Resolutions of Convex Geometries

TL;DR

This paper introduces a resolution operation for convex geometries, explores its properties, especially for special types like ordinal and affine, and characterizes primitive affine convex geometries.

Contribution

It defines and analyzes the resolution operation for convex geometries, showing it preserves convexity and characterizing primitive affine convex geometries.

Findings

Resolutions of convex geometries always produce convex geometries.

Resolutions of ordinal convex geometries remain ordinal.

Some affine convex geometries are not preserved under resolution.

Abstract

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions -- compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Mohring and Radermacher (1984) -- , resolutions of convex geometries always yield a convex geometry. We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.