Stephen Paul KIng This may be up your alley. I’m looking at it now.
“The Spectral Presheaf of an Orthomodular Lattice
Some steps towards generalized Stone duality
Sarah Cannon, Linacre College, University of Oxford
A thesis submitted for the degree of
Master of Science in Mathematics”
“We will map the elements of a complete orthomodular lattice
L to the algebra of clopen subobjects of the spectral presheaf of L; using the right adjoint of this map, we show that these clopen subobjects, modulo an equivalence relation, form a complete lattice isomorphic to L. This can be seen as a generalization of Stone’s representation theorem for Boolean
I find it interesting as I was looking for tie-ins from complete lattices (which I have a grasp on) and various areas.