IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-94-009-8179-9_11.html
   My bibliography  Save this book chapter

Lattices

In: Introduction to Algebra

Author

Listed:
  • R. Kochendörffer

    (University of Dortmund)

Abstract

If A and B are subsets of a given set M then so are A ⋂ B and A ⋃ B. So the set of all subsets of M can be regarded as an algebraic system with the operations ⋂ and ⋃ If A and B are subgroups of a given group G, then A ⋂ B is again a subgroup of G. Apart from trivial cases, however, the set theoretical union of A and B is not a subgroup of G. But if A ⋃B is defined as the subgroup generated by A and B, then the set of all subgroups of G can again be considered as an algebraic system with the operations ⋂ and ⋃. The same applies to the set of all normal subgroups of G. Similarly, the subfields of a given field form an algebraic system with the operations ⋂ and ⋃ where A ⋃B means the subfield generated by the subfields A and B. Other examples of such algebraic systems arise when we consider the set of all subspaces of a given vector space or the set of all ideals of a given ring. This suggests the study of algebraic systems with two operations ⋂ and ⋃ which have some but in general not all the properties of the set theoretical intersection and union.

Suggested Citation

  • R. Kochendörffer, 1972. "Lattices," Springer Books, in: Introduction to Algebra, chapter 11, pages 379-406, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-8179-9_11
    DOI: 10.1007/978-94-009-8179-9_11
    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

    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:978-94-009-8179-9_11. 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.