Convex sublattices of a lattice and a fixed point property
AbstractThe 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 Hausdor type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson .
This copyright statement was adapted from the statement for the University of Calgary Repository and from the statement for the Electronic Journal of Combinatorics (with permission).
The copyright policy for Contributions to Discrete Mathematics (CDM) is changed for all articles appearing in issues of the journal starting from Volume 15 Number 3.
Author(s) retain copyright over submissions published starting from Volume 15 number 3. When the author(s) indicate approval of the finalized version of the article provided by the technical editors of the journal and indicate approval, they grant to Contributions to Discrete Mathematics (CDM) a world-wide, irrevocable, royalty free, non-exclusive license as described below:
The author(s) grant to CDM the right to reproduce, translate (as defined below), and/or distribute the material, including the abstract, in print and electronic format, including but not limited to audio or video.
The author(s) agree that the journal may translate, without changing the content the material, to any medium or format for the purposes of preservation.
The author(s) also agree that the journal may keep more than one copy of the article for the purposes of security, back-up, and preservation.
In granting the journal this license the author(s) warrant that the work is their original work and that they have the right to grant the rights contained in this license.
The authors represent that the work does not, to the best of their knowledge, infringe upon anyone’s copyright.
If the work contains material for which the author(s) do not hold copyright, the author(s) represent that the unrestricted permission of the copyright holder(s) to grant CDM the rights required by this license has been obtained, and that such third-party owned material is clearly identified and acknowledged within the text or content of the work.
The author(s) agree to ensure, to the extent reasonably possible, that further publication of the Work, with the same or substantially the same content, will acknowledge prior publication in CDM.
The journal highly recommends that the work be published with a Creative Commons license. Unless otherwise arranged at the time the finalized version is approved and the licence granted with CDM, the work will appear with the CC-BY-ND logo. Here is the site to get more detail, and an excerpt from the site about the CC-BY-ND. https://creativecommons.org/licenses/
This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.