Automorphisms of decompositions

Abstract

Abstract. In [12] 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 Γ from the group of permutations of X to the automorphism group of Fact X.We show Γ 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| = 2 3 and when |X| = 33 . The bulk of our effort lies in a series of combinatorial arguments to show Γ 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.

Wigner's theorem for an infinite set

“…It also fails when n = 2 3 . The results of [8] shows that it holds when n = 3 3 . All other cases are unknown.…”

“…We have shown there is a positive answer for any infinite set X. What is known about the situation for finite sets is described in [8]. Wigner's theorem does not hold for sets X of finite cardinality n where n has only one or two prime factors.…”

“…In a series of papers [4,5,6,20] the automorphisms of M (2) were described in terms of automorphisms of the lattice M, and in the finite dimensional vector space setting this can further be refined using the fundamental theorem of projective geometry. The result most closely related to the Main Theorem is that of [8] where the same statement is established for a set X with 27 elements. This is a non-trivial result requiring a substantial amount of combinatorics.…”

“…This requires the observation that there is an n so that any element of the permutation group Perm(X) can be obtained as the composite of at most n involutions. In regards to this question, there is a discussion of similar issues in [8].…”

“…This is similar to the lattice of subspaces of R 3 . The sizes of these structures are also given in [8], and they grow quickly. For example, when |X| = 27, the orthomodular poset Fact(X) has 10,002,268,381,116,211,200,002 elements, but still has just four levels, i.e.…”