Cancellative Convex Semilattices

2025-07-15

AI Summary

This paper addresses the algebraic characterization of cancellativity in convex semilattices. Employing techniques from convex algebra, semilattice theory, and Riesz space (vector lattice) structures—augmented by categorical and universal-algebraic methods—the authors establish that a convex semilattice is cancellative if and only if it is isomorphic to a convex subset of some Riesz space, equipped with the standard convex operations. This result furnishes a precise structural correspondence between convex semilattices and ordered linear structures, analogous to Stone’s representation of Boolean algebras and Kneser’s characterization of convex cones. It constitutes the first complete structural theorem for cancellative convex semilattices. The isomorphism framework provides a rigorous foundation for algebraic modeling in probabilistic semantics and nondeterministic programming, marking a significant advancement in the algebraic theory of convex structures.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Uncertainty Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

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).

Problem

• Characterizing cancellative convex semilattices in Riesz spaces
• Extending Stone-Kneser theorem to convex semilattices
• Linking cancellative algebras to lattice-ordered vector spaces

Innovation

• Cancellative convex semilattices in Riesz spaces
• Isomorphism to convex subsets
• Lattice-ordered vector space operations