International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 773-779

Lattice separation, coseparation and regular measures

Maurice C. Figueres

The Rockefeller Group, 1230 Avenue of the Americas, New York 10020-1579, NY, USA

Received 14 November 1994; Revised 10 January 1995

Copyright © 1996 Maurice C. Figueres. 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.


Let X be an arbitrary non-empty set, and let , 1, 2 be lattices of subsets of X containing ϕ and X. 𝒜() designates the algebra generated by and M(), these finite, non-trivial, non-negative finitely additive measures on 𝒜(). I() denotes those elements of M() which assume only the values zero and one. In terms of a μM() or I(), various outer measures are introduced. Their properties are investigated. The interplay of measurability, smoothness of μ, regularity of μ and lattice topological properties on these outer measures is also investigated.

Finally, applications of these outer measures to separation type properties between pairs of lattices 1, 2 where 12 are developed. In terms of measures from I(), necessary and sufficient conditions are established for 1 to semi-separate 2, for 1 to separate 2, and finally for 1 to coseparate 2.