Idempotent convexity and algebras for the capacity monad and its submonads

TL;DR

This paper introduces idempotent convexity concepts and establishes categorical isomorphisms between algebras of the capacity monad and certain idempotent convex compacta, expanding the theoretical framework of capacity theory.

Contribution

It defines idempotent convexity and proves categorical isomorphisms between capacity monad algebras and idempotent convex compacta, linking algebraic and geometric structures.

Findings

Category of capacity monad algebras is isomorphic to $( ext{max}, ext{min})$-idempotent biconvex compacta

Category of sup-measure monad algebras is isomorphic to $( ext{max}, ext{min})$-idempotent convex compacta

Establishes a theoretical foundation connecting capacity theory and idempotent convexity

Abstract

Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine maps. It is also shown that the category of algebras for the monad of sup-measures ($(\max,\min)$-idempotent measures) is isomorphic to the category of $(\max,\min)$-idempotent convex compacta and their affine maps.