Interval-Valued Representability of Qualitative Data: the Continuous Case

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.

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