  Let G be a locally compact second countable Abelian group. Given a measure
preserving action T of G on a standard probability space, let M(T) denote the
set of essential values of the spectral multiplicity function of the Koopman
unitary representation of G associated with T. In the case when G is either a
discrete countable Abelian group or R^n, n>0, it is shown that the sets of the
form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive)
subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.
