Cancellative Convex Semilattices

TL;DR

This paper characterizes cancellative convex semilattices, showing they are isomorphic to convex subsets of Riesz spaces, extending known results from convex algebras to semilattice structures.

Contribution

It provides a new characterization of cancellative convex semilattices as convex subsets of Riesz spaces, generalizing classical results for convex algebras.

Findings

Cancellative convex semilattices are isomorphic to convex subsets of Riesz spaces.

Extension of Kneser and Stone's characterization from convex algebras to semilattices.

Provides a structural understanding relevant for probability and nondeterminism models.

Abstract

Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and nondeterminism, in particular by being the Eilenberg-Moore algebras of the nonempty finitely-generated convex subsets of the distributions monad. A convex semilattice is cancellative if the underlying convex algebra is cancellative. Cancellative convex algebras have been characterized by M. H. Stone and by H. Kneser: A convex algebra is cancellative if and only if it is isomorphic to a convex subset of a vector space (with canonical convex algebra operations). We prove an analogous theorem for convex semilattices: A convex semilattice is cancellative if and only if it is isomorphic to a convex subset of a Riesz space, i.e., a lattice-ordered vector space (with canonical convex semilattice operations).