  Harding showed that the direct product decompositions of many different types
of structures, such as sets, groups, vector spaces, topological spaces, and
relational structures, naturally form orthomodular posets. When applied to the
direct product decompositions of a Hilbert space, this construction yields the
familiar orthomodular lattice of closed subspaces of the Hilbert space.
  In this note we consider orthomodular posets Fact X of decompositions of a
finite set X. We consider the structure of these orthomodular posets, such as
their size, shape, and connectedness, states, and begin a study of their
automorphism groups in the context of the natural map \Gamma from the group of
permutations of X to the automorphism group of Fact X.
  We show \Gamma is an embedding except when |X| is prime or 4, and completely
describe the situation when |X| has two or fewer prime factors, when |X|=8 and
when |X|=27. The bulk of our effort lies in a series of combinatorial arguments
to show \Gamma is an isomorphism when |X|=27. We conjecture that this is the
case whenever |X| has sufficiently many prime factors of sufficient size, and
hope that our arguments here might be adapted to the general case.
