International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 49-59
doi:10.1155/S0161171293000055
On Alexandrov lattices
Department of Mathematics, New York Institute of Technology, Old Westbury 11568, New York, USA
Received 10 October 1991; Revised 20 January 1992
Copyright © 1993 Albert Gorelishvili. 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.
Abstract
By an Alexandrov lattice we mean a δ normal lattice of subsets of an abstract set X,
such that the set of ℒ-regular countably additive bounded measures is sequentially closed in the set
of ℒ-regular finitely additive bounded measures on the algebra generated by ℒ
with the weak
topology.
For a pair of lattices ℒ1⊂ℒ2 in X sufficient conditions are indicated to determine when ℒ1
Alexandrov implies that ℒ2 is also Alexandrov and vice versa. The extension of this situation is
given where T:X→Y and ℒ1 and ℒ2 are lattices of subsets of X and Y respectively and T is ℒ1−ℒ2
continuous.