Profinite orthomodular lattices

Abstract. If X is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in X again belongs to X. AMS subject classilkatioo (1980). 06-xX Key words. Orthomodular lattice, variety of algebras, MacNeille completion. The MacNeille completion of an ortholattice (abbreviated: OL) L carries a natur-al orthocomplementation, which is unique if one requires that it extends the orthocomplementation of L, cf. MacLaren [ 121. We always consider the MacNeille completion of an OL as an OL with this unique orthocomplementa-tion. We are interested here in the MacNeille completion of an orthomodular lattice (abbreviated: OML). Very little is known about it. The only positive re-sult we are aware of is due to Janowitz [9]. It states that the MacNeille comple-tion of an indexed OML is again an indexed OML. The only other result is negative: The MacNeille completion of even a modular OL is not necessarily an OML. An example of this (essentially, the only one known) can be constructed using