Convex sublattices of a lattice and a fixed point property

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 ∶ T → 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 �(ω) 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, Džamondja and Thompson 2010 [1].