# A functional representation of the capacity multiplication monad

## TL;DR

This paper introduces a new capacity monad called the capacity multiplication monad, which has a functional representation on compact spaces, extending previous max-min based models with multiplication operations.

## Contribution

It establishes a functional representation for the capacity multiplication monad, expanding the theoretical framework of capacity monads with new algebraic structures.

## Key findings

- Capacity monad has a functional representation.
- The space of capacities embeds into a space of functionals.
- Characterization of this space via properties of functionals.

## Abstract

Functional representations of the capacity monad based on the max and min operations were considered in \cite{Ra1} and \cite{Ny1}. Nykyforchyn considered in \cite{Ny2} some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations. We show that such capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum $X$ can be naturally embedded (with preserving of the monad structure) in some space of functionals on $C(X,I)$. We also describe this space of functionals in terms of properties of functionals.