Presenting convex sets of probability distributions by convex semilattices and unique bases

TL;DR

This paper establishes a unique base for finitely generated convex sets of finitely supported probability distributions and characterizes the monad of convex sets using convex semilattices, advancing the algebraic understanding of probabilistic structures.

Contribution

It introduces the concept of convex semilattices to present the monad of convex probability sets and proves the uniqueness of bases for finitely generated convex sets.

Findings

Unique base for finitely generated convex sets established

Monad of convex probability sets characterized by convex semilattices

Advances algebraic theory of probabilistic structures

Abstract

We prove that every finitely generated convex set of finitely supported probability distributions has a unique base, and use this result to show that the monad of convex sets of probability distributions is presented by the algebraic theory of convex semilattices.