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A proposal of the notions of ordered and strengthened ordered directional monotonicity for interval-valued functions based on admissible orders

Two of the main research lines in the theory of aggregation functions is the extension to more general domains and the relaxation of the monotonicity conditions. In this work, we discuss the state-of-the-art of the main introduced relaxed forms of monotonicity that can be found in the literature, i.e., weak, directional, ordered directional and strengthened ordered directional monotonicity. We pay special attention to the extension of a relaxed form of monotonicity to the interval-valued setting and we propose the concepts of ordered and strengthened ordered directional monotonicity for this general setting. Moreover, we study the main properties of the functions that satisfy the introduced properties and present some construction methods.