Interval-Valued Representability of Qualitative Data: The Continuous Case

Abstract

In the framework of the representability of ordinal qualitative data by means of interval-valued correspondences, we study interval orders defined on a nonempty set X. We analyse the continuous case, that corresponds to a set endowed with a topology that furnishes an idea of continuity, so that it becomes natural to ask for the existence of quantifications based on interval-valued mappings from the set of data into the real numbers under preservation of order and topology. In the present paper we solve a continuous representability problem for interval orders. We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions so that each element in X has associated in a continuous manner a characteristic interval or equivalently a symmetric triangular fuzzy number.

A valued Ferrers relation for interval comparison

“…Hence when the comparison rules of semiorders or interval orders are used in real life problems some information is lost. For instance, consider three intervals x = [0,6], y = [1,5] and z = [5,10]. For interval orders, intervals x and y are indifferent but also x and z are indifferent.…”

“…In this paper, if nothing is precised, we will refer to the standard negation which is ∀x ∈ [0,1], N(x) = 1 − x (for a discussion about such operators in the frame of fuzzy sets theory see [9]). …”

“…However the following proposition shows that some type of t-norms with zero divisors are not very intuitive for preference modeling since they do not allow the use of the whole range [0,1].…”

“…In this article we restrict our study to the case of a finite set of objects knowing that the cardinality of the object set plays an important role on the representation. We should mention the case of preference structures applied on infinite sets where topological structures are required (see [4,13] for relations between orders and topology and [1,2] for continuous representation of IO). In decision making, a numerical representation of a preference structure may be used for two different purposes.…”

“…We are going to use the symbols of T and S in order to represent t-norms (triangular norm) and t-conorms (triangular conorm) respectively as continuous representations of conjunction and disjunction operators in the case of continuous valuations. [19]) A triangular norm T is an increasing, associative and commutative [0,1] 2 [19]) A triangular conorm S is an increasing, associative and commutative [0,1] 2…”