On monotonicity of comonotonically maxitive functional
Taras Radul
TL;DR
The paper addresses whether comonotone maxitivity implies monotonicity for functionals on spaces of continuous functions $C(X,[0,1])$. It shows this implication fails in general by constructing a counterexample on a non-finite compactum, clarifying that normalization alone is not enough. It then highlights a positive result: if a functional is normalized, comonotonically maxitive, and $\ast$-homogeneous with respect to a continuous t-norm $\ast$ (i.e., belongs to $\mathcal{T}^\ast(X)$), monotonicity follows, in line with earlier work Rad1. The findings delimit the conditions under which monotonicity can be inferred from comonotone maxitivity, informing the characterization of fuzzy integrals and non-additive measures.
Abstract
The comonotonic maxitivity property of functionals frequently appears in the characterization of fuzzy integrals based on the maximum operation. In some special cases, comonotonic maxitivity implies monotonicity of functionals. The question of whether this implication holds in general was posed by T. Radul (2023). It was shown in that paper that the implication is valid for finite compacta. In this article, we provide a negative answer to the general problem and discuss additional properties that need to be imposed to ensure the implication holds.