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Archiv der Mathematik

, Volume 72, Issue 6, pp 418–425 | Cite as

Cellular lattices

  • Jürgen Reinhold
Article
  • 30 Downloads

Abstract.

Based on the work of P. Crawley and R. P. Dilworth on the decomposition theory for lattices we show that a strongly atomic \(\wedge \)-continuous lattice is modular iff for any two different upper (resp. lower) covers b, c of an element a the interval \([a,b\vee c]\) (resp.\([b\wedge c,a]\)) has height 2, and it is distributive iff for any two different upper (resp. lower) covers b, c of an element a the interval \([a,b\vee c]\) (resp.\([b\wedge c,a]\)) is isomorphic to 22. We then introduce the general notion of \({\cal C}\)-cellular lattices. These are lattices in which for any two different covers b, c of an element a the interval \([a,b\vee c]\) is isomorphic to a member of a fixed class \({\cal C}\) of lattices. For example, partition lattices are {22, M3}-cellular and the complete join-semilattice of all T0-topologies on a fixed set is {22, N5}-cellular. If \({\cal C}\) consists of finitely many finite lattices, then the width of all finite intervals in \({\cal C}\)-cellular lattices is bounded above by a function of the height of these intervals.

Keywords

General Notion Finite Interval Continuous Lattice Fixed Class Finite Lattice 

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Jürgen Reinhold
    • 1
  1. 1.Institut für Mathematik, Universität Hannover, Welfengarten 1, D-30167 Hannover, GermanyDeutschland

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