  Let G be a finite group acting by automorphism on a lattice A, and hence on
the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra
of multiplicative invariants.
  We investigate when algebras of multiplicative invariants are semigroup
algebras. In particular, we present an explicit version of a result of Farkas
stating that multiplicative invariants of finite reflection groups are indeed
semigroup algebras. On the other hand, multiplicative invariants arising from
fixed point free actions are shown to never be semigroup algebras. In
particular, this holds whenever G has odd prime order.
