Symmetric difference on orthomodular lattices and $Z_2$-valued states

Abstract

The investigation of orthocomplemented lattices with a symmetric difference initiated the following question: Which orthomodular lattice can be embedded in an orthomodular lattice that allows for a symmetric difference? In this paper we present a necessary condition for such an embedding to exist. The condition is expressed in terms of $Z_2$-valued states and enables one, as a consequence, to clarify the situation in the important case of the lattice of projections in a Hilbert space.

Keywords

• Symmetric
• difference
• orthomodular
• lattices
• $Z_2$-valued
• states