Bulletin of the Australian Mathematical Society

Research Article

Topological and order-topological orthomodular lattices

Zdenka Riecanováa1

a1 Department of Mathematics Electrotechnical Faculty of the Slovak, Technical University, Ilkovicova 3 CS-812 19, Bratislava, Czechoslovakia

Abstract

The necessary and sufficient conditions for atomic orthomodular lattices to have the MacNeille completion modular, or (o)-continuous or order topological, orthomodular lattices are proved. Moreover we show that if in an orthomodular lattice the (o)-convergence of filters is topological then the (o)-convergence of nets need not be topological. Finally we show that even in the case when the MacNeille completion S0004972700012168_inline1 of an orthomodular lattice L is order-topological, then in general the (o)-convergence of nets in S0004972700012168_inline1 does not imply their (o)-convergence in L. (This disproves, also for the orthomodular and order-topological case, one statement in G.Birkhoff's book.)

(Received December 05 1992)