Max-min measures on ultrametric spaces

TL;DR

This paper explores max-min measures on ultrametric spaces, establishing their relation to idempotent measures and analyzing the monads they generate, contributing to the understanding of measure theory in ultrametric contexts.

Contribution

It introduces a construction for max-min measures on ultrametric spaces and proves their functorial isomorphism with idempotent measures, clarifying their algebraic structures.

Findings

Max-min measures are functorially isomorphic to idempotent measures.

The monads generated by max-min and idempotent measures are not isomorphic.

Ultrametric measure constructions extend previous ultrametrization results.

Abstract

The ultrametrization of the set of all probability measures of compact support on the ultrametric spaces was first defined by Hartog and de Vink. In this paper we consider a similar construction for the so called max-min measures on the ultrametric spaces. In particular, we prove that the functors max-min measures and idempotent measures are isomorphic. However, we show that this is not the case for the monads generated by these functors.