CONVEX SUBLATTICES OF A LATTICE AND A FIXED POINT PROPERTY

The collection C-L (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 -> C-L(T) has a fixed point, that is x is an element of f (x) for some x is an element of 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 C-L (T) is complete. We show that for a lattice T, the fact that C-L (T) is complete is equivalent to the fact that T is complete and the lattice (omega) 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, Damondja and Thompson 2010 [1].