Normal Lattices and Coseparation of Lattices

Let X be an arbitrary non-empty set, and let be a lattice of subsets ofX such that }, X E . We first summarize a number ofknown conditions which are equivalent to being normal. We then develop new equivalent conditions in terms of set functions associated with/z E I(), the set of all noa-trval, zero-one valucl tely addittre measures on ttie a/geOra generated-6y L We ffar generalize all the above to the situation where 1 and 2 are a pair of lattices of subsets of X with ’1 c 2, and where we obtain equivalent conditions for to coseparate E2.


INTRODUCTION
Let X be an arbitrary non-empty set, and let be a lattice of subsets of X such that t, X E E. Various necessary and sufficient conditions for the lattice E to be normal are known (see [4,5,6]), and we summarize a number of these in section 2. We then give new necessary and sufficient conditions for the normality of E in section 3.These conditions are in terms of set functions associated with a E I(E), where I(E) is the set of non-trivial, zero-one valued, finitely additive measures on the algebra generated by .
Section 4 is devoted to the more general situation of a pair of lattices E and E with E1 c E2 and for which E1 coseparates E2.If E E, then E1 coseparates itself if and only if it is normal.We proceed, at first, to give necessary and sufficient conditions for E1 to coseparate E2 which extend known necessary and sufficient conditions for normal lattices which are summarized in section 2. Then we extend our new conditions for normality, to conditions both necessary and sufficient for E to coseparate E2 in terms of set functions associated with a # Jr(E ).
We begin in section 2 with a brief summary of the notation and terminology used throughout the paper.Related matters can be found in [2,4,6].We then turn our attention to normal lattices, and follow the program indicated above.

BACKGROUND AND NOTATION
Here we summarize briefly the notation and terminology that will be used throughout the paper Most of this is standard by now and follows that used in 1,3,4,7] for example.We will also assume for convenience that all lattices considered contain the and X.
X is an arbitrary non-empty set and E a lattice of subsets of X. A(E) denotes the algebra generated by E, and I(g) those non-trivial, finitely additive, zero-one valued measures defined on A().
Finally, we note that if 7"/is any collection of sets of/: with the finite imerseetion property, i.e. the intersection of any finite number of sets of 7"/is non-empty, then there exists a/ E In(/:) such that #(A) 1 for all A E 7. Io(/:) denotes those # 6 I(/:) such that / is a-smooth on /:, i.e. if L, E/: and L, 0 then /(L,) 0. There is a one-to-one correspondence between Io(/:) and all prime /:-filters with the countable intersection property.I () denotes these/ E I(/:) that are a-smooth on A(/:), or, equivalently, are eountably additive.I() Io(/:) N In(E.), and it is easy to see that if # e I.,(/:) then/ e I If # I(/:), we denote by #' the following set function defined on 79(X) for E C X, #'(E) inf{#(L')lE C L',L E / is a finitely subadditive outer measure.If #, v are set functions defined on/:, we write # _< (/:) if/(L) _< v(L) for all L E/:.It is now clear that # e In(Z:) if and only if # #'(/:). (2.2) A set function defined on/: is called modular if v(Lz U/-2) + v(L rq/.) v(Lz) + v(/.), for all L, L E/:.If v(L U L) + v(L1 Cl L) _< v(L) + v(L) for all L,/.E/:, then is called sub- modular, and superrnodular if the inequality is reversed.
Clearly if is c.p. then Io(') C Io(). is a normal lattice if for any A1, A2 E with A1 N A2 (Z) there exist B, B; E with A, C B i, A2 C B and B f'l B O.
We summarize some equivalent characterizations of normality in the following theorem (see [4,5,6]).TIiEOREM 2.1.is normal (where 0, X ) is equivalent to any of the following: 1) For each # E I(/:), there exists a unique v In(/:) such that # _< 2) For any I(/:) and v In(l;) such that # < v(/:) then/ < v v' 3) If# < v(/:) where/2 E I(/:), v E In(/:) then v(L') sup{v(L)l c L',L e } where L 4) If L C L U L where L, L,/,2 E/:, then L A U B where A, B and A C L, B C L.
Further characterizations of normality will be developed in section 3. We just note one consequence of normality.We denote by I(/:) those/ I(/:) such that/(L') 1 implies there exists an NORMAL LATTICES AND COSEPARATION OF LATI'ICES 555 C L', and/' () 1 where L .If L: is normal, then l(L:) IR(L:).The converse, however, is not tree in gen.
If we ve a p of las, , 2 of subss of X th 1 C 2, d if p I(2) then its refiion to (]) ll be doted by [ or simply p[ if thee is no biW.
In gen if we have a pr oflauis , 2 we de.e: 1 sstes 2 if for l A , B 2 such t A B , there es A] ] such thatBcAdAA=0.
Finely, v is a fite omer me (either fitely mbadditiv or coumably subadtive) defined on P(X), we dote by Sv the v-meurable s, so v is regular if for any A C X, there exists an E ,5 with A C E and v(A) v(E).If v is a regular outer measure which is finite then If v is any outer measure that just assumes the values zero and one, then v is clearly regular.

NORMAL LATTICES
In this section we wish to get characterizations of normal lattices in terms of certain set functions associated with a /2 I().In the presence of normality, these set functions have been investi- gated [2,4,5,6].We will summarize briefly these results, but we wish to go beyond this, and show that properties of these set functions can be utilized to give necessary and sufficient conditions for a lattice to be normal.
Since g is a finitely subadditive outer measure, we denote by ,., the g-measurable sets, i.e.
PROOF.a) Since g is a finitely additive measure on ,., it follows that glct(:) e I(E).
Let v be a set function defined on all subsets ofX.Recall v is submodular if and only if v(E1 E'2 + v(E1 CI E'2 <_ v(E1) + v(E2 for all El, E2 C X. It is easy to see that the following holds.LEMMA 3.4.If is a monotone set function defined on all sets E C X that assumes only the values 0 and 1, then v is finitely subadditive if and only if is submodular.Now we establish: THEOREM 3.5.If # E I(), then is normal if and only if is submodular (or equivalently if and only if is a finitely subadditive outer measure).PROOF.If is normal, then is a finitely subadditive outer measure by Theorem 3.1 b), and, therefore, submodular by the Lemma.
As our characterization theorem, we have TBXOREM a.6.If I(), then ; is normal if and only if PROOF.If'Z: is normal th =/z'() by Theorem 3.3 part b).Conversely, suppose =/z'(L;) and/2 is not normal.Then using the same notation and construction as in Theorem 3.5, we have #, (A) 0, while A1 C B', B implies/z(B') 1, but A1 c A. Thus (A1) 0, while clearly/z'(A1) 1.This contradiction proves the theorem.

COSEPARATION OF LATTICES
In the present seion, we will extend the results of sections 2 and 3 on normal lattices to a pair of lattices /21 and 2 such that 1 C 2, and where 1 coseparates coseparates itself if and only if it is normal.The work done here also extends that done in [2,5,6].
Note: Clearly Theorem 4.2 is equivalent to the following: Let 1 C 2 be lattices of subsets of X, and let # I(1) and v IR(2) be arbitrary with # < Y(1).Then 1 coseparates 2 if and only if v(B) 1, B 2, there exists an A 1 with A C B , and p(A) 1. Clearly this result extends Theorem 2.1 part 3. We now extend the comment following Theorem 2.1.Theorem 4.3.Let 1 and 2 be lattices of subsets of X such that 1 C 2. Also let 1 coseparate 9..Then, iffor/EI(1) and v I(2) with # < v(1) and with v(B')=sup{(A)[A C B',A 1}, B 2, we have v IR(2).
PROOF.Suppose v(B') 1, where B E 2. Then there exists A , A C B' and '(A) 1.
PROOF.If v q Io(), then there exists a sequence {B}, B,, 9. such that B'. $ 0 and v(B') 1 for all n.By the note after Theorem 4.2 there exists A, 1 with A, C B and (A,) 1 for all z.Clearly, we may assume A,, , so A, L 0, which implies 6 Io(1), a contradiction.THEOREM 4.5.Let 1 C u and 1 coseparates .I fv Iu() and if1 is almost countably compact then v E Io (22).PROOF.Let A vl,(:,).A In(1), and since 1 is almost countably compact, there exists a I, () such that Now, by Theorem 4.4, it follows that v I(), which completes the proof.REMARK.Under the assumption of Theorem 4.5, if in addition, I,(E6)C 1o(9.), then v I (9_), in which case 2 is countably compact.
Now we show TIIEOREM 3.2.If L: is normal and if p I(L:), then A() C .

Finally we have THEOREM 3 . 3 .
If is normal and if p E I() then a) 1) I(:), and b)