arXiv subjects

John Harding

Publications and source records attributed to John Harding.

21 records · Page 2Linked to original sources

arXiv2013

Sections in orthomodular structures of decompositions

There is a family of constructions to produce orthomodular structures from modular lattices, lattices that are M and M*-symmetric, relation algebras, the idempotents of a ring, the direct product decompositions of a set or group or topological space, and from the binary direct product decompositions of an object in a suitable type of category. We show that an interval [0, a] of such an orthomodular structure constructed from A is again an orthomodular structure constructed from some B built from A. When A is a modular lattice, this B is an interval of A, and when A is an object in a category, this B is a factor of A.

math.QA↗

arXiv2013

Automorphisms of decompositions

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 Γfrom the group of permutations of X to the automorphism group of Fact X. We show Γis 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 Γis 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.

math.CO↗

arXiv2010

Subalgebras of orthomodular lattices

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.

math-ph↗