Abstract
The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed measurement or a certain degree of similarity can be only determined between the objects being compared. Since Trillas introduced such kind of operators, many authors have studied their properties and applications. In particular, an intensive research line is focused on the metric behavior of indistinguishability operators. Specifically, the existence of a duality between metrics and indistinguishability operators has been explored. In this direction, a technique to generate metrics from indistinguishability operators, and vice versa, has been developed by several authors in the literature. Nowadays, such a measurement of similarity is provided by the so-called fuzzy metrics when the degree of similarity between objects is measured relative to a parameter. The main purpose of this paper is to extend the notion of indistinguishability operator in such a way that the measurements of similarity are relative to a parameter and, thus, classical indistinguishability operators and fuzzy metrics can be retrieved as a particular case. Moreover, we discuss the relationship between the new operators and metrics. Concretely, we prove the existence of a duality between them and the so-called modular metrics, which provide a dissimilarity measurement between objects relative to a parameter. The new duality relationship allows us, on the one hand, to introduce a technique for generating the new indistinguishability operators from modular metrics and vice versa and, on the other hand, to derive, as a consequence, a technique for generating fuzzy metrics from modular metrics and vice versa. Furthermore, we yield examples that illustrate the new results.
Highlights
Throughout this paper, we will use the following notation
We introduce a new type of operator, which we have called modular indistinguishability operator, which provides a degree of similarity or equivalence relative to a parameter and retrieves as a particular case fuzzymetrics and classical indistinguishability operators
We have explored the metric behavior of this new kind of operator in such a way that the new results extend the classical results to the new framework and, in addition, allow for exploring the aforesaid duality relationship when fuzzy metrics are considered instead of indistinguishability operators
Summary
In 1982, Trillas, E. introduced the notion of indistinguishability operator with the purpose of fuzzifying the classical (crisp) notion of equivalence relation (see [1]). Let us recall that an indistinguishability operator, for a t-norm ∗, on a non-empty set X is a fuzzy set E : X × X → [0, 1], which satisfies, for each x, y, z ∈ X, the following axioms:. In addition, E satisfies for all x, y ∈ X the following condition: Axioms 2017, 6, 34; doi:10.3390/axioms6040034 www.mdpi.com/journal/axioms (E1’) E( x, y) = 1 implies x = y, it is said that E separates points. According to [1] (see [2]), the numerical value E( x, y) provides the degree up to which x is indistinguishable from y or equivalent to y. The greater E( x, y), the more similar x and y are
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