Localization Relations are symmetric and T -transitive fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets. This paper shows that not all symmetric and T -transitive fuzzy relations may be obtained in such way, provides an easy condition for characterizing those that may, and states a representation theorem for this class of relations. The logical and geometric implications of such representation are analyzed. The paper also characterizes those (just) symmetric fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets.

Some methods to define operations with logics for typical fuzzy lists and multisets are given to operate lists of degrees with different length using the least common multiple method, four filling methods and two truncating methods. The properties of the logics are studied. Some examples are given.

Extensionality is explored form different points of view. Extensional fuzzy subsets from a fuzzy equivalence relation E are considered as observable subsets with respect to the granularity generated by E. Interestingly, they are characterized as the fuzzy subsets that can be obtained as combinations of the fuzzy equivalence classes of E. Extensional mappings are characterized topologically and the set of extensional mappings between two universes are algebraically determined. Specifying the results to fuzzy mappings from a universe X onto [0, 1] an interpretation of type-2 fuzzy subsets of X as fuzzification of its type-1 fuzzy subsets is provided.

Hesitant Fuzzy Sets are useful to represent the information given by several experts. However, this possibility usually require to deal with sets of values with different cardinality and the necessity to compare them, it is, not all experts evaluate all the elements in the universe. In this paper it is proposed a new nomenclature for Hesitant Fuzzy Sets called Hesitant Fuzzy Sets using Lists. It is proposed a new partial order relation between lists of membership degrees. This partial order relation allows to handle lists with different lengths. In addition, Hesitant Fuzzy Relations on Lists operations are defined and the T-transitive closure of a Hesitant Fuzzy Relation is given. It is proved this T-transitive closure always exists and it is unique. Finally, an algorithm to compute the T-transitive closure of a Hesitant Fuzzy Relation is proposed and several examples are given.

An interesting problem within the theory of indistinguishability operators is how to approximate an arbitrary fuzzy subset by a similar extensional one. In this paper, we aim to solve this question and provide three methods to find extensional approximations of fuzzy subsets μ. These methods are exhaustively explained for different Archimedean t-norms, and an example is provided to illustrate them.

In this paper we study the relationship between the cuts of a T-indistinguishability operator and the t-norm T chosen to define fuzzy transitivity. The key result is, strict Archimedean t-norms provide equivalence relations as their zero cuts, and that property characterizes such t-norms. As a consequence, a decomposition theorem for such T-indistinguishabilities is proved.

An isomorphism f between two continuous Archimedean t-norms T and T′ transforms a T-indistinguishability operator E into a T′-indistinguishability operator f ∘ E and many interesting properties of E are transfered to f ∘ E by f. This paper generalizes this result in order to relate indistinguishability operators with respect to two non isomorphic continuous Archimedean t-norms. This will allow us to transfer definitions and properties from strict to non-strict Archimedean t-norms and vice versa.

Fuzzy transitivity is a key property for many fuzzy relational structures, such as Fuzzy Preorders and Equivalences. Theoretical models for practical problems which are based on fuzzy relations make use of continuous scales, mostly the unit interval [0,1]. Practical implementation of these models though, involves their discretization into finite scales, which generally results in some loss of transitivity. In this paper we study if there are any transitivity preserving discretization strategies. Also, we evaluate the loss of transitivity in some commonly used discretization approaches

Given a T-indistinguishability operator, the corresponding set HEof extensional sets and the operators ΦEand ΨEthat provide upper and lower approximations of a fuzzy subset by extensional ones can be defined. Reciprocally, given any of these, the initial indistinguishability operator can be retrieved. In order to understand in depth how this relation works, it will be shown that there is an isomorphism between the corresponding lattices and how this isomorphism acts over the operations and orderings of the lattices will be studied.

Although Fuzzy Preorders and Fuzzy Equivalences are intended to model vague concepts, they are defined in terms of properties or axioms to be fulfilled in a crisp way. In this paper we present two different approaches to overcome this problem (the axiomatic and the similarity based approaches) and relate them both. New results concerning the similarity between relational structures are obtained. As a consequence, every arbitrary fuzzy relation will be considered to be a fuzzy preorder or a fuzzy equivalence, at least to some extent.

Although Fuzzy Preorders and Fuzzy Equivalences are well established types of fuzzy relations, they are defined in terms of properties to be fulfilled in a crisp way. In this paper we relax those requirements by making them gradual in some way. As a result, a number of relations which are not strictly reflexive, symmetric or transitive with respect to any t-norm T, would be regarded as (Approximate) Fuzzy Preorders or Equivalences.

Three ways to approximate a proximity relation$R$(i.e., a reflexive and symmetric fuzzy relation) by a$T$-transitive one where$T$is a continuous Archimedean$t$-norm are given. The first one aggregates the transitive closure$\overline{R}$of$R$with a (maximal)$T$-transitive relation$B$contained in$R$. The second one computes the closest homotecy of$\overline{R}$or$B$to better fit their entries with the ones of$R$. The third method uses nonlinear programming techniques to obtain the best approximation with respect to the Euclidean distance for$T$the$\Lstrok$ukasiewicz or the product$t$-norm. The previous methods do not apply for the minimum$t$-norm. An algorithm to approximate a given proximity relation by a${\rm Min}$-transitive relation (a similarity) is given in the last section of the paper.

A new map (ΛE) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of ΛE is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φE and ψE) are also reviewed, in order to compare them with ΛE. Interesting properties of the fixed points of φE and [Formula: see text] are studied. Among others, the fixed points of ΛE ( Fix (ΛE)) are proved to be the maximal fuzzy points of (X, E) and the fixed points of [Formula: see text] coincide with the Image of ΛE. An isometric embedding of X into Fix (ΛE) is established and studied.

Lipschitzian and kernel aggregation operators with respect to the naturalT-indistinguishability operator Et and their powers are studied. A t-normTis proved to be ET-Lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an Archimedean t-normTwith additive generator t, the quasi-arithmetic mean generated by t is proved to be the most stable aggregation operator with respect toT.

In this paper we study the fuzzy points with respect to a T-indistinguishability operator E stressing our attention to the maximal ones. This is due to the fact that normal fuzzy points are maximal and they coincide with the columns of E or, in other words, its equivalence classes. Fuzzy points express the granularity inherent with the indistinguishability and maximal ones give the maximal granularity generated by E. In general it is difficult to find the maximal fuzzy points of a universe X with a T-indistinguishability operator E, but in this paper we propose a geometrical approach that allows us to find them if X is a finite set.

This article gives a new approach to aggregating assuming that there is an indistinguishability operator or similarity defined on the universe of discourse. The very simple idea is that when we want to aggregate two values a and b we are looking for a value λ that is as similar to a as to b or, in a more logical language, the degrees of equivalence of λ with a and b must coincide. Interesting aggregation operators on the unit interval are obtained from natural indistinguishability operators associated to t-norms that are ordinal sums. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 857–873, 2006.

This paper gives an approach to the aggregation of a pair of values a and b in [0,1] assuming that there is an indistinguishability operator or similarity defined in this interval. The basic idea is to understand the aggregation of two numbers a and b as a value lambda that has the same similarity with respect to a than to b or more precisely, the degrees of equivalence of lambda with a and with b must coincide. Some interesting aggregation operators are obtained from natural indistinguishability operators associated to t-norms which are ordinal sums

A new way to generate indistinguishability operators coherent with the underlying ordering structure of the real Dine is given in the sense that this structure should be compatible with the betweenness relation generated by the relation. A new way to generate indistinguishability operators coherent with the underlying ordering structure of the real line is given in the sense that this structure should be compatible with the between ness relation generated by the relation.

This article studies the aggregation of transitive fuzzy relations. We first find operators that preserve transitivity and then extend the results to aggregating operators. As special cases, means and some kind of suitable ordered weighted averaging (OWAs) are used to aggregate transitive fuzzy relations with respect to an Archimedean t-norm. Three families of transitive relations that allow us to modify the entries of a given relation R continuously towards the smallest and the greatest ones in our universe are given. Aggregation of nonfinite families of transitive relations also is studied and applied to calculate the degree of inclusion or similarity of fuzzy quantities (fuzzy subsets of an interval of the real line). © 2003 Wiley Periodicals, Inc.

A new approach to normalizing fuzzy sets is introduced where it is assumed that the normalization method is compatible with a given t-norm. In this context it is proved that the most usual ways to normalize fuzzy subsets correspond to the most common t-norms. For a given fuzzy subset μ, the corresponding normalized fuzzy subset [Formula: see text] can be viewed as the distribution of μ conditioned on the (degree of) existence of its elements with maximal membership. From this view point we investigate the less specific normal fuzzy subset of X among the most similar fuzzy subsets to μ and the normal fuzzy subset generating the same fuzzy T-preorder as μ.

Given a T-indistinguishability operator E defined between some fuzzy subsets of a universe of discourse X, this paper studies three ways to generate an indistinguishability operator on X compatible with E inspired by the tools used in approximate reasoning and on the duality principle. Of special interest are the cases when the fuzzy subsets determine a partition and a hard-partition of the universe X. © 2002 Wiley Periodicals, Inc.

This paper presents a new method for approximating an unrestricted belief measure assuring that the "order" defined by the compatibility degree between evidence and the singletons set is preserved. Our approach, based on the concept of fuzzy T-preorder, also allows us to define several equivalence criteria over the set of all basic probability assignment functions on a given domain. Some others related aspects as uniqueness of the approximations are also addressed.

The most common ways used to generate indistinguishability operators, namely as transitive closure of reflexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, are related for archimedean t-norms introducing the notion of length of indistinguishability operators.

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators.