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On the functor of comonotonically maxitive functionals

Taras Radul

TL;DR

The paper defines a functor $S$ of comonotonically maxitive functionals and proves its isomorphism with the capacity functor $M$ via the max-plus fuzzy integral, providing an idempotent analogue of the Riesz representation. It proves a representation theorem: a normalized, monotone, comonotonically maximative, and plus-homogeneous functional $I:C(X)\to\mathbb{R}$ corresponds uniquely to a capacity $c\in MX$ by $I(\varphi)=\int_X^{\vee+} \varphi dc$, with $c$ constructed to ensure upper semicontinuity. This yields a homeomorphism $l_X: MX\to SX$, establishing $M\cong S$ as functors and situating $I$ as a subfunctor of $O$. The work situates these constructions in the category-theoretic monad framework on compacta, relating $O$, $I$, and $\Pi$ and outlining conditions under which $S$ can be extended to a submonad of $O$. Overall, it generalizes the Riesz-type correspondence to capacities via the max-plus integral and clarifies the categorical structure of these non-additive measures.

Abstract

We introduce a functor of functionals which preserve maximum of comonotone functions and addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and contains the idempotent measure functor as subfunctor. The main aim of this paper is to show that this functor is isomorphic to the capacity functor. We establish such isomorphism using the fuzzy max-plus integral. In fact, we can consider this result as an idempotent analogue of Riesz Theorem about a correspondence between the set of $σ$-additive regular Borel measures and the set of linear positively defined functionals.

On the functor of comonotonically maxitive functionals

TL;DR

The paper defines a functor of comonotonically maxitive functionals and proves its isomorphism with the capacity functor via the max-plus fuzzy integral, providing an idempotent analogue of the Riesz representation. It proves a representation theorem: a normalized, monotone, comonotonically maximative, and plus-homogeneous functional corresponds uniquely to a capacity by , with constructed to ensure upper semicontinuity. This yields a homeomorphism , establishing as functors and situating as a subfunctor of . The work situates these constructions in the category-theoretic monad framework on compacta, relating , , and and outlining conditions under which can be extended to a submonad of . Overall, it generalizes the Riesz-type correspondence to capacities via the max-plus integral and clarifies the categorical structure of these non-additive measures.

Abstract

We introduce a functor of functionals which preserve maximum of comonotone functions and addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and contains the idempotent measure functor as subfunctor. The main aim of this paper is to show that this functor is isomorphic to the capacity functor. We establish such isomorphism using the fuzzy max-plus integral. In fact, we can consider this result as an idempotent analogue of Riesz Theorem about a correspondence between the set of -additive regular Borel measures and the set of linear positively defined functionals.
Paper Structure (4 sections, 7 theorems, 13 equations)

This paper contains 4 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. A characterization of the max-plus integral
  4. Categorical aspects.

Key Result

Lemma 1

Let $\varphi$, $\psi\in C(X)$ be two comonotonic functions. Then we have $\varphi_t\subset\psi_t$ or $\varphi_t\supset\psi_t$ for each $t\in\mathbb R$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 1
  • proof
  • ...and 3 more