Projections on orthomodular lattices

Abstract

In terms of c-closure operators we give a necessary and sufficient condition for an orthocomplemented poset to be an orthomodular lattice. C-closure operators are closely related to projections and appear as a generalization of symmetric closure operators.

We show how a projection can be represented as a product of a SASAKI-projection and a symmetric closure operator. Finally, starting with a subset of an orthomodular lattice, we construct explicitly the symmetric closure operator that maps the lattice onto the commutant of that subset.