ON LATTICE-TOPOLOGICAL PROPERTIES OF GENERAL WALLMAN SPACES

Let X be an arbitrary set and ℒ 
 a lattice of subsets of X such that ϕ,X∈ℒ.𝒜(ℒ) is the 
algebra generated by ℒ and I(ℒ) consists of all zero-one valued finitely additive measures on 
𝒜(ℒ). Various subsets of and I(ℒ) are considered and certain lattices are investigated as well as the topology of 
closed sets generated by them. The lattices are investigated for normality, regularity, repleteness and 
completeness. The topologies are similarly discussed for various properties such as 
T2 and Lindelof.


INTRODUCTION
Let X be an arbitrary set and a lattice of subsets of X such that , X E E. ,A() is the algebra generated by and I() denotes those non trivial, zero-one valued, finitely additive measures on Various subsets of I() are considered and certain lattices in these subsets are investigated as well as the topology of closed sets generated by them.The lattices are investigated for normality, regularity, and for a variety of repleteness and completeness conditions.The topologies are similarly investigated for various properties such as T2 and Lindeltf.Necessary and sufficient conditions for these properties to hold can effectively be given in terms of measure conditions on the original lattice.Some investigations in these matters have been begun in [2], [3] and [8].We go beyond these results, and introduce new subsets of I() and their lattices to investigate.
] and [10]), and is added mainly for the reader's convenience.
We then proceed in the subsequent sections to analyze in detail the lattice-topological structure of various Wallman spaces as indicated above.

BACKGROUND AND NOTATIONS
Let X be an arbitrary nonempty set, and a lattice of subsets of X.It is assumed throughout the paper that , X E .
We adhere to the customary lattice-topological definitions which can be found for example in ], [2], [4], [7] and 10].Here, we just note some of the measure theoretic equivalents.For this purpose we introduce the following notations: ,A() denotes the algebra generated by , and () the set of C VLAD non-trivial zero-one valued finitely additive measures on 4() IR() the set of -regular measures of I(), where # E I() is -regular if for any A E 4()#(A) sup{#(L)/L C A,L L} sequences {L} of sets of with L, 4),/(L,) 0. ia() the set of(7-smooth measures of I() on , where # I() is -smooth on if for all ofr-smooth measures on 4() of 1() I() the set of -regular measures of Ia().r()={H, defined on , non-trivial, monotone, and H(ANB) II(A)H(B),A,B } the set of all premeasures on 7ro() is the set of all pre- measures on which are (7-smooth on Note that there exists a one-to-one correspondence between -filters .T and elements of 7r() given by H(L) 1 iff L .T.
-filters with countable intersection property and ro ().All elements of I() and all prime -filters, given by: for any # E 1() we associate the prime -filter given by .Y" {A ./#(A)= 1}.
REMARK.If is disjunctive and if for each II 6 7r(), there exists a v 6 I() such that II _< v() then I(), -)'V,() is Lindelff This result is known and appears e.g. in [8].

ON PRIME COMPLETE AND COUNTABLY COMPACT LATTICES
In this section we investigate the equivalence and consequences of stronger lattice completeness assumption.
b) Let 12 be disjunctive, regular, LindelOf, almost countably compact and let To(L) be prime complete.Then is countably compact.
b) 12 regular and Lindel0f implies 12 mildly normal and by the above result, it follows that 12 is countably compact.
We finly note at the conditions of Theorem 5.4 that 4 with/A 6. STRONGLY or-SMOOTH MEASURES Here we consider another Wallman space and analyze the relevant lattice in detail.DEFINITION 6.1 A measure # I() is strongly g-smooth on L: iff for any sequence {L}, L , L , if f L then #( L) inf#(L) =lirnoo #(L).We denote J(L:) the set of strongly g-smooth nontrivial zero-one valued measures on .DEFINITION 6.2 The lattice L: is weakly prime complete if for Now define the following condition"