International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 474356, 13 pages
Research Article

Lattice Operators and Topologies

Department of Computer and Information Science, Brooklyn College, 2900 Bedford Avenue, Brooklyn, NY 11210, USA

Received 16 March 2009; Accepted 13 October 2009

Academic Editor: Robert Redfield

Copyright © 2009 Eva Cogan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.