IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-3-319-44236-5_9.html
   My bibliography  Save this book chapter

Lattice Theory of the Poset of Regions

In: Lattice Theory: Special Topics and Applications

Author

Listed:
  • N. Reading

    (North Carolina State University, Department of Mathematics)

Abstract

Hyperplane arrangements (collections of codimension-1 subspaces) have long been an object of study in combinatorics, topology, and geometry. This chapter explores the lattice theory of the poset of regions of a (real) hyperplane arrangement. We discuss the open problem, first posed by Björner, Edelman, and Ziegler [70], of characterizing by local geometric conditions which posets of regions are lattices. We give a local geometric characterization (“tightness”) of which posets of regions are semidistributive lattices. Along the way, we discuss a local condition for checking that a partially ordered set is a lattice, along with analogous local conditions for determining lattice-theoretic properties. In the case of simplicial arrangements (which are in particular tight), we characterize the regions of the arrangement in terms of two notions of combinatorial convexity.

Suggested Citation

  • N. Reading, 2016. "Lattice Theory of the Poset of Regions," Springer Books, in: George Grätzer & Friedrich Wehrung (ed.), Lattice Theory: Special Topics and Applications, chapter 0, pages 399-487, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-44236-5_9
    DOI: 10.1007/978-3-319-44236-5_9
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-3-319-44236-5_9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.