Infinite Convex Geometries with Lower semi-modularity and Join Semi-distributivity

Abstract We construct a class K of infinite convex geometries by taking unions of chains of finite convex geometries, and prove that every member of K is both lower semi-modular and join semi-distributive. These properties, while automatic in the finite case, can fail for infinite convex geometries in general. We also show that not all infinite convex geometries arise via this construction, so that K is a proper subclass of the class of all infinite convex geometries.