Convex sublattices of a lattice and a fixed point property
Résumé
The collection $\mathcal{C}_{L}(T)$ of nonempty convex sublattices of a lattice $T$ ordered by bi-domination is a lattice. We say that $T$ has the \emph{fixed point property for convex sublattices} (CLFPP for short) if every order preserving map $f:T\rightarrow \mathcal{C}_{L}(T)$ has a fixed point, that is $x\in f(x)$ for some $x\in T$. We examine which lattices may have CLFPP. We introduce the \emph{selection property for convex sublattices} (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that $\mathcal{C}_{L}(T)$ is complete. We show that for a lattice $T$, the fact that $\mathcal{C}_{ L}(T)$ is complete is equivalent to the fact that $T$ is complete and the lattice $\powerset (\omega)$ of all subsets of a countable set, ordered by containment, is not order embeddable into $T$. We show that for the lattice $T:=\mathcal {I}(P)$ 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 \cite{abraham-all}.