The class semigroup of local one-dimensional domains

Let R be a local one-dimensional domain. We investigate when the class semigroup s(R) of R is a Clifford semigroup. We make use of the Archimedean valuation domains which dominate R, as a main tool to study its class semigroup. We prove that if s(R) is Clifford, then every element of the integral closure (R) over bar of R is quadratic. As a consequence, such an R maybe dominated by at most two distinct Archimedean valuation domains, and (R) over bar coincides with their intersection. When s(R) is Clifford, we find conditions for s(R) to be a Boolean semigroup. We derive that R is almost perfect with Boolean class semigroup if, and only if R is stable. We also find results on s(R), through examination of [V/P : R/m] and v(m) where V dominates R, and P, m are the respective maximal ideals. (c) 2008 Elsevier B.V. All rights reserved.