Valuations on Distributive Lattices II
Ladnor Geissinger · Birkhäuser Boston eBooks · 2009
We continue the study begun in part I [Arch. Math. 24, 230–239 (1973)] of the valuation ring of a finite distributive lattice. We show that it is the Mobius algebra of the set of join.irreducible elements and we derive Solomon’s formula for idempotents. We use the duality between posets and distributive lattices given in part I to derive mapping properties of Möbius algebras. From this we get theorems on extending finitely additive measures, theorems of Rota concerning Möbius functions, an identity due to Klee, a factorization theorem of Stanley and Greene, and results on the characteristic valuation of a distributive lattice.