ON NORMAL LATTICES AND WALLMAN SPACES

Let X be an abstract set and a lattice of subsets of X. The notion of R being mildly normal or slightly normal Is investigated. Also, the general Wallman space with an alternate topology is investigated, and for not necessarily disjunctive, an analogue of the Wallman space is constructed.

concepts and notations that will be used throughout the paper for the convenience of the reader.We then precede to the consideration of mildly normal and slightly normal lattices in section 3, and then to analogues of the general Wallman space in sections 4 and 5.
We flnally note that most of the results hold equally well for abstract lattices.
a) Let X be an abstract set and fl a lattice of subsets of X.We shall always assume, without loss of generality for our purposes, that , X E .The set whose general element L' is the complement of an element L of is d.,ed by Q '.
G.M. EID said to be complement generated if f, for every L of there exists a squence L'.
is separating if for any two elements {Ln}n= in such that L =I,l=1 n x v of X, there exists an element I, such that E L and y L. is T 2 if, for any two elements x y of X, there exlst, A,B .2 such that x A', y e B' and A' B' . is said to be disjunctive if f, for every x E X and L if x L then there exists an E i, such that x L and L O L . is regular if f, for every x E X and every L E , if x L then there exists L I, L 2 fl such that x LI,' L CL and L OL 2' . is normal Iff, for any L|, L2g , if LI, L and L L 2' . is LIfL2 then there exists L! L 2 such that LlC L2C lindelof Iff, for every L ; A, if L then for a countable subcollecton (La of {L}; i 1 Li= '" fl is compact tff, for everv Lcze , e A, if L then for some finite subcol[ecttou (L of {L}; L .Next, consider any two lattices ill' 2 of subsets of X. fll is said to semi-separate or for abbreviation ( s.s.q 2) iff, fr every ^IL E QI and every L2E if Llfl L2= 0 then there flL .I is said to separate if for exists L IE such that L2C L and L any L 2, 2 , if L 2 flL 2 there exist L I, L such that L 2 L I, L2c L and L! 0 .We denote by I :, qet whose general element is the intersecti,, arbitrary subsets of fl.b) Let A h, ,.,y algebra of subsets of X.A measure on A is defined to be a function, from A R such that is bounded and finitely additive.The algebra of subsets of X generated by fl is denoted by A(.).If x X, if x.A then x is the measure concentrated at x so u..'.\) [0 if x#A where A The set whose general element is a measure on A() is denoted M().Note that, since every element of M(fl) is equal to the difference of nonnegatlve elements of M(),, without loss of generality, we may work excl..'i voly with nonnegative elements of M(fl).Let eM(fl), is ft-regular if for any A e A(); (A) sup{(L);LCA, L E fl}.
The set whose general element is an element of M()which is -regular is denoted by MR(fl).An element E M(fl) is o-smooth on fl, if Lne fl, n 1,2,... and Ln then (L 0. The set whose general element is an element of M(fl) w:[ t, i, o-smooth on fl is denoted by M ().We say that is o-smooth on A(fl) if A A(fl), n 1,2... o n and A + t' iCA 0. e set whose general element Is an element of M(fl) which n smooth on A(fl) is denoted by M(fl).Note that if MR(Q) then M (fl).l()IR(fl) I () l(fl) l(fl) are the subsets of the corresponding M's consisting of the non-trlvlal zero-one valued measures.
L Is replete iff, whenever e I() then S() # .A'premeasure on fl Is defined to a function from fl to {0,I} such that () 0, (A) < (B) for every A CB where A,B and If (A) (B) then (AB) I. H() denotes the set of all premeasures on G. c) an immediate consequence of Zorn's Lemma, we have, for every B I(), there exists an element v IR() such that U v on or simply (B v(fl)).Also, for any two lattices GI'G2 of subsets of X, if GIC , then for every E IR(G1 )' there exists a E IR() such that vIA(G and such that a v is unique if G1 separates . Moreover, is normal iff, for I(), < I(G) v2(G)where I,2 IR( then I u2" R is regular iff, for any i,2 e l(fl); I B2(fl) then S( I) S(2).The result xe IR(fl) Iff fl is disjunctive leads us to the Wallman Topology which is obtained by taking the totality of all W(L) { e IR(fl); (L) for L e fl} as a base for the closed sets on IR(fl).For a disjunctive fl, IR(fl) with the W(fl) of closed sets is a compact T space and will be T 2 Iff fl is normal and is called the general Wallman space associated with X and fl.Also, for a disjunctive fl and A,B W(A) is a lattice with respect to union and intersection.Moreover,W(A') (W(A))', W(A(R)) -A(W(fl)), W(A) -W(B) iff A B and W(A)c W(B) iff AcB.Now, we note that, if is disjunctive so is W(R), and in addition to each E M(R), there exists a E M(W(fl)) defined by (A) -(W(A)) for all A A(R) such that the map is one-to-one and onto; moreover 3. ON NORMAL LATTICES.
In this section, we elaborate on the notion of a normal lattice, investigate lattices which satisfy weaker conditions, and discuss their interrelations under the extension and restriction properties.
Throughout this section R'RI' and will denote lattices of subsets of the set X. Note that, for Lle R; i=1,2,3; O is normal iff LICLUL , then L AIUBI; AL, BICL;; AI,BIE then^L^G .In addition,^I f^L I'vL G, UL and by the ^nrmallty of , L LIUL2 with (L) B(LI) + (2 then either (L I) or (L 2) and so either L G or Lv G. Thus, G is a prime ' filter.
THEOREM 3.2.Suppose i C and RI separates .Then, I is normal iff is normal.
PROOF.(i) Suppose fl is normal then by the separation.It [s clear that is normal.
(ii) Suppose R 2 is normal.
LEMMA 3.1.If is complement generated, then is slightly normal.
PROOF.If is complement generated, then Io fl' )C 1R(fl), that is, if Io ( fl' there exists a unique v IR(); (fl).Thus, is slightly normal.
In the nexf theorem, we will see when a partial converse of theorem 3.9 is true.
THEOREM 3.10.Suppose I separates f12" If f12 is mildly normal and countably paracompact then fll is slightly normal.
PROOF.Let E Io (I '); I(I )' 2(I);!' 2 IR(I).Extend to I(2).Consider Bn +' BnE then BnC A'n +' AnE I since 2 is countably paracompact and I separates 2 therefore 2 is I -countably paracompact and (Bn) (A) --(A) 0. Since E I(i), then la(2).Now extend l,V2 to TI, T2 E IR(2) then i(), 2(2)bY separation and so I 2 since is mildly normal.Thus I 2 and so I is slightly normal.REMARK 3.1.It is not difficult to give similar conditions (as in theorem 3.10) to obtain the other partial converse of theorem 3.9.

ON SPACES RELATED TO THE GENERAL WALLMAN SPACE.
We recall from section 2 that for an arbitrary lattice R of subsets of X, IR() with the topology W() of the closed set, is a compact T space.Also, if is disjunctive and separating then X can be embedded in IR().Moreover, if is disjunctive, so is W(), and IR()with the topology W()is T 2 iff is normal.In this section, we consider alternate topologies on IR().THEOREM 4.1.Consider z W(') for a base of closed sets W(L');L'E P Then IR() with the topology is T 2.
THEOREM 4.3.The sets of W() are clopen in the topology.
PROOF.Since W(L) is disjunctive then by remark 4.2 (li) for any L e , W(L) =r.W(L'), W(L)CW'(L ),L is closed in the -topology.
But, W(L) is also open in -topology since W(L) (W(L')').Thus, the sets of W()are closed in the -topology.
PROOF.(i) Since the sets W(fl) are clopen by Theorem 4.3 then W() e 6 and so W() 6.Now, if 6 TW(.) then since W() is compact so is 6 and so is an algebra by theorem 4.4.(ll) The converse is clear.
In the next theorem, we glve another equivalent condition for fl to be an algebra.
is an algebra iff W(fl') is a disjunctive lattice in IR(). PROOF.

ON NON-DISJUNCTIVE LATTICES.
We next consider the case where is not necessarily disjunctive.We begin, by introducing the notion of an -convergent measure and some related results and then proceed to the construction of an analogue of the Wallman space.DEFINITION 5.1.
e I() is said to be -convergent if there exists an x X such that x ( )" THEOREM 5.1.
THEOREM 5.2.Suppose "I 2 ()' for all I' 2 e I().Then a) If I is -convergent so is 2" b) If ' is regular and 2 is f-convergent then I is it-convergent.PROOF.a) Suppose I is -convergent, then x UI() for some x, but I 2 () then Bx 2 () and so 2 is -convergent.b) Suppose 2 is f?-convergent, then x 2 () for some x and so 2 x (') and x e S( 2) on ' but I U2 () then 2 I ' and since ' is regular x e S(2 S(BI on '.Thus, S(I) on ' and so I is it-convergent.THEOREM 5.3.Suppose ' is T 2 and is f-convergent where u e I().Then, there exists a unique x e X: x (')" PROOF.If ' is T 2 and if x#y then there exists L I, L 2 e ; x e LI, y e L2; LIL2 and x(Ll) I, x(L2) I. Now, if x U(n)and y B() then u(LI) u(L2) but LI0 L 2 which is a contradiction and so the desired result is true.U2 I, l then there exists an L such that say, I(L) I, u2(L) 0. ^2 Therefore Ul W(L), U2 W(L') and so W(L) Is To.PROOF.Suppose (i) (U (La) O (W(L)) 0" Let A =(ULa) e a, B (L 8) e a.
Since a U() then (UW(Lo))cW(A) and (UW(LB) cW(B).If AnB I then L nL for some , B-Let x e L 0L B then x and x e W(L )and x e W(L B) which contradicts (i).us, A B 0, W(A)W(B) 0, and the desired result is now clear.Now, we note that If U() then I with W() is generalized absolutely eIed and t absolutely closed if is 2" u, tff we ensider X and (a) and 2' then I, 0 is an absolute elure X tnce ne can easily obere that X (X). REK.
analogous construction can now be done for i o, o.Where IR() g is convergent.We note that the constructions here genraite the ork Lut [6].
o {x: x e X} U{ e l(fl): Is not convergent} and (A) { e I O, and one can show that () Is weakly replete. is weakly replete O iffor any