Infinite convex geometries with lower semi-modularity and join semi-distributivity

TL;DR

This paper investigates infinite convex geometries that are lower semi-modular and join semi-distributive, demonstrating how certain classes can be constructed from finite geometries and exploring properties preserved in the infinite case.

Contribution

It introduces a class of infinite convex geometries built from finite ones via chain unions and analyzes the preservation of specific lattice properties in this construction.

Findings

Certain infinite convex geometries can be constructed from finite ones.

Lower semi-modularity and join semi-distributivity are preserved in these constructions.

Not all infinite convex geometries can be obtained through chain unions.

Abstract

The following article treats about convex geometries which are lower semi-modular and join semi-distributive lattices. Firstly, it is shown that there is a class $K$ of infinite convex geometries which can be build out of finite ones by using the construction of a union of a chain. Then it is shown that elements of $K$ preserve lower semi-modularity and join semi-distributivity which are not default properties in the infinite setting. It is also discussed that not all of the infinite convex geometries may be obtained by the means of the union of a chain.