Polarization theorems for arbitrary DMCs

A polarization phenomenon in a special sense is shown for an arbitrary discrete memoryless channel (DMC) by imposing a quasigroup structure on the input alphabet. The same technique is used to derive a polarization theorem for an arbitrary multiple access channel (MAC) by using an appropriate Abelian group structure. These results can be used to construct capacity-achieving polar codes for arbitrary DMCs with a block error probability of o(2^-N1/2-ε), and an encoding/decoding complexity of O(N log N), where N is the block length.