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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Leaning on the mathematical concept of an interval order, we show that intransitivities that appear in several chemical processes, mainly related to mixing and competition, can actually be located and handled within a thermodynamical setting whose basis is the classical axiomatics due to Carathéodory, now using two intertwined entropy functions. Interdisciplinary comparisons to other similar theories (e.g., Utility Theory) are also made, pointing out the common mathematical background based on the numerical representability of total preorders and interval orders.

total preorders, continuous utility representations, separating scales, decreasing scales, C 60, D 11, 54 F 05, 06 A 06,

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