Order algebras
Robert L. Davis · Bulletin of the American Mathematical Society · 1970
A partially ordered set P in which every pair of elements has a greatest lower bound is a semigroup, with pq^pAq, and hence is naturally associated with a semigroup algebra Z[P] over the integers.For finite P Solomon has given [3 ] a marvelously ingenious construction of an analogous sort of algebra even when P is not a semilattice and so cannot be made into a semigroup.Semigroup algebras and Solomon's "Möbius algebras" have applications in combinatorial problems involving the underlying orders.Now in a recent study [2] of valuations and Euler characteristics on lattices Rota introduced an ostensibly quite different sort of algebra he called a "valuation ring" which, rather surprisingly, plays a role like that of a semigroup algebra.More surprising, in view of their entirely different genesis and description, is that Rota's valuation ring can be shown to include Solomon's Möbius algebra as a special case.Rota's construction, when used to associate such an algebra to a partial order P (which is only one outgrowth of his inquiry), leads in stages through several different structures.The results implicitly provide a recursive procedure for computing products in the valuation ring V(P), but give no direct formula.Solomon, on the other hand, defined his Möbius algebra by giving an explicit, if rather complicated, formula to express products of elements of P as linear combinations of P-elements.The purpose of this note is to determine from Rota's construction an explicit formula for products in VÇP) which depends only on the order structure of P.This will show at once that Rota's construction includes Solomon's, and it can be recast in a particularly simple form that clarifies further consequences and applications.