ON NORMAL LATTICES AND SEPARATION AND SEMI-SEPARATION OF LATTICES

This present paper is concerned with two main conditions, that of normality of a lattice, and separation and semi-separation of two lattices, both looked at using measure theoretic techniques. We look at each property using {0,1} two valued measures and associated {0,1} valued set functions.


INTRODUCTION
In this paper we consider necessary and sufficent conditions for a lattice of subsets of an abstract set to be normal,in terms of measure theoretic conditions.We also consider conditions when two lattices separate or semi-separate each other,again using measure theoretic methods.
In the first part of the paper,we consider consequences of a lattice L of subsets of an abstract set X being normal.This is is equivalent as is well known ,(and which we prove), to each element of IxeI(L),the set of non-trivial finitely additive{ 0,1 two valued measures having a unique regular extension velR(L) st v>l.t (L).We then extend this work to look at relations with various classes of measures I$(L),IW(L), set functions kt',kt",and side conditions on the lattice such as cg,, and look at necessary and sufficent conditions that a lattice of subsets have the normal property.
In the second part of the paper we investigate when two lattices L1 ,L2 of an abstract set X L2_L1 ,L1 either separates or semi-separates L2,as well as consequences of separation or semii- separation of two lattices.We again, investigate these properties in some detail in a measure theoretic setting,where they are equivalent to the existence and uniqueness of extensions or restrictions of regular measures on the two lattices.
We also include a section on notation,terminology ,basic backround,and references for the readers convenience.In addition other notions are introduced as needed in the sections in which they occur.

BACKROUND AND NOTATION
We begin by reviewing some notation and terminology which is fairly standard (see,for example, Alexsandroff [1], Camacho [2], Grassi [3], and Szeto [4]).We supply some backround and notation for the readers convenience.
Let X be an abstract set and L a lattice of subsets of X st J,XeL.A delta lattice is one that is closed under countable intersections,and the delta lattice genereated by / is denoted 5(t-) .A lattice is complement generated iff for every Le/there exists a sequence of subsets Anet_ n= 1,2 such that L=An'(' denotes complement).I. is countably paracompact if for every sequence Lnel.and Ln,], then there exists Ln"/ st Ln"_L n and Ln-'$.A tau lattice is one that is closed under arbitrary intersections,and the tau lattice generated by / is denoted :L.
Let I(L) denote the set of non-trivial two valued {0,1 fintely additive measures on the algebra A(L) generated by {k}.Also let teI(o*,t-) denote those elements of I(k) that are sigma- smooth on I_,i.e.Lnet-Ln,I,o ,laeI(*,l) then limla(Ln)=0.I$(k denotes those elements of I(*,k) such that if Lnek I.teI$(/),Ln,l,, and Ln=Le/then la(L)=limla(Ln)" I(o,k) will denote those measures that are sigma-smooth on A(t.),i.e. if AneA(/) An,l,J then liml.t(An)=0.Note that this is equivalent to countable additivity.IR(L) will stand for those measures on A(L) that are l..A lattice is said to be disjunctive if for any xeX and Let.such that x L then there exists a LleL st xeL1 and LLI=O.A lattice is said to be normal if for L1,L2eL and LlL2=O,there exists L3,L4eL such that L3'_L1 L4'_L2 and L3'L4'=O.A lattice is said to be T2 if for x,yeX there exists L ,L2eL such that xeL I',yeL2' and L 'L2'=O A fact we will use throughout this paper is that there exists a 1-1 correspondence between prime L-filters and elements of I(L),and a one to one correspondence between L-ultrafilters and elements of IR(L).This correspondence is set up by letting bmI(L) and H={LeL la(L)= }.Then H is a prime L-filter and conversely if H is a prime L-filter there exists a measure associated with H such that if LeH (L)=I.A similiar correspondence holds for H and bielR(L) in which case H is an L-ultrafilter.
We will assume in discussing H(L) for convenience, that / is disjunctive ,although it will be clear that this assumption is not always necessary.

ON NORMAL LATYIC'F
In this section we extend the work of Eid [5].andHuerta [6],and consider further consequences of a lattice being normal as well as new equivalent characterizations of normality.First we have the following measure theoretic characterzafion of normality: THEOREM 3.1: A lattice L. is normal iff for I.teI(L.)and Vl,v2elR(L.) st I.t<Vl (L.) t.t<v2 (L.) implies that v l=V2.
We then have the following theorem.
THEOREM 3.2:_If L is normal and cp then L is cc iff L ace.Proof: Assume L is cc,then let I.telR(L') which implies that I.tel(L) .But since L is cc this implies that I.td(t*,L).(NoteL cc implies / ace without any other conditions on the lattice).Conversely let L be normal cp and ace.Then let I.mI(L).This implies that laeI(L') and since every filter is contained in an ultrafilter ,there exists an associated velR(L') st I.t< v (L') or Iv (L).Since L is ace veI(t*,L) ,and also since L is normal and cp there exists a vlelR(o,L) st V<Vl (L).Thus because L is normal this implies that v<l.t<Vl (L) ,IxeI(*,L) and L is cc.
THEOREM 3.4: Let L be cg and normal,and I.teI$(L) then l.telR(L).
Note that It" is an outer measure.

LATrlCE SEPARATION
In this section we study and characterize separation and semi-separation between pairs of lattice in a measure theoretic fashion,and give some applications of these results .We first give some definitions.DEFINITION 4.1: Let L1 ,t-2 be lattices st L2L1 .Then L1 is said to semi-separate 12 if for L1 eL and L2eL2 and L c'tL2=O,there exists a L 1-eL st L l"L2 and L c"tL 1"=0.DEFINITION 4.2: Let L1 ,L2 be lattices such that L2L1 then L1 is said to separate L2 if for L2,L2"eL2 and L2c"d_,2~=O,then there exists L 1,Ll~eL st L lL2 L 1~L2~a nd L -xL 1"=.DEFINITION 4.3: Let L and L2 be lattices such that L2L l, then if IJ.eI(L2) the restriction of to A(L1) will be noted by I.tl ,and leI(L 1).
We now proceed to look at what separation and semi-separation implies about the relationship between IR(L1 and IR(L2).THEOREM 4.1: Let L1 and L2 be lattices such that L2L1 and L1 semi-separates L2.Then if veIR(L2) we have that =v (L and IJ.eIR(L ).
We have from theorem 4.2 that if L1 semi-separates I-2 then :IR(L2)--)IR(t.1)the restriction map is defined .A converse holds for special lattices in the next theorem.