The class semigroup of local one-dimensional domains

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