Convex Sublattices of a Lattice and a Fixed Point Property

TL;DR

This paper investigates the fixed point property for convex sublattices in lattices, characterizing when such lattices have this property based on completeness and structural conditions of the underlying posets.

Contribution

It introduces the selection property for convex sublattices (CLSP), links it to fixed point properties, and characterizes lattices with CLFPP through completeness and the absence of certain sublattice embeddings.

Findings

Complete lattices with CLSP have CLFPP.

CL(T) is complete iff T is complete and certain sublattices are not embeddable.

For lattices of initial segments, properties are equivalent to the underlying poset having no infinite antichain.

Abstract

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application of a wonderful Hausdorff? type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010]. Key words and phrases. posets, lattices, convex sublattice, retracts, fixed point property