Let *m* be a positive integer, *S* be a finite set, and *T* be a
multiset whose elements are subsets of *S*, each having a multiplicity
less than *m*. We say that *T* is a *mod-**m** parity
multistructure* for *S* if, for each subset *b* of *S*, the number *m*
divides
,
where, in this intersection,
each subset of *b* is counted as many times as it occurs in *T*.
If *T* actually is a set (that is, has all multiplicities at most 1),
then we say that it is a *mod-**m** parity structure*.

In order to understand mod-*m* parity multistructures, we use the ring

We define a map from the collection of multisets of subsets of

We can also generalize Theorem 2 and Corollary 2. Again, the proofs are similar to the ones we have for parity structures, but for the second, there is a bit of calculation with binomial coefficients.

For a collection *T* of subsets of a finite set *S* and a sequence of
natural numbers
,
write
for the number of elements
of *T* having size equal to *a*_{i} for some value of *i*.

- a)
- For any subset
*b*of*S*and any multiple*k*of*m*, the number*m*divides . - b)
- For any subset
*b*of*S*of size divisible by*m*and any natural number*k*, the number*m*divides , where (*m*,*k*) is the greatest common divisor of*m*and*k*.

It is harder, on the other hand, to understand mod-*m* parity structures.
We know of two basic families: For any subset *c* of *S* of size
congruent to 1 modulo *m*, the collection of subsets of *S* obtained
by adding *m*-1 new elements to *c* forms a mod-*m* parity structure.
And for any subset *c* of *S* of size congruent to 2 modulo *m*,
the collection of subsets of *S* obtained by adding either *m*-2
or *m*-1 new elements to *c* forms a mod-*m* parity structure.
Of course, any disjoint unions of collections of the above types will
also be mod-*m* parity structures, but for *m* greater than 2,
these are the only examples we know. It would be nice to know
whether there are more, and we pose the following open problem: