http://hdl.handle.net/2117/3772 2013-05-24T12:38:32Z http://hdl.handle.net/2117/19246 Title: Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator Authors: Soto, De A R; Recasens Ferrés, Jorge Abstract: This work shows a method to obtain a hierarchy of partitions on the universe [0,1] in such a way that each of them is compatible with a refinement of Lukasiewicz indistinguishability operator. The classes of the partition at a given level present a relation of antonymy between them. Moreover, the partition at a certain level can be seen as the refinement of a previous level by means of a class of linguistic modifiers. Due to this fact, they seem appropriate for modeling linguistic labels of a linguistic variable. The associated indistinguishability operators show the increasing granularity when the number of classes rises up. 2013-05-15T11:39:15Z http://hdl.handle.net/2117/19242 Title: A reformulation of entropy in the presence of indistinguishability operators Authors: Recasens Ferrés, Jorge; Hernández, E Abstract: This paper deals with the measurement of entropy when an indistinguishability relation on the set of events has been defined. Our approach states that entropy could be measured in terms of the observed distinguishability of the set of events. In this sense, the “observer paradigm” is introduced, and definitions for joint and conditional entropy under this paradigm are given. We also present some interesting properties and relationships with Shannon's entropy measure. 2013-05-15T10:58:59Z http://hdl.handle.net/2117/19241 Title: On a geometric combinatorial problem Authors: Recasens Ferrés, Jorge Abstract: The study of the betweenness relations defined by metrics leads to a geometric problem that yields an upper bound to Turán's number T(n,5,3). 2013-05-15T10:49:35Z http://hdl.handle.net/2117/19240 Title: Finite-valued indistinguishability operators Authors: Mayor, Gaspar; Recasens Ferrés, Jorge Abstract: Fuzzy equality relations or indistinguishability operators generalize the concepts of crisp equality and equivalence relations in fuzzy systems where inaccuracy and uncertainty is dealt with. They generate fuzzy granularity and are an essential tool in Computing with Words (CWW). Traditionally, the degree of similarity between two objects is a number between 0 and 1, but in many occasions this assignment cannot be done in such a precise way and the use of indistinguishability operators valued on a finite set of linguistic labels such as small, very much, etc. would be advisable. Recent advances in the study of finite-valued t-norms allow us to combine this kind of linguistic labels and makes the development of a theory of finite-valued indistinguishability operators and their application to real problems possible. 2013-05-15T10:41:27Z http://hdl.handle.net/2117/19237 Title: Fixed points and generators of fuzzy relations Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: The fixed points of a T-indistinguishability operator are characterized as its generators in the sense of the Representation Theorem of Valverde. The geometric description of the set of fixed points of a reflexive and symmetric fuzzy relation based on this characterization gives a way to explicitly calculate it. 2013-05-15T10:27:44Z http://hdl.handle.net/2117/19235 Title: E-T-lipschitzian and E-T-kernel aggregation operators Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: Lipschitzian and kernel aggregation operators with respect to natural T-indistinguishability operators ET and their powers are studied. A t-norm T is proved to be ET-lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to T. 2013-05-15T10:10:56Z http://hdl.handle.net/2117/19234 Title: Aggregation of T-transitive Relations Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: 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. 2013-05-15T10:05:21Z http://hdl.handle.net/2117/19228 Title: The group of isometries of an indistinguishability operator Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: This paper studies some geometric aspects of indistinguishability operators (also called similarities and fuzzy equivalences). Concretely, it will be focused on the (geometric) group associated to a T-indistinguishability operator E on X (i.e., the group of all bijective maps View the MathML source such that E(x,y)=E(h(x),h(y))∀x,y∈X). The cases for E being one dimensional and invariant under translations on the real line will be completely studied. This last property will be generalized to any group and there will be stated a bijection between indistinguishability operators invariant under translations on a group and its normal fuzzy subgroups. 2013-05-15T08:28:50Z http://hdl.handle.net/2117/19227 Title: One dimensional indistinguishability operators Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: The main result of the paper is an algorithm that allows us to decide when a given fuzzy relation is a one-dimensional T-indistinguishability operator for some archimedean t-norm T (in the sense of the Representation Theorem of Valverde (Fuzzy Sets and Systems 17 (1985) 313–328)). The algorithm also finds all t-norms with this property. 2013-05-15T08:20:17Z http://hdl.handle.net/2117/19226 Title: Fuzzy t-transitive relations: eigenvectors and generators Authors: Jacas Moral, Juan; Recasens Ferrés, Jorge Abstract: After some preliminaries, the set of generators of a generalized equality relation (T-indistinguishability operator) is studied. This set is identified with the set of the eigenvectors of the relation. The relation between the fuzzy and “metric” topologies derived from these equalities is stablished. The concept of basis is introduced and the construction of a procedure in order to calculate explicitly a basis of a T-indistinguishability operator, for T archimedean, is proposed. 2013-05-15T08:08:39Z http://hdl.handle.net/2117/19223 Title: How to make T-transitive a proximity relation Authors: Garmendia, L; Recasens Ferrés, Jorge Abstract: 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 R macr of R with a (maximal) T-transitive relation B contained in R . The second one computes the closest homotecy of R macr 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 Lukasiewicz 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 min-transitive relation (a similarity) is given in the last section of the paper. 2013-05-15T07:39:50Z http://hdl.handle.net/2117/19204 Title: A model for CAGD using fuzzy logic Authors: Jacas Moral, Juan; Monreal Pujadas, Amadeo; Recasens Ferrés, Jorge Abstract: This paper presents a first approach to a system based on fuzzy logic for the design of curves and surfaces in the context of computer aided geometric design. Bézier curves and surfaces can be seen as particular cases of this system. 2013-05-14T11:56:35Z http://hdl.handle.net/2117/19203 Title: Indistinguishability relations in Dempster-Shafer theory of evidence Authors: Hernandez, E; Recasens Ferrés, Jorge Abstract: Each theory or model implicitly defines its inherent notion of equality for the objects in question. In turn, this equality, and its counterpart, the mathematical concept of equivalence, provides the basis on which to establish classification mechanisms for the domain at hand. 2013-05-14T11:50:53Z http://hdl.handle.net/2117/19202 Title: Generating indistinguishability operators from prototypes Authors: Hernández, E; Recasens Ferrés, Jorge Abstract: 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. 2013-05-14T11:42:57Z http://hdl.handle.net/2117/19191 Title: Extensionality based approximate reasoning Authors: Boixader Ibáñez, Dionís; Jacas Moral, Juan Abstract: The aim of this paper is to analyze Approximate Reasoning (AR) through extensionality with respect to the natural T-indistinguishability operator, by considering the indistinguishability level between fuzzy sets as a formal measure of its degree of similarity, resemblance or closeness, having in all these terms an intuitive meaning. 2013-05-14T08:34:13Z