T-indistinguishability Operator Research Articles

Aggregation of indistinguishability operators

The paper studies the aggregation of pairs of T-indistinguishability operators. More concretely we address the question whether A(E,E′) is a T-indistinguishability operator if E,E′ are T-indistinguishability operators. The answer depends on the aggregation function, the t-norm T, and the chosen T-indistinguishability operators. It is well-known that an aggregation function preserves T-transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm TM in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of TM-indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property.

Aggregation of partial T-indistinguishability operators and partial pseudo–metrics

In this contribution we address our attention on the aggregation of partial T-indistinguishability operators (relations) and partial pseudo-metrics. A characterization of those functions that allow to merge a collection of partial T–indistinguishability operators into a new one was provided by Calvo et al. in [10] by means of (T,TM)-tuples, but here we present another characterization in terms of (+,max⁡)-tuples. Also, we analyze the aggregation of a collection (Ei)i=1n of partial Ti-indistinguishability operators. Moreover, we provide that a generalized inter–exchange composition functions condition is a sufficient condition to guarantee that a function merges partial Ti-indistinguishability operators into a single one. In addition, we give different expressions of those aggregation functions that are object of our study, most of them are defined by means of the additive generators of the corresponding t-norms and another particular function. We see that the functions, that merge partial S-pseudo-metrics into a new one, are related to the functions that aggregate partial pseudo-metrics. Finally, we show the relation between the functions, that merge partial T-indistinguishability operators and the functions that preserve the partial T⁎-pseudo-metrics in the aggregation process.

Characterization of unidimensional averaged similarities

A T-indistinguishability operator (or fuzzy similarity relation) E is called unidimensional when it may be obtained from one single fuzzy subset (or fuzzy criterion). In this paper, we study when a T-indistinguishability operator that has been obtained as an average of many unidimensional ones is unidimensional too. In this case, the single fuzzy subset used to generate E is explicitly obtained as the quasi-arithmetic mean of all the fuzzy criteria primarily involved in the construction of E.

INDISTINGUISHABILITY OPERATORS WITH RESPECT TO DIFFERENT t-NORMS

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.

Permutable indistinguishability operators, perfect vague groups and fuzzy subgroups

Permutability between T-indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures. It will be shown that the sup −T product E∘F of two T-indistinguishability operators is also a T-indistinguishability operator if and only if E and F are permutable T-indistinguishability operators (i.e., E∘F=F∘E). This property will be related to the study of fuzzy subgroups, fuzzy normal subgroups and vague groups. The aggregation of fuzzy subgroups will also be analyzed.

A MAP CHARACTERIZING THE FUZZY POINTS AND COLUMNS OF A T-INDISTINGUISHABILITY OPERATOR

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.

ET-lipschitzian and ET-kernel aggregation operators

Lipschitzian and kernel aggregation operators with respect to natural T-indistinguishability operators E T and their powers are studied. A t-norm T is proved to be E T -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.

Indistinguishability relations in Dempster–Shafer theory of evidence

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. Nevertheless, equivalence relations have not been proved to be sufficiently suited to capturing the underlying structure when dealing with domains pervaded with uncertainty. Therefore, the need for a more flexible definition of equality brings the concept of T-indistinguishability operator on to the scene. In this paper, we study the notion of indistinguishability within the context of the Dempster–Shafer theory of evidence and provide effective definitions and procedures for computing the T-indistinguishability operator associated with a given body of evidence. We also show how these procedures could also be adapted in order to provide a new method for tackling the problem of belief function approximation based on the concept of T-preorder.

The group of isometries of an indistinguishability operator

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 h : X→X 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.

Generating indistinguishability operators from prototypes

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.

Maps and isometries between indistinguishability operators

In this paper, some geometric aspects of indistinguishability operators are studied by using the concept of morphism between them. Among all possible types of morphisms, the paper is focused on the following cases: Maps that transform a T-indistinguishability operator into another of such operators with respect to the same t-norm T and maps that transform a T-indistinguishability operator into another one of such operators with respect to a different t-norm T′. The group of isometries of a given T-indistinguishability operator is also studied and it is determined for the case of one-dimensional operators, in particular for the natural indistinguishability operators ET on [0, 1]. Finally, the indistinguishability operators invariant under translations on the real line are characterized.

One-dimensional indistinguishability operators

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.

Extensionality based approximate reasoning

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.

Fuzzy T-transitive relations: eigenvectors and generators

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.

Fixed Points and Generators of Fuzzy Relations

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.