characterization of fuzzy preference structures through łukasiewicz triplets in this paper we characterize the reconstruction of a fuzzy set from its subsets by means of lukasiewicz triplets this result allows us to introduce a new definition of fuzzy strict preference which is also satisfied in the crisp framework the usual definitions of fuzzy indifference and fuzzy incomparability together with this one enable to construct and to characterize fuzzy preference structures from a reflexive fuzzy binary relation

emergence of fuzzy preferences for risk in a birkhoff von neumann logics environment we show that if a portfolio of a financial derivative asset and a stock is put in an environment where the value of an asset besides its price is formalized as a superposition of price states such portfolio may not be risk free and fuzzy preferences for risk premia may exist we argue for a modification of the classical brownian motion process as used in option pricing this modification on the classical brownian motion we call the email protected brownian motion and one specific format of this email protected brownian motion can be shown to have a connection with the quantum physical schrodinger equation

time reversal in stochastic processes and the dirac equation we consider the motion of a classical particle in 1 1 dimensional space time four probability distributions govern the trajectory of the particle these give the probability of moving to the left or right in space while moving backwards or forwards in time if these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner then the master equations for these probabilities give rise to the dirac equation without recourse to direct analytic continuation in contrast when a particle always moves forward in time an analytic continuation is required to recover the dirac equation

on the fuzzification of the classical definition of preference structure this paper is divided into two main parts in the first part we propose an appropriate definition for fuzzy preference structures the definition is based on two main features the first feature consists in using a strong de morgan triplet as a model for the classical de morgan triplet the second feature consists in choosing a particular completeness condition such that when it is combined with some other conditions all the generalized completeness conditions are satisfied these two features of our definition are justified by a set of remarkable theorems and propositions proving some equivalencies we can write this definition with only three minimal and independent conditions in the second part we use the new definition to analyse two methods of construction of fuzzy strict preference indifference and incomparability relations from a given reflexive fuzzy relation on the one hand we prove that the triplets of fuzzy strict preference indifference and incomparability relations constructed from a reflexive fuzzy relation by using a same t norm are not necessarily fuzzy preference structures on the other hand we prove that those generated by the method of fodor and roubens are fuzzy preference structures according to the proposed definition

an axiomatic approach to fuzzy preference modelling abstract we propose an axiomatic approach to the definition of fuzzy strict preference p indifference i and incomparability j associated with a fuzzy preference relation r expression for p i and j are given via solving two systems of functional equations

a valued ferrers relation for interval comparison the paper deals with the valued comparison of intervals for decision making interval orders are classical preference structures where the comparison of intervals is done in an ordinal way in this paper we focus on valued comparison where more information especially the distance between end points of intervals is used in order to have more sophisticated preference structures the generalization of an interval order as a valued structure requires the choice of de morgan triplets we propose a valued outranking relation for interval comparison and show that it satisfies different definitions of valued interval orders using different de morgan triplet the decomposition of our outranking relation into preference and indifference provides a valued preference structure where the preference is t transitive and monotone

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

on the rationality of knorr s virtually irreducible lattices abstract it was established by reinhard knorr that the half of brauer s height zero conjecture which assumes an abelian defect group would follow from the nonexistence of certain integral representations of abelian p groups we establish nonexistence of such representations of rank less than 14 for the elementary abelian group of order 8 and believe these to be the first results of this kind the result is made possible by imposing a simple rationality condition

examples counterexamples and measuring fuzzy associations this paper examines the measurement of the degree to which tuples in a database support a relation among attributes based on a comparison of the number of examples and counterexamples of the relation in particular we are concerned with associations that represent imprecise constraints placed upon the value of one attribute by those of other attributes associations of this form may be described by fuzzy rules and the analysis requires an assessment of the degree to which a tuple satisfies the imprecise constraint specified by the rule standard measures of rule validity are extended to fuzzy associations based on the degree that the tuples are examples counterexamples or irrelevant to imprecise relations a scaling of the relevance of a tuple is proposed to minimize the impact of the accumulation of small membership values on the confidence based validity measures

convex combinations in terms of triangular norms a characterization of idempotent bisymmetrical and self dual compensatory operators it is observed that most of the well known compensatory operators are defined in terms of convex transformations of the values of triangular norms and conorms or other operators in this paper the characterization and the applicability of parametrized operators constructed again in terms of triangular norms and conorms but as transformations of their arguments are studied it turns out that these operators are closely related to the ordinary convex combination moreover when idempotency is required these operators can be characterized algebraically as bisymmetrical and self dual operators that satisfy the cancellation law

valued preference structures abstract an approach to the axiomatics of valued preference modelling is suggested in the framework of this approach general definitions of strict preference indifference and incomparability relations associated with a valued preference relation are established via solving a system of functional equations any solution of this system satisfies most of the classical properties characterizations of important particular solutions are also proved

construction of global fuzzy preference structures from two parameter single criterion fuzzy outranking relations in this paper we deal with the construction of global fuzzy preference structures from two parameter single criterion fuzzy outranking relations proposed in 10 and which are based on the aggregation of outranking indices associated with a cuts we consider two ways of constructing global fuzzy preference structures i aggregate the two parameter single criterion fuzzy outranking relations and then construct the global fuzzy preference structures and ii construct global fuzzy preference structures and then aggregate the single criterion fuzzy preference indifference and incomparability relations we also investigate the case where these two ways of proceeding lead to the same results

distributivity inequalities of monotonic operations the aim of this work is to complement recent results about distributive binary operations our investigation was motivated by non distributive logical connectives and their generalizations in fuzzy set theory in spite of the title we neither solve the concerned functional inequalities nor look for characterization of the solutions of such inequalities we only lay bare conditions for their fulfilment for monotonic operations with neutral or zero element we present the sufficient conditions to obtain the subdistributivity or superdistributivity in the case of the lack of distributivity for the considered pairs of operations in particular we describe families of increasing binary operations subdistributive or superdistributive with respect to idempotent uninorms and nullnorms

fuzzy sets and decision analysis abstract this survey points out recent advances in multiple attribute decision making methods dealing with fuzzy or ill defined information fuzzy maut as well as fuzzy outranking methods are reviewed aggregation procedures choice problems and treatment of interactive attributes are covered trends in research and open problems are indicated

on the compositional characterization of complete fuzzy pre orders complete pre orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations in this paper we investigate this characterization in a fuzzy setting we consider two types of completeness weak completeness and strong completeness and decompose a fuzzy pre order by means of an indifference generator in particular a frank t norm in the weakly complete case we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them in the strongly complete case we lay bare a stronger type of transitivity of the strict preference relation we conclude the paper with a rather negative result there is no hope to obtain a compositional characterization of weakly complete fuzzy pre orders and hence also not of fuzzy pre orders in general

traces of fuzzy binary relations abstract in this paper the right and left trace of a fuzzy binary relation is defined and investigated basic properties of a fuzzy relation such as reflexivity different kinds of transitivity linearity etc are characterized by means of its traces

fuzzy interval and semi orders abstract the seminal paper by luce econometrica 24 1956 178 on intransitive indifference provoked a rich body of research on how to model such a type of indifference interval orders and semi orders play a central role in the modelization of intransitive indifference this paper is an attempt to fuzzify interval orders and semi orders we use an additive fuzzy preference structure to argue for a fuzzification of interval and semi orders we show that fuzzy interval orders are to a degree semi orders however against expectations we cannot show that fuzzy interval graphs are to a degree fuzzy indifference graphs

the distributivity property of increasing binary operations abstract this paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory in particular some assumptions namely associativity and commutativity and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified moreover some other properties e g componentwise convexity concavity for special operations from the class of quasi copulas is considered

systems of ordinal fuzzy logic with application to preference modelling abstract in this paper we survey several many valued propositional logics in which the truth functions in the real unit interval 0 1 of their connectives are definable only from the natural ordering of the scale without using any richer algebraic structure in particular we describe a complete notion of proof for weighted formulas of the implication free fragment of godel logic with involution the usefulness of this logical framework for the formalization of ordinal preference modelling is also shown

parametric classes of generalized conjunction and disjunction operations for fuzzy modeling it is argued that inference procedures of fuzzy models do not always require commutativity and associativity of the operations used this raises the possibility of considering nonassociative and noncommutative conjunction and disjunction operations such operations are investigated in this paper and different methods for their generation are proposed a number of new types of conjunction operations that are simpler than the known parametric classes of t norms are given and as an application example the approximation of a function by a fuzzy inference system is considered

fuzzy confirmation measures this paper proposes a number of measures for assessing the quality of fuzzy association rules the focus is on a particular class of measures which are fuzzy counterparts of probabilistic confirmation measures the distinguishing feature of such measures is that they are based on probabilistic in dependence and so a corresponding definition of fuzzy in dependence is proposed this forms the basis for the definition of a fuzzy confirmation measure and for proposing a number of such measures based on a well known fuzzy confidence measure various properties are considered to narrow the field of viable fuzzy confirmation measures with one particular measure being found to have a number of advantages

the law of large numbers for fuzzy variables under a general triangular norm extension principle this paper presents a law of large numbers for t noninteractive weakly noninteractive fuzzy variables when using a general t norm extension principle the result is that convergence to a crisp set is obtained for all archimediean t norm intersection operators this generalises a previous result of badard the result is proved by deriving it from a similar result for the law of large numbers for dependency bounds of sums of random variables dependency bounds are used when nothing is known of the joint distribution of two random variables apart from their marginals the bridge used to connect dependency bounds with fuzzy number operations is used to give a probabilistic interpretation of fuzzy number operations some numerical examples are presented and a new method of numerically calculating the sup t convolutions of fuzzy numbers is presented this new method generalises the well known result for the calculation of sup min convolutions in terms of the level sets of membership functions

knorr lattices their characters and some subsets of fpd examples of positive height cyclic knorr lattices for elementary abelian p groups are constructed and bounds relating the height of such lattices to the size of the group are given the problem is formulated in terms of finite geometry when p 2 certain examples are shown to arise from difference sets

managing non homogeneous information in group decision making within the frame of decision aid literature decision making problems with multiple sources of information have drawn the attention of researchers from a wide spectrum of disciplines in decision situations with multiple individuals each one has his own knowledge on the alternatives of the decision problem the use of information assessed in different domains is not a seldom situation this non homogeneous information can be represented as values belonging to domains with different nature as linguistic numerical and interval valued or can be values assessed in label sets with different granularity multi granular linguistic information decision processes for solving these problems are composed by two phases aggregation and exploitation the main problem to deal with non homogeneous contexts is how to aggregate the information assessed in these contexts in this paper taking as base the 2 tuple fuzzy linguistic representation model we shall develop an aggregation process for dealing with non homogeneous contexts in first place we shall develop an aggregation process for combining numerical interval valued and linguistic information afterwards we shall propose different extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi granular linguistic information

adaptation and optimal chemotactic strategy for e coli extending the classic works of berg and purcell on the biophysics of bacterial chemotaxis we find the optimal chemotactic strategy for the peritrichous bacterium ital e coli in the high and low signal to noise ratio limits the optimal strategy depends on properties of the environment and properties of the individual bacterium and is therefore highly adaptive we review experiments relevant to testing both the form of the proposed strategy and its adaptability and propose extensions of them which could test the limits of the adaptability in this simplest sensory processing system copyright ital 1998 ital the american physical society

some properties of choice functions based on valued binary relations abstract choice functions based on t norms of valued binary relations are introduced strict preference is also specified with the use of a t norm properties of the choice functions are investigated and rationality conditions are studied some classical particular cases are presented

learning valued preference structures for solving classification problems this paper introduces a new approach to classification which combines pairwise decomposition techniques with ideas and tools from fuzzy preference modeling more specifically our approach first decomposes a polychotomous classification problem involving m classes into an ensemble of binary problems one for each ordered pair of classes the corresponding classifiers are trained on the relevant subsets of the transformed original training data in the classification phase a new query is submitted to every binary learner the output of each classifier is interpreted as a fuzzy degree of preference for the first in comparison with the second class by combining the outputs of all classifiers one thus obtains a fuzzy preference relation which is taken as a point of departure for the final classification decision this way the problem of classification is effectively reduced to a problem of decision making based on a fuzzy preference relation corresponding techniques which have been investigated quite intensively in the field of fuzzy set theory hence become amenable to the task of classification in particular by decomposing a preference relation into a strict preference an indifference and an incomparability relation this approach allows one to quantify different types of uncertainty in classification and thereby supports sophisticated classification and postprocessing strategies

weak and strong fuzzy interval orders abstract the purpose of this paper is to introduce and investigate the fuzzification of the classical interval order one of the most interesting classical preference structures without incomparability in this study we consider fuzzy preference structures as defined by de baets et al fuzzy preference structures without incomparability receive special attention their fuzzy preference and fuzzy large preference relations share certain types of the t n ferrers property two special types of the j n ferrers property are introduced the weak ferrers and strong ferrers properties the classical interval order is briefly reviewed t fuzzy interval orders are introduced and it is shown that their fuzzy preference relation is sup t transitive two special types of t fuzzy interval orders are considered weak and strong fuzzy interval orders corresponding to transforms of w and to m the particular intermediate role of the weak ferrers property of the fuzzy preference relation of a fps without incomparability is demonstrated on the one hand it is a necessary condition for the fps to be a strong fuzzy interval order while on the other hand it is a sufficient condition for this structure to be a weak fuzzy interval order finally the concept of an cut of a fps is introduced leading to an interesting characterization of the strong ferrers property of a fps without incomparability

approximate reasoning for production planning abstract approximate reasoning for production planning could effectively be based on an interval valued representation of linguistic variables an interval valued representation provides a framework that i incorporates a dispersion about membership values and ii produces a robust result compared to classical results the proposed representation and inference is applied to a set of production problems in prototype studies the result of one such prototype study is summarized

construction of a two parameters fuzzy outranking relation from fuzzy evaluations at the early stages of design process several design concepts are proposed as potential solutions the selection of the most appropriate one is based on their evaluation with respect to several conflicting criteria in this paper we consider the case where the evaluations of design concepts with respect to different criteria are represented by means of fuzzy numbers we deal with the construction of a two parameters fuzzy outranking relation from fuzzy evaluations it is based on the aggregation of outranking indices associated with alpha cuts this fuzzy outranking index allows the decision maker to adopt different strategies by choosing a degree of optimism and a degree of aggressiveness

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

fuzzy rationality measures fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives it is therefore natural to extend to fuzzy preferences the notion of rationality or consistency with this goal in mind we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality specifically we establish a collection of conditions that any fuzzy rationality measure must satisfy some examples of fuzzy rationality measures are then given and analyzed

elicitation of fuzzy association rules from positive and negative examples the aim of this paper is to provide a crystal clear insight into the true semantics of the measures of support and confidence that are used to assess rule quality in fuzzy association rule mining to achieve this we rely on two important pillars the identification of transactions in a database as positive or negative examples of a given association between attributes and the correspondence between measures of support and confidence on one hand and measures of compatibility and inclusion on the other hand in this way we remove the mystery from recently suggested quality measures for fuzzy association rules

comparing fuzzy partitions a generalization of the rand index and related measures in this paper we introduce a fuzzy extension of a class of measures to compare clustering structures namely measures that are based on the number of concordant and the number of discordant pairs of data points this class includes the well known rand index but also commonly used alternatives such as the jaccard measure in contrast with previous proposals our extension exhibits desirable metrical properties apart from elaborating on formal properties of this kind we present an experimental study in which we compare different fuzzy extensions of the rand index and the jaccard measure

a survey of weak connectives and the preservation of their properties by aggregations the aim of this paper is to choose diverse definitions of generalized logical connectives and to present them in a coherent order from the weakest to the richest such a rich list of notions allows us to consider the problem of admissible aggregations in the presented classes of unary and binary operations this gives a contribution to the discussion of tolerance analysis in soft computing decision making approximate reasoning and fuzzy control first we present a short survey of the development of mv logic s connectives next we discuss postulates used for generalized logical connectives finally we describe families of weak connectives and indicate the preservation of their properties by some aggregation functions

indicators of fuzzy relations in this paper first we give the definitions of various indicators of fuzzy relations and their basic properties then we investigate the relationships between these indicators particularly between those of t transitivity negative s transitivity t s semitransitivity and t s ferrers property the investigation is divided into three parts results in the general case results under a condition proposed by peng et al and results under special t norms and t conorms on account of our research certain results on fuzzy relations in the literature are extended to the degree description

strict preference relations based on weak t norms abstract in this paper we introduce and investigate weak t norms and related concepts in order to give a general representation of valued strict preference relations associated with a given valued preference relation all the well known forms of valued strict preference relations are particular cases of our results moreover when some types of rationality assumptions are made our unified approach seems to be the only possible one

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

some cyclic virtually irreducible lattices abstract virtually irreducible lattices which satisfy a simple rationality condition and have positive height are constructed for elementary abelian 2 groups the non existence of such lattices would have implied part of brauer s height zero conjecture a characterization of rank 6 lattices for the elementary abelian group of size 16 is given

closing reciprocal relations w r t stochastic transitivity we equip the set of reciprocal relations with an appropriate order relation capturing the bipolar semantics turning this set into a complete join semilattice we also introduce the notion of a compatible family of reciprocal relations and show that such a family has an infimum moreover we discuss the one to one correspondence between the set of 3 valued reciprocal relations and the set of complete crisp relations we refine the theorem of bandler and kohout concerning the existence of closures of elements of a poset w r t some given property rendering it applicable to the above join semilattice the paper is solely dedicated to the transitivity property although uniquely defined for 3 valued reciprocal relations general reciprocal relations can exhibit various types of transitivity we characterize the mappings g and h for which the corresponding g stochastic and h isostochastic transitive closure exists for any reciprocal relation in particular it follows that weak and strong stochastic transitive closures always exist while this is not the case for moderate stochastic transitivity moreover max isostochastic transitivity turns out to be the only practically relevant type of isostochastic transitivity finally we provide algorithms realizing each of these transitive closures

reflections on an old problem that of preserving the logical forms and symmetry abstract this paper is devoted to analyze and discuss some of the new aspects about the preservation of the logical forms appearing in ordinary reasoning and fuzzy logic considering a basic fuzzy algebra bfa as the basic formal model for representing imprecise predicates several classical properties of logical forms which can be not valid depending on the representation of the logical connectives in a bfa are studied special emphasis is put on the general symmetry property or its absence in some logical forms that in some particular cases and in some sense can become symmetrical as they are for instance the linguistic negation the law of perfect repartition or the additive law of measures in any case the difference between what is supposed in a mathematical model and what can be checked in the reality cannot be forgotten since any reality and the study of the ordinary reasoning in natural language is a clear example of that is but an observed reality anyway if what is in a mathematical fuzzy model is not always symmetrical and since these models represent linguistic statements affected by imprecision and uncertainty it can be concluded that in general neither plain language nor commonsense or ordinary reasoning show the same level of symmetry exhibited by the specialized language and the deductive reasoning allowing mathematical proofs

theory and methodologysome properties of choice functions based on valued binary relations choice functions based on t norms of valued binary relations are introduced strict preference is also specified with the use of a t norm properties of the choice functions are investigated and rationality conditions are studied some classical particular cases are presented

fuzzy connectives via matrix logic abstract an approach to the definition of fuzzy logical connectives is proposed using matrix logic this matrix representation of boolean logical operators enables us to link together conjunction disjunction and negation differently from the traditional de morgan law in this framework the frank family appears as the only possibility in addition the parameters in conjuntion and in disjunction are reciprocal to each other the usefulness of this case is illustrated by two applications

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

the use of fuzzy outranking relations in preference modelling this paper deals with preference modelling in the context of decision aid in this framework conflicting systems of logic uncertain knowledge ambiguous positions are always present in order to tackle this problem a multiple criteria methodology is proposed mainly based on fuzzy outranking relations introduced both at one dimensional and multi dimensional levels some properties of outranking relations are investigated such relations are then combined using fuzzy logical connectives to generate relational systems of fuzzy preferences that are shown to be very useful to reflect the vagueness of information in the various preference situations that may be considered in the modelling process

preservation of fuzzy relation properties based on fuzzy conjunctions and disjunctions during aggregation process in this contribution necessary and sufficient conditions for n argument functions to preserve properties of fuzzy relations during aggregation process are presented the proposed properties depend on binary operations including t norms and t conorms and more generally fuzzy conjunctions and disjunctions generalizations of the following properties are studied antisymmetry asymmetry connectedness transitivity negative transitivity semitransitivity and ferrers property

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

a dynamic consensus scheme based on a nonreciprocal fuzzy preference relation modeling in this paper we propose a dynamical consensus scheme which makes use of an index of comparability and of an index of concordance to regulate the information flow in the discussion among the experts in addition to the traditional use of a concordance index to identify the discordant expert who can be invited to review his opinion or to explain his judgment to the other experts here an index of comparability is utilized to identify the experts that have faced difficulties in expressing their preferences with this information at hand a moderator can invite the experts that are confident on their opinions to support the experts that are hesitating in the construction of the fuzzy preference relations

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

approximate evaluations based on aggregation functions aggregation of real valued evaluations of elements x y from a set u is investigated in order to find an approximate evaluation of the composition of x and y performed by a binary operation defined on u the notions of a lower bound and an upper bound of an evaluation function with respect to a binary operation are introduced and illustrated with examples approximate evaluations of composite concepts described by the union of granules of a finite fuzzy set are studied in detail

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures ii the identity case abstract it has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de morgan triplet involved contains a continuous archimedean triangular norm having zero divisors or hence a transform of the ukasiewicz triangular norm in this second part additional arguments for this statement are supplied in what we call the identity case is the identity mapping and the involutive negator is the standard negator first it is shown that the use of a continuous non archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible secondly using such a triangular norm implies that at least in the square 0 4 5 2 it actually behaves as the ukasiewicz triangular norm furthermore an important transformation theorem indicates that any fuzzy preference structure with respect to a de morgan triplet containing such a continuous non archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard ukasiewicz triplet these additional arguments conclude in a convincing way our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

dimension of valued relations abstract the classical notion of dimension of a partial order can be extended to the valued setting as was indicated in a particular case by ovchinnikov 1984 ovchinnikov s v 1984 representations of transitive fuzzy relations in skala h j termini s trillas e eds aspects of vagueness reidel boston pp 105 118 relying on valverde s result 1985 valverde l 1985 on the structure of f indistinguishability operators fuzzy sets and systems 17 313 328 on the transitive closure of a valued relation we define the dimension of a valued quasi order building then on fodor and roubens 1995 fodor j roubens m 1995 structure of transitive valued binary relations mathematical social sciences 30 71 94 we also show that the definition can be generalized to all valued relations by using valued biorders instead of valued weak orders as one dimensional relations interesting combinatorial questions about the new dimension concept arise and are investigated here in particular we aim at a characterization of valued quasi orders of dimension two

fuzzy preference structures without incomparability abstract in this paper we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability this study is carried out for the fuzzy preference structures introduced recently by de baets van de walle and kerre a set of remarkable theorems gives detailed insight in the relationships between the sup t transitivity of the fuzzy preference and indifference relations and the sup t transitivity of the fuzzy large preference relation several paths of thought involving t norms with or without zero divisors are explored and where required illustrative counterexamples confirm the falsity of certain implications finally we introduce the t n ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this ferrers property

effects of cross diffusion on turing bifurcations in two species reaction transport systems we study the turing bifurcation in general two species reaction transport systems where particle dispersal is governed by diffusion including cross diffusion by performing a linear stability analysis we find conditions for the turing instability and compare the results with the standard turing conditions we apply our results to two model systems the lengyel epstein model and the brusselator and find strong effects of cross diffusion on the turing bifurcation

subdistributivity and superdistributivity of binary operations recently the property of distributivity of binary operations on 0 1 is of great interest both for practical as well as fundamental reasons in this paper we summarize some facts connected with distributivity and non distributivity of binary operations on 0 1 motivated by non distributive logical connectives in fuzzy set theory we show some new results concerning the above property a new direction of investigations mainly concerns the distributivity inequalities for triangular norms and conorms and their diverse generalizations in particular we look for examples of binary operations fulfilling the distributivity inequalities

on mixed de morgan triplets abstract in this paper we find the general solution of the functional equation t f x y f x n y h x where t is a t norm or t conorm f is an aggregation function n is a strong negation on the unit interval and h is an increasing function

measurement theoretic justification of connectives in fuzzy set theory abstract the problem of representing intersection and union in fuzzy set theory is considered there are various proposals in the literature to model these concepts the possibility of using continuous triangular norms and conorms including min and max are taken up in a measurement theoretic setting the conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed these conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful it is emphasized that the archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations

continuity of left continuous triangular norms with strong induced negations and their boundary condition first a characterization is given for the continuity of a class of left continuous triangular norms it is established that a left continuous triangular norm with strong induced negation is continuous if and only if it is strictly increasing on the domain where its value is positive second we show that axioms of a left continuous triangular norm with strong induced negation are not independent namely the boundary condition follows from the other axioms

on some solutions of the distributivity equation abstract the aim of this paper is to present some interesting results on the distributivity equation for which several pairs of binary operations on 0 1 are considered moreover different methods to solve distributivity equations are used in this work

on fuzzy strict preference indifference and incomparability relations abstract the existence of fuzzy models for strict preference indifference and incomparability satisfying all classical properties is established a particular class of such models is completely described

a formal study of linearity axioms for fuzzy orderings abstract this contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings different existing concepts are evaluated with respect to three fundamental correspondences from the classical case linearizability of partial orderings intersection representation and one to one correspondence between linearity and maximality as a main result we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case if we do not require a one to one correspondence between linearity and maximality however we obtain that an implication based definition appears to constitute a sound compromise in particular if ukasiewicz type logics are considered

the solution set of a fuzzy relational equation with sup conjunctor composition in a complete lattice this paper deals with a fuzzy relational equation with sup conjunctor composition in a complete lattice we investigate its properties and describe its solution set in particular we give a sufficient and necessary condition that the solution set is nonempty and a sufficient condition that there exist minimal elements in the solution set

randomly curved runs interrupted by tumbling a model for bacterial motion received 23 february 2005 published 24 august 2005small bacteria are strongly buffeted by brownian forces that make completely straight runs impossible amodel for bacterial motion is formulated in which the effects of uctuational forces and torques on the runphase are taken into account by using coupled langevin equations an integrated description of the motion including runs and tumbles is then obtained by the use of convolution and laplace transforms the propertiesof the velocity velocity correlation function of the mean displacement and of the two relevant diffusioncoef cients are examined in terms of the bacterial sizes and of the magnitude of the propelling forces forbacteria smaller than

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures i general argumentation abstract the generalization of the concept of a classical or crisp preference structure to that of a fuzzy preference structure expressing degrees of strict preference indifference and incomparability among a set of alternatives requires the choice of a de morgan triplet i e of a triangular norm and an involutive negator the resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences more specifically one of the degrees of strict preference indifference or incomparability should always be unconstrained to the preference modeller this intuitive requirement is violated when choosing a triangular norm without zero divisors since in that case fuzzy preference structures reduce to classical preference structures and hence none of the degrees can be freely assigned furthermore it is shown that the choice of a continuous non archimedean triangular norm having zero divisors is not compatible with our basic requirement the sets of degrees of strict preference indifference and incomparability in 0 1 are always bounded from above by a value strictly smaller than 1 these fundamental results imply that when working with a continuous triangular norm only archimedean ones having zero divisors are suitable candidates these arguments sufficiently support our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

on transitivity of strict preference relations abstract a general notion of a strict preference relation associated with a preference relation valued in a semilattice with difference is introduced if a preference relation is valued in a linear scale then the associated strict preference relation is always transitive in contrast for some nonlinear scales transitivity of a preference relation does not imply transitivity of the associated strict preference relation the identity x y z x z y z is characteristic for the class of lower semilattices with difference for which associated strict preference relations preserve transitivity of preference relations

a general model for deriving preference structures from data in this paper we comment upon the integrated model for valued preferences introduced by fodor ovchinnikov and roubens in particular while on one hand we revise basic assumptions and point out their intuitive meaning on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results but also a functional characterization of the whole family of solutions

an alternative definition for fuzzy interval orders the class of interval orders is one of the most studied classes of preference structures without incomparability in the theory of classical preference modelling in this paper we propose a generalization of the classical definition of interval order and strict interval order we focus our attention on two important classes of interval orders and strict interval orders which are respectively defined by means of a strong de morgan triplet and a min max de morgan triplet

the frank inequality abstract we investigate a functional inequality for copulas that has emerged from our study of the comparison of a set of random variables pairwisely coupled by a same copula any copula satisfying this inequality is necessarily symmetric and radially symmetric moreover any associative copula satisfying this inequality is a solution to the well known frank equation for this reason the inequality is coined the frank inequality we fully characterize the associative copulas that satisfy the frank inequality they turn out to be either frank copulas or ordinal sums of a same frank copula with equidistant idempotent elements as a by product we observe that frank copulas are super additive on the unit square

a general approach to solving a wide class of fuzzy optimization problems results of research into the use of fuzzy sets for handling various forms of uncertainty in the optimal design and control of complex systems are presented a general approach to solving a wide class of optimization problems containing fuzzy coefficients in objective functions and constraints is described it involves a modification of traditional mathematical programming methods and is associated with formulating and solving one and the same problem within the framework of mutually conjugated models this approach allows one to maximally cut off dominated alternatives from below as well as from above the subsequent contraction of the decision uncertainty region is associated with reduction of the problem to multicriteria decision making in a fuzzy environment the general approach is applied within the context of a fuzzy discrete optimization model that is based on a modification of discrete optimization algorithms prior to application of these algorithms there is a transition from a model with fuzzy coefficients in objective functions and constraints to an equivalent analog with fuzzy coefficients in objective functions alone the results of the paper are of a universal character and are already being used to solve problems of power engineering

interval valued preference structures different languages that are offered to model vague preferences are reviewed and an interval valued language is proposed to resolve a particular difficulty encountered with other languages it is shown that interval valued languages are well defined for de morgan triples constructed by continuous triangular norms conorms and a strong negation function a new transitivity condition for vague preferences is suggested and its relationships to known transitivity conditions are established a complete characterization of intervalvalued preference structures is also provided

fuzzy strict preference and social choice in this paper we generalize some classical factorizations of a fuzzy relation into a symmetric component indifference and an asymmetric and regular component regular fuzzy strict preference from the above notions we establish two properties of any regular fuzzy strict preference of a max transitive fuzzy relation which are then used to obtain new fuzzy versions of gibbard s oligarchy theorem and arrow s impossibility theorem

conditional extensions of fuzzy preorders abstract the problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations and their applications several variations of the acclaimed szpilrajn s theorem have been provided inclusive of the case when some order conditions between elements are imposed on the extension we extend the analysis of that topic by alcantud to the fuzzy case by appealing to generators to decompose fuzzy preference relations into strict preference and indifference relations we give general extension results for the corresponding concept of compatible extension of a fuzzy reflexive relation then we investigate the conditions under which compatible order extensions exist such that certain elements are connected by the asymmetric part resp and certain other elements by the symmetric part to respective elements with degree 1

structure of left continuous triangular norms with strong induced negations iii construction and decomposition this paper is the continuation of jenei j appl non classical logics 10 2000 83 92 11 2001 351 366 where the rotation construction and the rotation annihilation construction have been presented respectively both constructions produce left continuous but not continuous triangular norms with strong induced negations here we show that these two constructions allow us to define indecomposability and that each decomposable left continuous triangular norm with strong induced negation can be derived as result of one of the above constructions in this way two general decomposition theorems arise in addition the decomposition of left continuous triangular norms with strong induced negations can be defined in a unique way namely when the decomposition is done by the minimal decomposition point

individual based and mean field modeling of direct aggregation we introduce two models of biological aggregation based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood in the first order model the location of each individual is subject to a density dependent random walk while in the second order model the density dependent random walk acts on the velocity variable together with a density dependent damping term the main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals instead aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density we formally derive the corresponding mean field limits leading to nonlocal degenerate diffusions then we carry out the mathematical analysis of the first order model in particular we prove the existence of weak solutions and show that it allows for measure valued steady states we also perform linear stability analysis and identify conditions for pattern formation moreover we discuss the role of the nonlocality for well posedness of the first order model finally we present results of numerical simulations for both the first and second order model on the individual based and continuum levels of description