  It is well known that the closed subspaces of a Hilbert space form an
orthomodular lattice. Elements of this orthomodular lattice are the
propositions of a quantum mechanical system represented by the Hilbert space,
and by Gleason's theorem atoms of this orthomodular lattice correspond to pure
states of the system. Wigner's theorem says that the automorphism group of this
orthomodular lattice corresponds to the group of unitary and anti-unitary
operators of the Hilbert space. This result is of basic importance in the use
of group representations in quantum mechanics.
  The closed subspaces $A$ of a Hilbert space $\mathcal{H}$ correspond to
direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the
Hilbert space, a result that lies at the heart of the superposition principle.
It has been shown that the direct product decompositions of any set, group,
vector space, and topological space form an orthomodular poset. This is the
basis for a line of study in foundational quantum mechanics replacing Hilbert
spaces with other types of structures. It is the purpose of this note to prove
a version of Wigner's theorem: for an infinite set $X$, the automorphism group
of the orthomodular poset Fact $(X)$ of direct product decompositions of $X$ is
isomorphic to the permutation group of $X$.
  The structure Fact $(X)$ plays the role for direct product decompositions of
a set that the lattice of equivalence relations plays for surjective images of
a set. So determining its automorphism group is of interest independent of its
application to quantum mechanics. Other properties of Fact $(X)$ are determined
in proving our version of Wigner's theorem, namely that Fact $(X)$ is atomistic
in a very strong way.
