Valuations on Distributive Lattices I
Ladnor Geissinger · Birkhäuser Boston eBooks · 2009
We continue the study, begun by G.-C. Rota, of the valuation ring of a distributive lattice. This ring is the representing object for all valuations on the lattice. In the locally finite case Rota established a connection with the incidence algebra of the set of join-irreducible elements, from which he derived interesting results about the Euler characteristic and Mobius function associated with some geometric objects. In this paper we give new proofs of some of his results, and extend others. In part I we discuss general properties of the valuation module and ring of a lattice, and determine their structure for a finite geometric lattice.