Finite intersection of valuation overrings of polynomial rings in at most three variables

Abstract

The group of divisibility of an integral domain is the multiplicative group of nonzero principal fractional ideals of the domain and is a partially ordered group under reverse inclusion. We study the group of divisibility of a finite intersection of valuation overrings of polynomial rings in at most three variables and we classify all semilocal lattice-ordered groups which are realizable over \(k[x_{1}, x_{2},..., x_{n}]\) for \(n\leq 3\).