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Author: William A. Stein
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We were led to consider parity structures while searching for a new class of simplicial complexes. We classified parity structures, but our original proof was based on a technical induction using binomial coefficients. Later we found a more conceptual proof that uses generating functions obtained from a free boolean algebra over the field  $\mbox{\bf F}_2$ of order 2. In Section 2 we describe how to view parity structures as elements of this boolean algebra, and we use this identification to classify parity structures. We should point out that our classification can be formulated and proved without this boolean ring with a comparable amount of work, but we find our method more interesting and more instructive. In Section 3 we use the classification to find another property of parity structures. We then prove in Section 4 that parity structures satisfy stronger versions of both the defining property and this new one. Finally, in Sections 5 and 6 we discuss some related issues involving an ideal  $\sigma R_{S}$ we define below, and we give a generalization of parity structures.

William Arthur Stein