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Valuations on Distributive Lattices III

  • Ladnor Geissinger
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

In Parts I and II [Arch. Math. 24, 230–239, 337–345 (1973)] we were principally interested in combinatorial applications of the valuation ring of a distributive lattice. We now show how this ring provides a natural setting for some elementary results in measure theory as well as some classical results on representations of distributive lattices. Specifically, in the valuation ring V(L) of a distributive lattice L it is easy to identify various extensions of L as well as prime ideals of L and so arrive at some theorems of Pettis, Birkhoff, and Stone. For any faithful representation of L as a lattice of sets the extension of V (L) by real (or complex) scalars is naturally isomorphic to the algebra of simple functions, and the sup norm on the functions comes from an intrinsic norm on VR(L). The Stone space of L corresponds to the spectrum of VR(L) with the Zariski topology.

Keywords

Prime Ideal Distributive Lattice Valuation Ring Combinatorial Theory Faithful Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Ladnor Geissinger
    • 1
  1. 1.Mathematics DepartmentUniversity of North CarolinaChapel HillUSA

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