent papers

attribute and size reduction mechanisms in multi adjoint concept lattices attribute reduction and size reduction in concept lattices are key research topics in formal concept analysis fca this paper combines both strategies in the multi adjoint concept lattice framework in order to simplify the information provided by the original context specifically we present three procedures which merge the attribute reduction and the size reduction by means of an irreducible cut concept lattice analyzing the obtained properties

basic theorem of fuzzy concept lattices revisited abstract there are two versions of the basic theorem of l concept lattices for l being a complete residuated lattice both proved by belohlavek the crisp order version and the fuzzy order version we introduce a third version equivalent to the fuzzy order version but simpler and related more closely to the classical wille s basic theorem of concept lattices then we use it to prove some new results on substructures of l concept lattices and show a simpler proof of a known result on factor structures of l concept lattices we show by means of several counterexamples that the crisp order version does not describe the structure of l concept lattices sufficiently we argue that in order to formulate and prove theoretical results on l concept lattices that are similar to those known from classical formal concept analysis it is essential to use the fuzzy order version of the basic theorem we also discuss the correspondence between the belohlavek s fuzzy order version of the basic theorem and the version introduced in this paper

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

grouping fuzzy sets by similarity the paper presents results on factorization of systems of fuzzy sets the factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a an obstacle to such factorization well known in fuzzy set theory is the fact that being similar at least to degree a is not an equivalence relation because in general it is not transitive as a result ordinary factorization using equivalence classes cannot be used this obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a we show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice this lattice plays the role of a factor structure for the original system of fuzzy sets particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices

evolution in time of l fuzzy context sequences in this work we consider a complete lattice l and we study l fuzzy context sequences which represent the evolution in time of an l fuzzy context to carry out this study in the first part of the paper we consider n ary owa operators in complete lattices which enable us to make a general analysis and a temporal analysis at any moment in time of l fuzzy context sequences after that evolution in time of the relationship between the objects and the attributes is considered in particular we analyze the concepts of trend and persistent formal contexts finally we illustrate our results with an example where we consider the particular lattice l j 0 1

typical gibbs configurations for the 1d random field ising model with long range interaction we study one dimensional ising spin systems with ferromagnetic long range interaction decaying as n 2 alpha in 0 frac 12 in the presence of external random fields we assume that the random fields are given by a collection of symmetric independent identically distributed real random variables which are gaussian or subgaussian with variance we show that when the temperature and the variance of the randomness are sufficiently small with overwhelming probability with respect to the random fields the typical configurations within intervals centered at the origin whose length grow faster than any power of 1 are intervals of spins followed by intervals of spins whose typical length is simeq theta frac 2 1 2 alpha for 0 1 2 and between e frac 1 theta and e frac 1 theta 2 for 1 2

a rough margin based support vector machine by introducing the rough set theory into the support vector machine svm a rough margin based svm rmsvm is proposed to deal with the overfitting problem due to outliers similar to the classical svm the rmsvm searches for the separating hyper plane that maximizes the rough margin defined by the lower and upper margin in this way more data points are adaptively considered rather than the few extreme value points used in the classical svm in addition different support vectors may have different effects on the learning of the separating hyper plane depending on their positions in the rough margin points in the lower margin have more effects than those in the boundary of the rough margin from experimental results on six benchmark datasets the classification accuracy of this algorithm is improved without additional computational expense compared with the classical n svm

triadic concept lattices in the framework of aggregation structures abstract triadic concept analysis is a method of analysis of three way tabular data describing objects by binary attributes they have under various conditions it aims to extract a set of three dimensional clusters so called triadic concepts from the data while the method was recently extended to handle the case of graded data this extension is only one of a variety of possible generalizations namely the one where the so called antitone concept forming operators are considered the generalizations not covered by the mentioned approach include isotone concept forming operators one sided concepts the use of truth stressing hedges and many others since all of the generalizations have their applications for example in factor analysis of relational data it is desirable to have a uniform mathematical foundations covering all of the cases in this paper we provide such foundations by using triadic aggregation structures as a scale of truth degrees we develop triadic concept analysis in a very general setting that covers a wide variety of its possible extensions in a uniform way

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

representing lattices using many valued relations this paper provides a representation theorem of lattices using many valued relations we show that any many valued relation can be associated to a unique lattice which is a meet sublattice of a product of chains conversely to any lattice we can associate a many valued relation such that its associated lattice is isomorphic to the initial one thereby we obtain a representation theorem of lattices using many valued relations moreover since several many valued relations might have the same associated lattice we give a characterization of the minimal many valued relation that can be associated to a lattice we then sketch a polynomial time algorithm which computes such a minimal relation from either a lattice or an arbitrary relation this representation presents several advantages it is smaller than the usual binary representation all known reconstruction algorithms working on binary relation can be used without loss of efficiency it can be used by existing data mining processes

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

rough set driven approach for attribute reduction in fuzzy formal concept analysis abstract the reduction of the set of attributes is an important preliminary challenge in order to obtain information from knowledge systems two remarkable formal tools for extracting such information are rough set theory rst and formal concept analysis fca as well as their fuzzy generalizations this work introduces a new method to reduce attributes in fuzzy fca considering the reduction philosophy given in rst and studies its main properties this method allows us to carry out a deeper study of the relation between these two theories moreover the proposed methodology has been compared with other existing reduction mechanisms

multi adjoint property oriented and object oriented concept lattices this paper presents a generalisation of the classical property and object oriented concept lattices to a fuzzy environment based on the philosophy of the multi adjoint paradigm these concept lattices are generalisations of rough set theory used to consider two different sets the set of objects and the set of attributes to apply the corresponding modal operators as in formal concept analysis first of all the paper presents several specific properties of adjoint triples in which the duals of several supports are considered these properties are then used to introduce the multi adjoint property and object oriented concept lattices in which different adjoint triples can be used in non linear sets as well as the corresponding representation fundamental theorems moreover these lattices are related to the multi adjoint concept lattice in which negation operators are not needed as a result this relation allows the properties to be translated from one to another

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

l concept analysis with positive and negative attributes we describe an extension of formal fuzzy concept analysis allowing a user to choose which attributes are viewed as positive and which are viewed as negative the two sets are then handled using a combination of previously studied antitone concept forming operators and isotone concept forming operators respectively the two main outputs of formal concept analysis namely concept lattices and attribute implications in the setting of positive and negative attributes are presented an analogy of the main theorem of concept lattices and a relationship between the new concept lattice and the previously studied concept lattices is showed we introduce basic syntactic and semantic notions for attribute implications called fuzzy containment implications we consider two settings one where the sets of positive and negative attributes are crisp sets and a generalization where the two sets are fuzzy sets

computing the lattice of all fixpoints of a fuzzy closure operator we present a fast bottom up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees along with the partial order on the set of all fixpoints fuzzy closure operators appear in several areas of fuzzy logic and its applications including formal concept analysis fca that we use as a reference area of application in this paper several problems in fca such as computing all formal concepts from data with graded attributes or computing non redundant bases of all attribute dependencies can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data the development of a general algorithm that is applicable in particular to these problems is the ultimate purpose of this paper we present the algorithm its theoretical foundations and experimental evaluation

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

adjoint negations more than residuated negations negation operators are useful operators which have extensively been studied and used in different fuzzy settings such as in fuzzy logic this paper introduces a new and flexible kind of negation operator based on adjoint triples and which is called adjoint negation relative to a fixed element besides proving several interesting properties the comparison with weak negations studied by trillas esteva et al and pairs of weak negations introduced by georgescu and popescu is presented furthermore adjoint negations are used in order to define the corresponding dual disjunctive to an adjoint conjunctor

a note on variable threshold concept lattices threshold based operators are reducible to classical concept forming operators the present paper deals with formal concept analysis of data with fuzzy attributes we clarify several points of a new approach of s q fan w x zhang variable threshold concept lattice inf sci accepted for publication which is based on using thresholds in concept forming operators we show that the extent and intent forming operators from s q fan w x zhang inf sci accepted for publication can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied e g in r belohlavek fuzzy galois connections math logic quarterly 45 4 1999 497 504 r belohlavek concept lattices and order in fuzzy logic ann pure appl logic 128 2004 277 298 s pollandt fuzzy begriffe springer verlag berlin heidelberg 1997 as a consequence main properties of the new operators from s q fan w x zhang inf sci accepted for publication including the properties studied in s q fan w x zhang inf sci accepted for publication can be obtained as consequences of the original operators from r belohlavek 1999 r belohlavek 2004 s pollandt 1997

parameterizing the semantics of fuzzy attribute implications by systems of isotone galois connections we study the semantics of fuzzy if then rules called fuzzy attribute implications parameterized by systems of isotone galois connections the rules express dependences between fuzzy attributes in object attribute incidence data the proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches we formalize the general parameterizations propose bivalent and graded notions of semantic entailment of fuzzy attribute implications show their characterization in terms of least models and complete axiomatization and provide characterization of bases of fuzzy attribute implications derived from data

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

on possible generalization of fuzzy concept lattices using dually isomorphic retracts we provide a generalization of fuzzy concept lattices based on so called weak galois connections generalization is that instead of dually isomorphic closure systems we consider dually isomorphic retracts of complete lattices we also give a generalization of the concept lattices with hedges proposed by belohlavek and vychodil based on composition of interior operators with galois connections

formal concept analysis based on fuzzy granularity base for different granulations this paper introduces granular computing grc into formal concept analysis fca it provides a unified model for concept lattice building and rule extraction on a fuzzy granularity base for different granulations one of the strengths of grc is that larger granulations help to hide some specific details whereas fca in a grc context can prevent losses due to concept lattice complexity however the number of superfluous rules increases exponentially with the scale of the decision context to overcome this we present some inference rules and maximal rules and prove that the set of all these maximal rules is complete and nonredundant thus users who want to obtain decision rules should generate maximal rules examples demonstrate that application of the method is valid and practicable in summary this approach utilizes fca in a grc context and provides a practical basis for data analysis and processing

some constructions of the join of fuzzy subgroups and certain lattices of fuzzy subgroups with sup property in this paper some new lattices of fuzzy substructures are constructed for a given fuzzy set m in a group g a fuzzy subgroup s m generated by m is defined which helps to establish that the set l s of all fuzzy subgroups with sup property constitutes a lattice consequently many other sublattices of the lattice l of all fuzzy subgroups of g like l s t l s n l s n t etc are also obtained the notion of infimum is used to construct a fuzzy subgroup i m generated by a given fuzzy set m in contrast to the usual practice of using supremum in the process a new fuzzy subgroup i m is defined which we shall call a shadow fuzzy subgroup of m it is established that if m has inf property then i m also has this property

on the definition of suitable orderings to generate adjunctions over an unstructured codomain given a mappingf a b from a pre ordered set a into an unstructured set b we study the problem of defining a suitable pre ordering relation on b such that there exists a mapping g b a such that the pair of mappings f g forms an adjunction between pre ordered sets the necessary and sufficient conditions obtained are then expressed in terms of closure operators and closure systems

concept lattices defined from implication operators abstract we introduce the work with two extensions of the concept definition of wille to the fuzzy theory showing the relationship between them we continue with the study of fuzzy material implications and their associated operators of kohout and bandler that lump together the definitions given until this moment to conclude we show an example where we compare the results obtained from the l fuzzy concepts using the fuzzy implications of lukasiewicz kleene dienes and godel brower

attribute reduction in multi adjoint concept lattices knowledge reduction is one of the key issues in formal concept analysis and there have been many studies on this topic the irreducible elements in a lattice are also very important since they form the basic information of a relational system moreover they are also important from the viewpoint of attribute reduction both topics are notably more complicated in a fuzzy setting since not only the size of the sets of attributes and objects influence the size of the fuzzy concept lattice but the truth value sets where the sets of objects attributes and the relation are evaluated are important this paper presents in the general fuzzy framework of multi adjoint concept lattices a characterization of the meet irreducible elements from which a classification of attributes and its application to attribute reduction is introduced