On compactness of lattices

In this paper, we present a study of various concepts pertaining to compactness properties of a lattice. Special cases include such notions as countably compactness, almost countably compactness, and countably paracompactness. Some investigations in these matters have been done in [2, 7, 14]. We go beyond these results and introduce new subsets of I(L) and their lattices to investigate. This way, we investigate the topological-like properties of lattices through measurability techniques. We consider X to be an arbitrary nonempty set and L a lattice of subsets of X such that ∅,X ∈ L. By A(L) we denote the algebra of subsets of X generated by L, and I(L) denotes those nontrivial, zero-one-valued, finitely additive measures on A(L). We begin with some standard notations and terminology that will be used throughout the paper. Our notations and terminology are consistent with those in the literature (see, e.g., [1, 2, 8, 14]) and are added mainly for the reader’s convenience. We then proceed to analyze interrelations between these various concepts as indicated above. In the last part of the paper we wish to extend some lattice compactness results to the important situation of two lattices L1 ⊂ L2 of subsets of X and investigate separation properties in connection to various aspects of compactness properties. Throughout the paper we treat almost all cases by general measure-theoretic techniques, the advantage being that much of the results obtained for the special case of zeroone valued measures can easily be extended to the case of arbitrary measures on A(L) which are nonnegative, finite, finitely additive, not necessarily two-valued measures of I(L) (as in the case of Theorem 5.4).