An isomorphism of idempotent monads

TL;DR

This paper explores the isomorphism between two idempotent measure monads, leading to the development of a fuzzy integral and insights into related convexities, bridging different algebraic structures in measure theory.

Contribution

It establishes an isomorphism between measure monads based on different operations and constructs a new fuzzy integral, advancing the theoretical understanding of idempotent measures.

Findings

Proves isomorphism between measure monads with max-addition and max-multiplication

Constructs a fuzzy integral based on max and addition operations

Analyzes convexities associated with these monads

Abstract

We consider isomorphism between the idempotent measure monad based on the maximum and the addition operations and the idempotent measure monad based on the maximum and the multiplication operations. A one of the consequences of this result is the construction of a fuzzy integral based on the maximum and the addition operation. We also investigate convexities related to these monads.