An incidence Hopf Algebra of Convex Geometries

TL;DR

This paper explores whether convex geometries, modeled as meet-distributive lattices, can be structured into an incidence Hopf algebra, linking combinatorial convexity with algebraic frameworks.

Contribution

It investigates the construction of an incidence Hopf algebra based on convex geometries and their associated meet-distributive lattices.

Findings

Establishes a bijection between meet-distributive lattices and convex geometries.

Proposes a framework for an incidence Hopf algebra of convex geometries.

Provides insights into the algebraic structure underlying convexity models.

Abstract

A lattice L is "meet-distributive" if for each element of L, the meets of the elements directly below it form a Boolean lattice. These objects are in bijection with "convex geometries", which are an abstract model of convexity. Do they give rise to an incidence Hopf algebra of convex geometries?