Parity structures and generating function from Boolean rings

Parity structures and generating function from Boolean rings

David Petrie Moulton and William A. Stein

Let S be a finite set and T be a subset of the power set of S.
Call T a parity structure for S if, for
each subset b of S of odd size, the number of subsets of b
that lie in T is even. We classify parity structures using generating
functions from a free boolean ring.
We also show that if T is a parity structure, then, for each
subset b of S of even size, the number of subsets of b of
odd size that lie in T is even. We then give several other
properties of parity structures and discuss a generalization.