Valuations on Distributive Lattices III
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.
KeywordsPrime Ideal Distributive Lattice Valuation Ring Combinatorial Theory Faithful Representation
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