Finite versus uncountable convex lattices from point configurations

TL;DR

This paper investigates the structure and enumeration of convex lattices generated by finite point configurations, revealing finiteness in relative lattices but uncountability in convex lattices for six or more points.

Contribution

It introduces the notion of point configurations via relative lattices and proves the uncountability of convex lattices for n ≥ 6.

Findings

Number of relative lattices is finite.

Number of convex lattices is uncountable for n ≥ 6.

Provides a combinatorial framework for analyzing convex lattices.

Abstract

We study the smallest convex lattice generated by a finite set of points. To analyze this structure, we introduce the notion of a point configuration, defined via the relative lattice. Under a suitable completeness condition, this lattice becomes a combinatorial counterpart of the convex lattice and is therefore easier to handle. We investigate the enumeration of these structures and prove that, while the number of relative lattices is always finite, the number of convex lattices is uncountable for $n \geq 6$.