[Paper Review] Convexities on ordered structures have their Krein--Milman theorem

Paul Poncet|arXiv (Cornell University)|Jan 4, 2013

Advanced Algebra and Logic41 references3 citations

TL;DR

This paper establishes analogues of the classical Krein–Milman theorem for convexities on ordered algebraic structures, particularly semilattices, using abstract convexity and continuous lattice theory. It proves that in a locally convex topological semilattice, every locally compact, weakly-closed, convex subset containing no line is the weakly-closed convex hull of its extreme points.

ABSTRACT

We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.

Motivation & Objective

• To extend the classical Krein–Milman theorem to non-linear, ordered algebraic structures such as semilattices and lattices.
• To investigate convexity structures on partially ordered sets, semilattices, and lattices, particularly subsemilattices and sublattices.
• To establish conditions under which convex subsets of ordered structures are generated by their extreme points, analogous to the Krein–Milman property.
• To explore the role of topological and order-theoretic properties—such as local convexity, compactness, and small semilattices—in enabling such theorems.
• To examine the interplay between convex geometry, Helly numbers, and the depth of semilattices in relation to extreme point structures.

Proposed method

• Uses abstract convexity theory and continuous lattice theory to define convex subsets as subsemilattices in semilattices.
• Applies the fundamental theorem of compact semilattices, linking algebraic and topological properties of semilattices.
• Employs the weak topology generated by continuous semilattice morphisms into [0,1] to define weak-closedness.
• Utilizes Wallace’s lemma on minimal elements in compact partially ordered sets to establish existence of extreme points.
• Applies domain-theoretic methods, including the concept of small semilattices (local convexity), to ensure structural control.
• Analyzes convex geometries and Helly numbers via the depth and breadth of semilattices, linking them to extremal structure.

Experimental results

Research questions

• RQ1 Can the Krein–Milman theorem be generalized to non-linear, idempotent algebraic structures such as semilattices?
• RQ2 What conditions on topological semilattices ensure that convex subsets are the weakly-closed convex hull of their extreme points?
• RQ3 How do convexity structures like subsemilattices, order-convex sublattices, and algebraic sublattices relate to convex geometries in ordered structures?
• RQ4 What is the role of the Helly number and breadth in characterizing the number of extreme points needed to generate a convex set?
• RQ5 Under what conditions does a distributive lattice with algebraic convexity satisfy the Krein–Milman property?

Key findings

• In a locally convex topological semilattice, every locally compact, weakly-closed, convex subset containing no line is the weakly-closed convex hull of its extreme points.
• The concept of 'line' is properly defined in the non-linear semilattice setting, generalizing the classical notion from linear spaces.
• For semilattices with finite breadth $b$, a Minkowski-type theorem holds: every point is the join of at most $b$ extreme points.
• The depth of a semilattice coincides with its Helly number, and this invariant governs the number of extreme points in compact convex subsets.
• A lattice with order-algebraic convexity is a convex geometry if and only if it is a chain, and such chains satisfy the Krein–Milman property when topologically compatible.
• The existence of doubly-irreducible elements—necessary for extreme points in algebraic convexity on lattices—is not guaranteed in general, even in finite distributive lattices.