Commutators and decompositions of orthomodular lattices

Abstract

We show that in any complete OML (orthomodular lattice) there exists a commutator c such that [0, c⊥] is a Boolean algebra. This fact allows us to prove that a complete OML satisfying the relative centre property is isomorphic to a direct product [0, a] × [0, a⊥] where a is a join of two commutators, [0, a] is an OML without Boolean quotient and [0, a⊥] is a Boolean algebra. The proof uses a new characterization of the relative centre property in complete OMLs. In a final section, we specify the previous direct decomposition in the more particular case of locally modular OMLs. © 1989 Kluwer Academic Publishers.