rewriting systems over similarity and generalized pseudometric spaces and their properties we present a study of confluence and related properties of fuzzy relations defined over similarity spaces the ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems this paper is a continuation of our previous paper where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees leaving the ordinary notions a particular case when the underlying structure is the two valued boolean algebra in this paper we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation we also present definitions of confluence and related properties of relations on a generalized pseudometric spaces using the well known link between generalized pseudometrics and similarities we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces the introduced notions are further illustrated by example of rewriting based on graded if then rules

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

mixing times for the mean field blume capel model via aggregate path coupling we investigate the relationship between the mixing times of the glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure for this we consider the mean field blume capel model one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure within a two dimensional parameter space there exists a curve at which the model undergoes a second order continuous phase transition a curve where the model undergoes a first order discontinuous phase transition and a tricritical point which separates the two curves we determine the interface between the regions of slow and rapid mixing in order to completely determine the region of rapid mixing we employ a novel extension of the path coupling method successfully proving rapid mixing even in the absence of contraction between neighboring states

closedness of the category of liminf complete fuzzy orders it is shown that the category of liminf complete fuzzy orders and liminf continuous maps valued in the complete residuated lattice based on a continuous t norm on the unit interval is monoidal closed and that this category is cartesian closed if and only if the t norm is the minimum

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

fuzzy implication operators for difference operations for fuzzy sets and cardinality based measures of comparison in this paper we determine by means of fuzzy implication operators two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively the obtained operations allow us to construct as in b de baets h de meyer transitivity preserving fuzzification schemes for cardinality based similarity measures european journal of operational research 160 2005 726 740 cardinality based similarity measures which are reflexive symmetric and transitive fuzzy relations and to propose two classes of distances metrics which are fuzzy versions of the well known distance of cardinality of the symmetric difference of crisp sets

a note on approximate equality versus the poincare paradox the purpose of this note is to comment on a paper by m de cock and e e kerre that was recently accepted for publication in fuzzy sets and systems the paper in contrast to the view of several other researchers states that fuzzy equivalence relations are not always able to resolve the poincare paradox this brief paper discusses this controversial contribution from the present author s subjective point of view

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

building a class of fuzzy equivalence relations in this paper we propose a practical method given a strict triangular norm with a convex additive generator for deriving a fuzzy equivalence relation whose reflexivity condition generalizes ruspini s definition of fuzzy partitions the properties of the relations their comparison their transitivity the construction of fuzzy equivalence relations on cartesian products are presented a large part of the paper is devoted to applications with fuzzy partitions defined on the real line several examples including the fairy tale problem from de cock and kerre on un suitable fuzzy relations to model approximate equality fuzzy sets and systems 133 2 2003 137 153 the comparison of colored objects and comfort situations are proposed

complementation and embeddings of c0 i in banach spaces we investigate in this paper the complementation of copies of this yields a new characterization of a class of injective banach spaces 2000 mathematical subject classification 46b20 46b26

triangular norms position paper iii continuous t norms this third and last part of a series of position papers on triangular norms for parts i and ii see e p klement r mesiar e pap triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press e p klement r mesiar e pap triangular norms position paper ii general constructions and parameterized families submitted for publication presents the representation of continuous archimedean t norms by means of additive generators and the representation of continuous t norms by means of ordinal sums with archimedean summands both with full proofs finally some consequences of these representation theorems in the context of comparison and convergence of continuous t norms and of the determination of continuous t norms by their diagonal sections are mentioned

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

rigorous results on the bipartite mean field model we consider a bipartite mean field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to we rigorously compute the value of the limiting pressure per particle using tail estimation techniques we study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical magnetic field perturbations are considered

on continuous triangular norms the problem of converging to any given continuous triangular norm by a sequence of continuous archimedean triangular norms is studied such a sequence is built up in a constructive way the class of well founded triangular norms is introduced it is proved that a sequence of triangular norms converges to a given continuous triangular norm if and only if this convergence is uniform finally we prove that every continuous triangular norm is a uniform limit of continuous archimedean triangular norms

on the construction of fuzzy betweenness relations from metrics abstract we consider the problem of constructing a fuzzy betweenness relation from a metric more precisely given a continuous archimedean triangular norm we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm in case the metric is bounded and given a 1 lipschitz continuous triangular norm we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm since the ukasiewicz and product triangular norms are both continuous archimedean and 1 lipschitz continuous all three construction methods may be used interestingly the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the ukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm we end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

the aggregate path coupling method for the potts model on bipartite graph in this paper we derive the large deviation principle for the potts model on the complete bipartite graph kn n as n increases to infinity next for the potts model on kn n we provide an extension of the method of aggregate path coupling that was originally developed in the work of kovchegov otto and titus j stat phys 144 5 1009 1027 2011 for the mean field blume capel model and in kovchegov and otto j stat phys 161 3 553 576 2015 for a general mean field setting that included the generalized curie weiss potts model analyzed in the work of jahnel et al markov process relat fields 20 601 632 2014 we use the aggregate path coupling method to identify and determine the threshold value s separating the rapid and slow mixing regimes for the glauber dynamics of the potts model on kn n

products of elements in vague semigroups and their implementations in vague arithmetic vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in demirci internat j uncertainty fuzziness and knowledge based systems 10 1 2002 25 internat j general systems 32 2 2003 157 177 to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers the main motivating problem of this paper is to introduce the notion of vague product sum of a finite number of real numbers in vague arithmetic and to point out their fundamental properties from a more abstract mathematical point of view the vague product sum of a finite number of real numbers in vague arithmetic and their properties can be considered as the vague product of a finite number of elements in vague semigroups and their relevant properties for this reason a large part of this paper is devoted to the vague product of a finite number of elements in vague semigroups and their elementary properties as a direct implementation of the present results it is shown that the vague product sum of a finite number of real numbers in vague arithmetic can be easily evaluated in terms of the underlying many valued equivalence relations furthermore various non trivial examples for the vague product sum of a finite number of real numbers in vague arithmetic are designed and a simple technique for the construction of such non trivial examples is stated

symmetric difference operators based on thresholds abstract here we study two formulas that define the symmetric difference operators of fuzzy sets in fuzzy logic we call this an exclusive or operator an interesting open problem was to find a certain class of operator systems where the two well known definitions of the symmetric difference operator are equivalent and it is associative here we state a functional equation which characterizes the equivalence of these two forms we present a general class of operators which fulfills the associativity requirement and we show that the two definitions are equivalent we generalize the symmetric difference operator to the n variable case and based on the latter the weighted form can be obtained we will give concrete forms of these operators and we will also show that they have different types of transitivity properties

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

the generalized associative law in vague groups and its applications ii as a continuation of part i vague integral powers of elements in vague groups and their representation properties are introduced in this paper thereafter some rudimentary algebraic properties of vague integral powers of elements obtained from the generalized vague associative law formulated in part i are established the present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic for this reason special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic and some practical applications related to the discrete structure of measurement instruments are also given

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

t evaluators and s evaluators the scalar evaluation of the aggregation of elements from a complete lattice via the operation meet join is compared with the aggregation of scalar evaluations of these elements via a t norm t conorm based on this comparison the notion of a t evaluator s evaluator is introduced and its basic properties are studied the relationship between t s evaluators fuzzy measures and generalized filters is investigated an application of t evaluators in fuzzy preference modelling is presented

similarity based fuzzy limits in this paper the concept of similarity based fuzzy limits in metric spaces is introduced and it is shown that it is an application of limit maps of topologies on 0 1 valued sets introduced by hohle and sostak furthermore it is observed that the similarity based setting introduced is a more general approach to fuzzy limits than the one proposed by burgin

some results on the degree of symmetry of fuzzy relations abstract in this paper we investigate the degree of the symmetry of fuzzy relations on a set x based on the fuzzy e equality derived from uninorms with unit element e the degree of the symmetry of fuzzy relations are defined by two different approaches the type i degree of symmetry and the type ii degree of symmetry the main work of this paper include first we discuss some basic properties of those two degrees especially for the relationship between them in particular for continuous t norms we give a necessary and sufficient condition such that the type i degree of symmetry and the type ii degree of symmetry are equal and then we obtain that the type ii degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type i degree of symmetry with respect to whether they preserve the fuzzy e equalities or not in the meantime for a special case with continuous t norms we provide a necessary and sufficient condition such that the type i degree of symmetry preserves the fuzzy equalities finally for conjunctive left continuous idempotent uninorms with neutral element e 0 1 we find out a symmetric fuzzy relation s which is close enough to the given fuzzy relation r with respect to fuzzy e equality such that the degree of fuzzy e equality of s and r is the type ii degree of symmetry of r in particular we obtain analogous results for continuous t norms and one class of left continuous t norms

fuzzy submonoids fuzzy preorders and quasi metrics let l be a residuated lattice then given a monoid m we define a galois connection from the lattice of the compatible l preorders in m and the lattice of l submonoids of m given a set s we define a galois connection between the lattice of the l preorders in s and the lattice of l submonoids of the monoid ss i a link with the notion of quasi metric is also established

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

on automatic continuity and three problems of the scottish book concerning the boundedness of polynomial functionals in this paper we introduce and study the notions of isotropic mapping and essential kernel in addition some theorems on the borel graph and baire mapping for polynomial operators are proved it is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

a distance metric for a space of linguistic summaries producing linguistic summaries of large databases or temporal sequences of measurements is an endeavor that is receiving increasing attention these summaries can be used in a continuous monitoring situation like eldercare where it is important to ascertain if the current summaries represent an abnormal condition it is therefore necessary to compute the distance between summaries as a basis for such a determination in this paper we propose a dissimilarity measure between summaries based on fuzzy protoforms and prove that this measure is a metric we take into account not only the linguistic meaning of the summaries but also two quality evaluations namely the truth values and the degrees of focus we present examples of how the distance metric behaves and show that it corresponds with intuition

glauber dynamics for the mean field ising model cut off critical power law and metastability we study the glauber dynamics for the ising model on the complete graph also known as the curie weiss model for 1 we study metastability in particular we show that the glauber dynamics restricted to states of non negative magnetization has mixing time o n log n

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

coreflective hull of finite strong l topological spaces abstract this paper is concerned with the interaction between the logic features of the table of truth values and categorical properties of l topological spaces and l co topological spaces on one hand it is shown that for each unital quantale l the category of alexandroff strong l co topological spaces is the coreflective hull of finite strong l co topological spaces on the other hand in the case that the quantale l is the unit interval 0 1 equipped with a continuous t norm it is shown that the category of alexandroff strong 0 1 topological spaces is the coreflective hull of finite strong 0 1 topological spaces if and only if the continuous t norm is an ordinal sum of the ukasiewicz t norm whose set of idempotent elements is a well ordered subset of 0 1 under the usual order

on the compactness of admissible transformations of fuzzy partitions in terms of t equivalence relations this paper analyzes constraints in terms of fuzzy logical concepts for transforming families of fuzzy sets in order to guarantee linguistically interpretable solutions of fine tuned or optimized fuzzy systems and further to make the whole process of fine tuning a well posed problem for this purpose transformations for fuzzy sets are introduced which are constructed by means of the compositional rule of inference it is investigated under which conditions on these transformations various characteristics of fuzzy partitions and fuzzy sets like redundancy convexity and topological aspects remain invariant it is argued that further restrictions like the compactness of the set of admissible transformations are required in order to make the process of fine tuning well posed as main result it is demonstrated that compactness of this function space endowed with the uniform norm can be characterized in terms of purely fuzzy logical concepts that is by means of fuzzy equivalence relations

zeroes of real polynomials on c k spaces for a compact hausdorff topological space k we show that the function space c k must satisfy the following dichotomy i either it admits a positive definite continuous 2 homogeneous real valued polynomial ii or every continuous 2 homogeneous real valued polynomial vanishes in a non separable closed linear subspace moreover if k does not have the countable chain condition then every continuous polynomial not necessarily homogeneous and with arbitrary degree has constant value in an isometric copy of c0 for some uncountable

scaling limits for multi species statistical mechanics mean field models we study the limiting thermodynamic behavior of the normalized sums of spins in multi species curie weiss models we find sufficient conditions for the limiting random vector to be gaussian or to have an exponential distribution of higher order and compute the covariance matrix in terms of model parameters

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

algebras with fuzzy equalities an algebra with fuzzy equality is a set with operations on it that is equipped with similarity i e a fuzzy equivalence relation such that each operation f is compatible with described verbally compatibility says that each f yields similar results if applied to pairwise similar arguments on the one hand algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and have been studied from this point of view before on the other hand they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones the present paper aims at developing fundamental points of algebras with fuzzy equalities we introduce the notion of an algebra with fuzzy equality present natural examples compare the notion with other approaches and introduce and develop basic structural notions subalgebras morphisms products direct unions in a follow up paper vychodil direct limits and reduced products of algebras with fuzzy equalities submitted for publication we deal with advanced topics in algebras with fuzzy equalities direct limits reduced products

zeroes of polynomials on l abstract consider a polynomial p on complex l space we prove that if p vanishes at zero then it also vanishes on a non separable subspace for a polynomial on real l space we prove that if p vanishes on a copy of c 0 then it necessarily vanishes on a non separable subspace

remarks on uninorm aggregation operators it is proved that continuous uninorm aggregation operators are only t norms or t conorms some general constructions about uninorm aggregation operators are given

dominance in the family of sugeno weber t norms the dominance relationship between two members of the family of sugeno weber t norms is proven by using a quantifier elimination algorithm further it is shown that dominance is a transitive and therefore also an order relation on this family of t norms

a generalization of the mulholland inequality for continuous archimedean t norms it is well known that dominance between strict t norms is closely related to the mulholland inequality which can be seen as a generalization of the minkowski inequality however strict t norms constitute only one part of the class of continuous archimedean t norms the basic elements from which all continuous t norms are composed in this paper dominance between continuous archimedean t norms is shown to be related to a generalization of the mulholland inequality we provide sufficient and necessary conditions for its fulfillment

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

measures of specificity over continuous spaces under similarity relations we introduce the concept of specificity and indicate its importance as a measure of uncertainty for information represented using fuzzy sets or possibility distributions we provide formal measures of specificity for variables whose domain is an interval of the real line similarity relations are discussed and formalized the class of width based similarity relations is introduced we extend the measure of specificity to allow for the effect of an underlying similarity relationship the manifestation of this extension for a number of different similarity relations is investigated particularly notable results are obtained for width based similarity relationships more generally we note the connection between the size of a granule of information and its uncertainty motivated by this we investigate the effect of a similarity relationship on the perception of distance in the underlying space and use this to develop an extension of the specificity measure