On conditional monotonicities of interval-valued functions

This paper introduces the concept of conditional monotonicity and other related relaxed monotonicities within the framework of intervals equipped with admissible orders. It generalizes the work of Sesma-Sara et al., who introduced weak/directional monotonicity on intervals endowed with the Kulisch–Miranker order, and the work of Santiago et al., who introduced the notion of g-weak monotonicity in the fuzzy setting. The paper also explores properties of conditional monotonicities, introduce the notion of ordinal sum for a family of functions and examines the connections between conditional monotonicity, ordinal sums and implications.