Ergodic theory meets polarization II: A foundation of polarization theory for MACs

An open problem in polarization theory is to determine the binary operations that always lead to polarization when they are used in Arıkan style constructions for multiple access channels (MAC). This paper solves this problem by providing a necessary and sufficient condition for a sequence of binary operations to be polarizing. We show that a sequence of binary operations is MAC-polarizing if and only if the inverse of each binary operation in the sequence is strongly ergodic. We extend the ergodic theory of binary operations and study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly ergodic if and only if all the operations in the sequence are strongly ergodic.