DECOMPOSITIONS IN QUANTUM LOGIC

We present a method of constructing an orthomodular poset from a relat ion algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomodular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several kn own methods of producing orthomodular posets are shown to be special c ases of this result. These include the construction of an orthomodular poser from a modular lattice and the construction of an orthomodular poset from the idempotents of a ring. Particular attention is paid to decompositions of groups and modules. We develop the notion of a norm on a group with operators and of a projection on such a normed group. We show that the projections of a normed group with operators form an orthomodular poser with a full set of states. If the group is abelian and complete under the metric induced by the norm, the projections for m a sigma-complete orthomodular poset with a full set of countably add itive states. We also describe some properties special to those orthom odular posets constructed from relation algebras. These properties are used to give an example of an orthomodular poset which cannot be embe dded into such a relational orthomodular poset, or into an orthomodula r lattice. It had previously been an open question whether every ortho modular poset could be embedded into an orthomodular lattice.