Automorphisms of decompositions

TL;DR

This paper studies the automorphism groups of orthomodular posets formed by decompositions of finite sets, providing detailed results for specific set sizes and conjecturing broader applicability based on prime factorization.

Contribution

It characterizes the automorphism groups of these posets for various set sizes and establishes when the natural permutation group embedding is an isomorphism.

Findings

d7 is an embedding except when |X| is prime or 4

d7 is an isomorphism for |X|=27

Detailed combinatorial analysis for specific set sizes

Abstract

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.