  Let P be a semigroup that admits an embedding into a group G. Assume that the
embedding satisfies a certain Toeplitz condition and that the Baum-Connes
conjecture holds for G. We prove a formula describing the K- theory of the
reduced crossed product A \rtimes{\alpha},r P by any automorphic action of P.
This formula is obtained as a consequence of a result on the K-theory of
crossed products for special actions of G on totally disconnected spaces. We
apply our result to various examples including left Ore semigroups and
quasi-lattice ordered semigroups. We also use the results to show that for
certain semigroups P, including the ax + b-semigroup for a Dedekind domain R,
the K-theory of the left and right regular semigroup C*-algebras of P coincide,
although the structure of these algebras can be very different.
