AVANN, S. P. Math. Annalen 175, 320--336 (1968)

Locally Atomic Upper Locally Distributive Lattices S. P. AVANN It is well known that the meet-closed subsets of a partially ordered set P, ordered by set inclusion, form a compactly generated, dually compactly generated, distributive lattice, which we denote by LM(P). Moreover, LM(P ) is atomic if and only if P satisfies the Descending Chain Condition. In this paper we define a closure relation D--, D on LM(P ) in terms of an equivalence relation 0 on P satisfying specified conditions whereby the bar-closed sets of L~(P) form an upper locally distributive lattice LMo(P ). We thereby obtain an "upper distributive" theory that corresponds to the theory for LM(P ).

t. Notation and Terminology Lattice or set inclusion symbols are 2, 3, and > for inclusion, proper inclusion, and covering respectively. The zero and unit elements of a lattice L are z and u respectively. The quotient sublattice a/b is a/b = {c e L la 3=c 3=b }. We write c/bZ=ra/d when c/b is an upper transpose of a / d : c = a w b and d = a ~ b . We use w and U for lattice join; c~ and 0 for meet; + , ~, for set addition; (-), I~ for set intersection; and A - B for set difference. The empty set is 0. In a partially ordered set P the J-closure of a subset So__p is SS= {x~PIx~=s} for some s e S ; and a J is the J-closure of {a}, a e P ; dually for M-closure S M and a N. LM(P) is the complete lattice of M-closed subsets D = D M of P. For an equivalence relation 0 on P {x} = {x~ e P lx, Ox}. In a lattice L, x ~ L is completely join-irreducible, hereinafter abbreviated c-join-irreducible, if and only if x = U S implies x = s e S for So__L. We let P(L), or simply P whenever no ambiguity arises, denote the set of c-joinirreducible elements of L, partially ordered by the inclusion relation of L. We shall apply without reference the following characterization of a c-joinirreducible: x > a = ~ { d ~ L I d C x } . Dual statements may be made for the partially ordered set P ' = P'(L) of c-meet irreducibles y ~ L. DCC will stand for the Descending Chain Condition.

2. The Closure Operator Definition 2.1. Let 0 be an arbitrary equivalence relation on the elements of a partially ordered set P. Define D for D = D u E LM(P) as follows: ={xePl{x~}'D4=0

for some

x,____x}

where {x~} is the 0-equivalence class to which x~ belongs.

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Theorem 2.1. The mapping D ~ D is a closure operator on LM(P) determining the complete lattice LMo(P) of bar-closed sets in which (']~r C~=[I,~T C~ and U , ~ r C ~ = Z , ~ r C ~. It is readily observed that D is M-closed; that D = D =/~; that D = E implies D ~ E; and that LM(P ) has unit element P = P. The conclusions of the theorem therefore follow. It is evident that without further restriction on the equivalence relation 0 many choices of P and 0, even with one or the other fixed, will determine the same lattice LMo(P) within isomorphism. We now introduce suitable restrictions on 0 which, as we shall later prove, will make P isomorphic to the partially ordered set P(LMo(P)) of c-joinirreducible elements of LMo(P) under the natural mapping x*-~x M. Definition 2.2. We shall call the equivalence relation 0 on a partially ordered set P an upper locally distributive equivalence relation, henceforth abbreviated ULDER, on P if and only if 0 satisfies the following two conditions. (F) x M = x M for all x ~ P . (G) x M - x = ~ for all x e P . In Section 8 we shall consider the interdependence of (F) and (G) with several other conditions on 0. For the determination of the structure of LMo(P) we shall at present consider only the following additional condition. (H) x 10x2 + x l implies X 1 ~1) X 2 (and by symmetry x2 ~)xO. Lemma 2.t. (G) implies (H). Proof. Assume (H) is false: xl 3x2Oxj. Then xl Ox2 e ( x f - x 0 ; and for x, C Xl we have x, Ox~ e (x~ - xO. This yields the contradiction Xa e xl--~- xl X1M - - X 1.

Notation. For C e L M o ( P ) let C * = { { x } l x e C } ; and let L~ao(P) = {C*[ C e LMo(P) } with ordering in L~0(P) given by set inclusion. The reader should carefully distinguish C* from {x= e P[ x=0x e C}. Theorem 2.2. For unrestricted 0 LMo(P) is isomorphic to L~o(P) under the mapping C ~ C * . Also ( U ~ r E~)* = ~,~T E*~. Proof. The correspondence is onto L*o(P) by definition. Let D*= C*. For arbitrary x e D and arbitrary x_~ x we have x~e D also, since D = D M. By hypothesis x, Ox,~ ~ C. Hence x ~ C = C. Thus D ___C. Trivially D ~ C implies D'c= C*. The second statement of the theorem follows from Definition 2.1 and Theorem 2.1. Lemma 2.2. For unrestricted 0 and C ~ D in LMo(P) (a) implies (b) and (b) implies (c) where (a) C = D + x M = D + x for x minimal in C - D (and {x} • D = 0 necessarily). (b) C * - D * contains just one element (O-equivalence class). (c) C > D. I f 0 satisfies (H), then (a), (b) and (c) are equivalent. Proof. The first statement follows from Theorem 2.2. Now let 0 satisfy (H), and consider C > D. Since C - D ~: D = D, there exists x e (C - D) such that {x} ¢ O*. Assume that (x M - x) - D +- O. Then there exists x, ~ ((x M - x) - D)

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such that not only x, ¢ {x} by (H), but also {x~} ¢ D*; otherwise (x M - x ) - D _ D = D, a contradiction of our assumption. Now {x} is not represented in /~ + xff, for it is represented neither in D nor in x y by condition (H). By Theorem 2.2 we obtain C = C ~ D + x M 3 D + x y 3 D contrary to our hypothesis. Hence (x M - x ) - D = 0, that is, x is minimal in C - D. Moreover D = D + x with C* - D* consisting of just the one 0-equivalence class {x}. Thus (c) implies (a) as desired. Without imposition of condition (H) on 0 (c) does not imply (b) and (b) does not imply (a), although (b) implies C = D +"x~ for x not necessarily minimal in C - D . Definition 2.3. A lattice L is locally atomic (henceforth frequently abbreviated LA) if and only if e 3 d in L implies there exist r, s ~ L for which c _~r > s = d. L is atomic if and only if c 3 d implies there exists r e L for which

c3=r>d. Our definition of atomicity is not universally accepted. Some authors use the term relatively atomic and reserve the term atomic for the case d = z. Theorem 2.3. I f 0 satisfies (H), then LMo(P) is locally atomic. Proof. Let C 3 D in LMo(P). Then by Theorem 2.2 C * 3 D * . Consider x ~ ( C - D ) for which {x} ¢ D*. Then by Theorem 2,2 and (H) we have C

~ D + x-~=Dwx-~> O + ( x ~ t - x)~=O. In Section 4 we shall consider the property of atomicity in L~o(P ) in some detail. Definition 2.4. A lattice is upper semi-modular, henceforth abbreviated USM, if and only if L satisfies (USM) a > ac~b implies a wb > b. Theorem 2.4. I f 0 satisfies (H), then L~o(P) is USM. Proof: Let A > A c ~ B = A . B = D in LMo(P). Then A = D + x = D w x M for x minimal in A - D by Lemma 2.2. Since x ~ B, x is minimal in P - B . Again by Lemma 2.2 B < I T - 4 - x = B w Z ~ = B ~ D w x - ~ = B w A . Corollary 2.4.t. I f 0 is unrestricted but P satisfies the DCC (Descending

Chain Condition), then LMo(P) is USM. Proof. For applying Lemma 3.2 the existence of x minimal in A - D is guaranteed rather by the DCC in P. Definition 2.5. For an arbitrary element d of a lattice L let Va = {c, ELlc~> d} and let ua = ~ Va. We define L to be upper locally distributive (henceforth frequently abbreviated ULD) if and only if L satisfies: (ULD) The set of elements Vd generates under join the entire quotient lattice uJd and ua/d is a complete, atomic Boolean algebra whose points are the c~. Thus the elements U S all exist, are distinct for distinct subsets So= Va, and exhaust all of ua/d. Particularly ( ~ S)c~(~ T ) = t,) (S- T). This definition is more general than that of DILWORTH and CRAWLEr [6] in the fact that atomicity is not required in L. Indeed an element need not have any covering elements, in which case u,/a is understood to be a/a = a. Theorem 2.5. I f 0 satisfies (H), then LMo(P) is ULD.

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Proof. Let V9= {C~eLMo(P)fC~>DeLMo(P)} , and let Cs= ~) {C~lC~e SC__VD} . By L e m m a 2.2 C, = D + x, for x~ minimal in P - D, with C, = C a if and only if x, Ox,. By Theorem 2.2 Cs~--~C~(--*S is a 1 - t - 1 correspondence determining the triple isomorphism of the partially ordered set of elements Cs with the partially ordered set of corresponding 0-equivalence class sets C* represented in them, and with the complete atomic Boolean algebra of all subsets S of VD. Since {C*ISC VD} constitutes a quotient sublattice of L*Mo(P), then {CstS c=VD} constitutes a quotient sublattice of LMo(P). Theorem 2.6. Let 0 satisfv (H). Then Y is a c-meet-irreducible in LMo(P) if and only if Y = P - ~ {x~l ~ e T} jbr some entire O-equivalence class {x~le ~ T}. Hence there exists a 1 - 1 correspondence from the set of c-meetirreducible elements of LMo(P)onto the set ()fall O-equivalence classes {x} in P. Proof of sufficiency. For an arbitrary 0-equivalence class {x~_lee T} we show first that Y is an element of LMo(P). For if there exists v ~ ( Y - Y), then v ~ x~ for some c~~ T; moreover, by M-closure of 17, x~ e Y contrary to the fact that {x~} is not represented in Y. Thus Y - Y is empty and Ye L~0(P). The set {x~lee T} constitutes all of the minimal elements of P - Y = ~ {x~lc~ e T} so that Y < B = Y + x , = P - ~ { x ~ - x~te e T}. Moreover if Y C C, then there exists x~ e C for c( ~ T, so that C ~ Y w x ~ = B. Hence Y is c-meet-irreducible in LMo(P). Proof of necessity. Given Y < B and Y C C implies B c C, i.e., Y is c-meetirreducible. By L e m m a 2.2 B = Y + x for x minimal in B - Y = P - Y , and { x } C = P - Y . Thus ~ {x~l~e T } _ _ c P - Y by M-closure of Y. For arbitrary v ~ ( P - Y) we have YC Y w v M= Y + v m. Thus Y+vM~= Y + x, and by Theorein 2.2 v ~=x~Ox. Thus v e ~ x~J so that P - y__c ~ x ~ . Hence equality follows and Y = P - ~ x~. We shall henceforth refer to the x~Ox as being associated with Y. Theorem 2.7. Let 0 satisfy (H). Then X e LMo(P) is c-join-irreducible implies X = x-u jot some x e P. Proof. For a c-join-irr__educible X = X M = X e L M o ( P ) we always have X = ~ {xMlx~ e X} = U {xMlx: ~ X}. Hence X = x M by c-join-irreducibility. The converse of Theorem 2.7 fails as shown by the following example. Example 2.1. P = {x a Ox2 > s < x, > r}. In LMo(P) -~2 ~ s-~ = ~ ~ x M. We note that condition (F) fails, though (G) and (H) are valid. In view of the foregoing counterexample we are at this stage motivated to require conditions (F) and (G) on the equivalence relation 0. This will render valid the converse of Theorem 2.7, and more importantly P will turn out to be, within the natural isomorphism x--*x M, the partially ordered set of c-joinirreducible elements of LMo(P). Theorem 2.8. Let 0 be an ULDER on P. Then X ~ LMo(P) is c-join-irreducible if and only if X = x M = x-~ for some x e P. Proof. Necessity follows from Theorem 2,7 and condition (F). N o w consider x---~ = x M > x M - x = x M - x. Clearly x M ~ C implies x M - x ~ C.

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Corollary 2.8.1. Let 0 be an ULDER on P. Then P ~- P(LMo(P)) under the natural mapping x ~ x - ~ = x M, where P(LMo(P)) is the partially ordered set of all c-join-irreducible elements of Luo(P). Definition 2.6. A lattice L is smooth if and only if L satisfies condition (S): (S) c = U { x e P ( L ) lxC=c} for all c • L . Lis dually smooth if and only if L satisfies (DS) c= f]{y•P'(L)ly~=e} for all cEL. Theorem 2.9. I f 0 is an ULDER on P, then LMo(P) is smooth. Proof. F o r C = C M = C E L~o(P ) we always have C = ~ {x~-~ = x MJx ~ C} = U {xMIx • C } , where the x M are c-join-irreducible by Theorem 2.8. Theorem 2.10. I f 0 is an ULDER on P, then LMo(P) is dually smooth. Proof. F o r C•LMo(P) we have CO= Y = P - ~ {xJlx~6 {x}) if and only if ( ~ x~a) • C = 0 if and only if {x}- C = 0, since C = C u. Each such Y is c-meetirreducible according to Theorem 2.6. Thus

Cc__.(~ { Y = P - ~ {xJ~lx~ {x}} ] {x}.C=O} = l~ Y= P - ~ { ~ {x~lx~e {x}} I {x}. C = 0} = G •LMo(P ). But C (~ G by Lemma 2.2, since there is excluded from G all representative of each 0-equivalence class not represented in C. Thus C = (~ { Y I{x}- C = 0}. Definition 2.7. In a lattice L let Q be the partially ordered set of an equivalence class of projective prime quotients ordered by transposition. We say that L is upper splitting, henceforth abbreviated USP, if and only if each Q satisfies (USP) Q has a greatest element b/y; each c / d • Q contains a minimal element xJa~ c rc/d; and L is partitioned by y and the x~ into two disjoint sets: L=yM+~x~

and

yM.(~xa)=0.

Lemma 2.3. In Definition 2.7 y is c-meet-irreducible and each x~ is c-join-

irreducible, Proof. Let c 3 y. Then c ~ yM and c 2 x~ for at least one of the x~. But xJa~C=rb/y so that c ~ x ~ u y = b . Hence y is e-meet-irreducible. Now let d C x# for b/y ~=rX#/a# and assume d ~ y. Then xa 3 d ~ xa where b/y 3 rXffa6. Now aa = x~c~y = x~c~xac~y = xac~aa < x~, and ap C x~waa c=xa. Hence xouaa = xa. We obtain xo/a~ C TXp/ap, contrary to the minimality of xa/aa in Q. Our assumption is false so that d= y and de= xac~y= aa < xa. Hence x~ is c-joinirreducible. Theorem 2.11. I f 0 is an ULDER on P, then L~uo(P) is USP. Proof. By Lemma 2.2 a prime quotient C/D in LMo(P) uniquely determines a 0-equivalence class {x, I• • T} = C* - D* where C = D + x ~ for xp a minimal element of C - D. Moreover B / Y ~ rC/D ~=rXp/Ap where Xa = x~, P-

2 {x~J[a• T } = Y < B = P - - ~ ( x ~ - x ~ l c t • T} = Y u X ~ ,

and X~ > x f - x~ = A~ = Yc~X~ for all fl • T. Thus B/Y is the (unique) greatest element of the partially ordered set (ordered by transposition) of projective prime quotients C/D for which C* - D* = {x}, and X~/A~ is a lower bound of C/D that is clearly minimal in this set. Conversely, a 0-equivalence class {x~}

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in P determines a projectivity class of prime quotients with greatest element B/Y. Now {x~} is represented in an arbitrary element E ~ LMo(P), say by xe, if and only if E = E M ~ x}~= X~. On the other hand {x~} is not represented in E: { x , } ' E = 0 if and only if E = E M ~ Y = P - ~ { x ~ 1 ~ e r}. Thus LMo(P) = y M + ~ {X~tc~ e T} and y M . y , {X~lc~ ~ T} = 0 where yM is the M-closure of Y and X~ is the J-closure of X, in LMo(P). The proof is now complete. Corollary 2.11.1. The triple correspondence D~--~C/D+-+C is a 1 - 1 - 1 correspondence from ~ { Y / A ~ I a e T , A ~ = X , - x , } onto the set of prime quotients C/D for which C * - D * = {x~[ae T} and into ~ {B/X~lae T} yielding a triple isomorphism of the three partially ordered sets involved. Proof. Y ~-D ~-A~ = x ~ - x~ if and only if x~ is minimal in P - D. Then C = D + x~ = D u x~~ > D with C* - D* = {x,} and B / Y ~=r C/D 2 T X j A , . Corollary2.tt.2. I f E D F and E ~ Y ~ F in L~I0(P), then there exists R, S ~ Luo(P) such that E 3=R > S ~=F and B~ Y 3=TR/S 3=rX~/A~ .for some c~~ T. Proof. E ~ X~ > A, for some e e T . Hence A~ C=E c~ Y = S C__Y with X, > A~ = X~c~S. Thus E 3=R = X ~ u S > S 3=F by upper semi-modularity.

3. Atomieity and Unique Meet-irreducible Representations When 0 is equality, the following three conditions are equivalent in LM(P). (DCC) P satisfies the DCC. (AT) LM(P) is atomic. (UMR) Each D E LM(P) has unique reduced meet-irreducible representation. Since LM(P) is distributive, a fourth condition is trivially implied, namely (ND) There does not exist any modular non-distributive sublattice of order five (or equivalently, every modular sublattice is distributive). When 0 is an arbitrary U L D E R (always assumed in this section), we shall show that (DCC) merely implies the second and third conditions, which are still equivalent; and that these three conditions now non-trivially imply the fourth condition (ND). Definition 3A. Let P' be the partially ordered set of c-meet-irreducible elements of a lattice L. Then d ~ L is said to have a reduced meet-irreducible representation if and only if d = f'l {Y~~ P't c~E T} but d C ~'t {Y~e P'I e E S C T} for all S C T. In the following let the 0-equivalence class associated with a c-meetirreducible Y~ ~ LMo(P ) be {x~plfl E S~}, so that Y~= P - ~ {x~p[[3 ~ S~}. Theorem 3A. Let Y~ be a c-meet-irreducible containing D in LMo(P). Then Y appears in every meet-irreducible representation of D if and only if there exists a particular x~y that is minimal in P - D, 7 ~ S~. On the other hand Y~ is redundant in every meet-irreducible representation of D in which it appears if and only if x~B is non-minimal in P - D, for all [3 ~ S~. Proof. Since the last parts of these two equivalences are negations of one another, logically we need only prove the reverse implications. Hence let x ~ be minimal in P - D. Since__x~ - x ~ _c_D, we cannot also have x~, ~ D for any [3 e S~; for otherwise x ~ e D = D, a contradiction. Hence {x~,l [3 e S~} • D = 0

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J and D = Y, = P - ~ x,a. If D itself is c-meet-irreducible, D = Y, and by its own irreducibility Y~ must appear in every meet-irreducible representation of itself. Otherwise we consider D C Y~ and are to show that Y, is a member of an arbitrary meet-irreducible representation D = (~ { Y~16 ~ T} C Yo for all 6 e T. Since T must have more than one element, we may consider Ya in this representation that is arbitrary except that it is distinct from Y~. Assume x,~ ¢ Y6. Then x ~ __3xa~ for some xa~ associated with Y6. But Y~-t= Y~ requires x,~, D x6,, putting xa~ in D. This is contrary to {xa~}- D = 0, the equivalent of Ya D D. Hence x ~ e Y 6 for all Ya+Y~. Therefore x , ~ I I { Y 6 1 6 e T , Yo-I=Y,} = N { Y 6 1 6 e T , Y~-+Y~}3D, since x,.e¢D. Hence Y~ must belong to the representation. We now prove the second reverse implication and start with non-minimal x,p 3 v,p e ( P - D) for each fi ~ S~. Consider an arbitrary meetirreducible representation D = ~ { Y~t5 e T} that includes Y~= P - ~ x~¢. Now P - D = P - H { Y~16 ~ T} = ~ (P - Y~) = ~a~T ~, {XJ~t a e So}. Then for each fi we have x ~ D ~,'~ x~e~ for some particular 5p ~ T and a e Sa~. From this and (H) we have Y~p-t=Y, for all fl e S~. We also have x~t~(~ Y~, or Y~ c p - x~p for all fle S,. Hence

Thus finally

0 { r o l f e T-c~} = Y , ~ 0 {Yot lie T - T } = 0 {Yalbe T} = D , showing that Y, is redundant. Corollary 3.1.1. I f D ~ LMo(P) has any reduced representation at all, that reduced representation is unique, namely O = (-], Y~, where Y~ is the c-meetirreducible associated with x~r and where x ~ ranges over all minimal elements of P - D . Corollary 3.t.2. If P - D has no minimal elements, every Y~ D D is redundant in any meet-irreducible representation of D in which it appears. Another version of Corollary 3.1.1 is the following. Theorem 3.2. Let D=gP in L~to(P). D has a reduced meet-irreducible representation, necessarily unique, if and only D satisfies condition (N): (N) For every x ~ ( P - D ) there exists x ~ ~ ( P - D ) and x~;. minimal in P - D such that x 3=x~aOx=~. Proof. By the Corollary D has reduced meet-irreducible representation, necessarily unique, if and only if D = 0 ~ Y,, where the ~ correspond to the x,r that range over the minimal elements of P - D. This is equivalent to P - D = P - 1-Is Y~= P - I1~ (P - ~t~ xS~) = P - ~,~ Z~ x~J~" This in turn is equivalent to saying that x e (P - D) if and only if x ~=x:~Ox, r for x,r minimal in P - D, which is condition (N). Corollary 3.2.t. If every maximal chain of P - D has a minimal element, then D has unique reduced meet-irreducible representation in LMo(P). Definition 3.2. A lattice L is atomic over d e L if and only if c 3 d implies there exists e such that c = e > d. Thus L is atomic if and only if L is atomic over d for every d e L.

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Theorem 3.3. LMo(P) is atomic over D if and only if D has unique reduced meet-irreducible representation. LMo(P) is atomic if and only if every element has unique reduced meet-irreducible representation. Proof. Let D have unique reduced meet-irreducible representation, and consider arbitrary C 3 D in LMo(P). For arbitrary x ~ ( C - D ) we have x____~ with x,;~ minimal in P - D by Theorem 3.2. Then C~_D + x M

_ ~ .

Since x~,-x,,C_DC_D+x~ and x,~Ox~.~(V+x~), we have

x - ~ ~ . Thus ' D < E ' D ~ x ~ = ~ C D ' + x ~ C C , so that LMo(P) is atomic over D. Now start with D failing to have unique reduced meetirreducible representation. Then there exists x ~ ( P - D ) such that for all x ~ __zx {x~} has no element that is minimal in P - D . Assume there exists E ~ LMo(P) such that D w x ~t = D + x M ~=E > D. Then E is of form D + ~ where v is minimal in P - D , { v } . D = 0 , and v E D + x ~. Thus v O v a ~ ( D + x M) with Vl E (x M - D) necessarily. But then x ~ v~ Ov with v minimal in P - D, a contradiction. Hence L~to(P) is not atomic over D. Theorem 3.4. I f P satisfies the DCC, then LMo(P) is atomic. Proof. Let C D D. Then immediately there exists a minimal element x of C - D , and C ~ D + x M = D w x M > D . The converse of Theorem 3.4 fails as shown by the following counterexample. Example 3.1. Let P = {x 1 > x2 > ---; v~ > t 1, x~ > v2 > t2, x2 > v3 ~ t 3 . . . . } where XnOV, ( n = 1, 2, ...). This example is readily verified to be atomic, but contains an infinite descending chain. We note that when LMo(P) is atomic every element of P contains a minimal element, and every minimal element constitutes a one-element 0-equivalence class. Theorem 3.5. If LMo(P) is atomic over D, then there exists no modular non-

distributive sublattice of order five with zero element D. LMo(P) is atomic implies that there exists no modular non-distributive sublattice of order five. Proof. Let LMo(P) be atomic, and assume there does exist such a sublattice E 3 A, B, C 3 D. Then there exists A o such that A =~A o with A0 = D + x M where x is minimal in A - D and {x} • D = 0. Moreover, {x} • B = 0, otherwise x ~ B = B leading to the contradiction x ~ A. B = D. Similarly, {x}" C=O. Then by Theorem 2.2 we obtain {x} ¢ (B* + C*) = E*, which is contrary to {x} ~ A* C E*. Hence no such sublattice exists. The converse of Theorem 3.5 fails. When 0 is equality, LMo(P ) = LM(P) is distributive, and all sublattices are distributive. But atomicity of LM(P) is equivalent to the D C C in P. We summarize the results of this section in the following theorem. Theorem 3.6. Atomieity of LMo(P) and existence of unique reduced meet-

irreducible representations for all elements are equivalent conditions. These are implied by the DCC in P and imply that every modular sublattice of L~o(P ) is distributive (there exists no modular non-distributive sublattice of order five). The following example shows that when 0 is other than equality, LMo(P) may possess a modular non-distributive sublattice of order five.

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Example 3.2. P = {vl > v2 > ""; wl > N2 > --'; x I ~>x2 > "''} with v~Ow~+~, w~Ox~ +1, and x~Ovi +~ for odd i. P 9 v~, w~, x~t 9 0 is a modular non-distributive sublattice of order five in LMo(P). 4. The Converse Problem

In Section 2 we proved that when 0 is an U L D E R in P, L~o(P ) is complete (Theorem 2.1), LA (Theorem 2.3), USM (Theorem 2.4), U L D (Theorem 3.5), smooth (Theorem 2.9), dually smooth (Theorem 2.10), and USP (Theorem 2.11). In this section we shall show that the conditions of completeness, local atomicity, and upper splitting in a lattice L actually imply the other four conditions and moreover make L naturally isomorphic to LMo(P) under the mapping determined by the mapping of c-join-irreducible elements x*-,x M. The condition of completeness is independent of the other six conditions as shown by the following example. Example 4.t. L is isomorphic to ~o: L = {z < al < az < ' - ' } . Completeness is the only condition not satisfied. Likewise the (USP) condition is also independent of the others as seen in the next example. Example 4.2. Let ~0' be the dual of the ordinal co. Let the direct product of the ordinals ~0 + 1 and its dual 1 + ~' share with a copy of this direct product their minimal ~ + 1 chain. To this partially ordered set adjoin a unit element. The resulting lattice fails to satisfy only the (USP) condition. Without local atomieity, the (USM), (ULD), and (USP) conditions can be satisfied vacuously as seen by the following example. Example 4.3. L is an interval on the real line under natural ordering. It is therefore reasonable to require local atomicity along with the indispensable conditions of completeness and upper splitting. Moreover, simple examples show that in a LA lattice each of the five conditions completeness, (USM), (ULD), smoothness, and dual smoothness is independent of the other four. Lemma 4.t. I f a lattice L is complete, LA, and USP, then L is smooth and dually smooth. Proof. Let c 9 d in L. Then there exist r, s ~ L such that c =9r > s =9d. There exist a minimal prime lower transpose x/a and a maximal prime upper transpose b/y of r/s. The c-join-irreducible x satisfies x c x~;s = r c=c; but x ~ d c s c y, since x ~ y. Now let c 4=z be arbitrary and let C = {x e P(L) Ix c__c} ,i: O. ~) C exists by completeness, and ~) C c c. Assuming U C C c, there exists a c-joinirreducible x such that c =9x q: U C contrary to U C __9x e C. Thus Q) C = c. By a dual proof c = ~ C ' = ~ {y~U(L)ly=gc}. Lemma 4.2. If a lattice L is LA and USP, then L is USM. Proof. Let c > d = c c ~ e . Then there exist r, s e L such that euc=gr>s=ge. Then r/s has a maximal prime upper transpose b/y. Now c ¢ y, since otherwise in combination with e c_s _c y there obtains r c=c u e ~ y contrary to r ~ y = s < r. Then by hypothesis and Lemma 2.3 there exists a c-join-irreducible x c c for which x/a is a minimal prime lower transpose of b/y. Since c3cc~yg=cc~e

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= d < c , y n e = d . Hence x c ~ d = x n y c ~ c = a n c = a . Also d c x u d C c > d requires x ~ d = c. Thus c/d ~=rx/a c=Tb/y ~=Tr/s, and b/y is the unique maximal prime upper transpose of the equivalence class of projective prime quotients to which c/d and r/s belong. Now r/s can be chosen arbitrarily in cue/e. Hence the assumption c u e d r would similarly lead to another prime quotient p/q for which c u e =~p > q = r and c/d ~=vxl/al c Tbl/yl ~=TP/q" But then b/y = bl/y 1 by uniqueness, which yields the contradiction s = r n y = r n p n y = r . Thus c • r = r. By a similar proof e ¢ s, so that e = s < r = c w e. To guarantee the (ULD) condition a lattice must be complete, as shown by the next example. Example 4.4. L is the Boolean algebra of all finite subsets of an infinite set and their complements. L is not complete and fails to be ULD, though all the other five conditions are satisfied. Lemma 4.3. If a lattice L is complete, LA, and USP, then L is ULD. Proof. Let { G ] ~ e T} be the set of all elements covering an arbitrary element d 4=u in L. Let c g = U {G [~ ~ R c=T}. We must first show that cR C Cs if and only if R C S for R, S _c_T. If ca ~ c R for 6 ¢ R, then ca/d c=rc R +o/CR with CR+a > CR by L e m m a 4.2. Assume there exists S C T and ~ s ( T - S ) for which c a c=Cs, that is, c s -- Cs+~. Consider first S as finite and without loss in generality of minimal order s, s > 2. Then by (USM) for distinct fl, 7 e S we obtain the modular non-distributive sublattice cs > Cs-,, Cs-~, Cs+~-~-.~ > Cs-,-~,. But (USP) requires that all six projective prime quotients have a c o m m o n maximal prime upper transpose, an impossibility in a lattice. Thus S must be infinite, and for all finite subsets F C S we have cJd c rcr+®/cv with c~.+~> cv. These projective prime quotients must all have a common maximal prime upper transpose b/y with cF c=y :p G, CF+~. Thus y _DU {cv] F c s, F finite} = c s, and y 3~ U {Cv+~IF C S, F finite} -- Cs+~ D cF+~, a contradiction of our assumption. Hence ~ e ( T - S) implies c s < Cs+,. Indeed it now readily follows that R C S in the Boolean algebra of all subsets of T if and only if CR C Cs in L, and that one inclusion is a covering if and only the other is a covering. It remains to show that the complete Boolean algebra sublattice L d generated over d by the c, is actually the quotient sublattice cT/d of L. Consider arbitrary s such that dCsCCT, and let S = {a~ T[c~C=s}. Then csC__s. For arbitrary f i e ( T - S ) we have c ¢ > d = s n c t 3 , and by (USM) S
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while (LA) fails. Hence at least one of the conditions of smoothness or dual smoothness must be present to guarantee the existence of prime quotients before local atomicity can be expected. One combination of conditions implying local atomicity is given below, Theorem 4.2. I f L is USP, dually smooth, and USM, then L is LA. Proof. F o r c 3 d in L there exists a c-meet-irreducible y such that y ~ d and y~:c, where y < b only. I f y c c , then c~=b>yZ=d. I f y ¢ c, then y $ c and c~=x for some c-join-irreducible x > a only with x/a c=rb/y. Then xn(yc~c) = ac~c = a < x . By (USM) c3__xu(yc~c)> ync~=d. Related closely to the latter result is the following. Theorem 4.3. I f L is complete, USP, and dually smooth, then L is smooth. Proof. We refer to the proof of Theorem 4.2. In the first case y C c we have b/y ~=rx/a. Here x is c-join-irreducib!e and x c b _cc, whereas x ~ d since x ~ y. In the second case y ~ c again x = c and x ~ d. By completeness U E exists for the non-empty set E = {x s P(L)Ix ~ e}, and U E c e is contradictory. It is not our purpose here to pursue further questions of dependency a m o n g the seven conditions. We complete this section with the full answer to the converse problem. Theorem 4.4. Let P be the partially ordered set of c-join-irreducibles of a

complete, LA, USP, lattice L (satisfying aft seven conditions). Then L is isomorphic to LMo(P) under the natural mapping d ~ D = { x ~ P t x C = d } , where xa Oxe in P if and only if x 1 and x2 correspond to the same c-meet-irreducible y in the sense tha~ b/y is the unique common maximal upper transpose of the prime quotients xl/a I and x2/a 2. Moreover 0 is an ULDER. Proof. The sets D are clearly M-closed in P. Also 0 is trivially an equivalence relation in P. Hence the complete lattice LMo(P} is determined. We shall show for E = E M that E = / 7 if and only if E = {x ~ P lx ~ e} for some e e L. Assume there exists d e L such that d ~ D C D. By smoothness d = U D c x w d UD~L for x ~ ( D - - O ) and x > a only. Suppose x ~ a u d . Then x > a = x ~ ( a u d ) , and by (USM) a w d < x u ( a u d ) = x u d . Thus x/aC__TXUd/aud c=rb/y for maximal prime upper transpose b/y. But xOx o ~ D, so that x 0 = d a u d c=y contrary to xo/a o c=rb/y. Hence x c=a w d and a u d = x u d D d. By hypothesis there exist r , s ~ L such that U D~_audZ=r>sZ=d, and there exists a maximal prime upper transpose bl/y 1 ~=rr/s. Now a ~ Yl, otherwise r _ ~ a u d ~ a u s C = a ~ y l =Yl, contrary to r u y l =b~. Hence by (USP) we have a = xl with xl > al only and xl/a ~ c=rbl/Yi 3_rr/S. Since xl ~ Yt __Dd, x~ ~ D. Neither can we have x~ Oxz ~ D, since xl/a ~ ~=Tba/y~ 3-_rx2/a~ conflicts with xz c_d c_s c_Yv Therefore x > a ~ x~ ~/5 = D~ so that x q~D, contrary to our initial selection of x. Hence our initial assumption is false. Hence for each d ~ L we have d ~ D =/5. Next consider E ~ = E = E in P, and let U E = g --, G = G. Then G ~ E. Assume G D E. There exists v ~ (G - E) such that {v} . E = O ; otherwise G - E ~ _ E = E contrary to our assumption. Hence for each wa~E w~: v, for each v, ~ {v}, andby(USP)w¢ _ y~ where y~ is the c-meetirreducible associated with {v}. We obtain v = g = U E = U_~ wa ~ yv, contrary to vC_y~,. Hence U E = g ~ G = E . Ifd~D=D and e ~ E = E with D = E , then

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d= ~ D= ~ E=e. The correspondence d ~ D = D is therefore 1 - 1 from L onto LMo(P), It obviously preserves inclusion, so that L is isomorphic to LMo(P). For x e P x - ~ X , which is obviously x u. Recalling that x > a = U { c e L l c C x } for x e P , a ~ A = x U - x . Hence x M = x- ~ and x M - x = ~ - x. Therefore 0 is an ULDER, and the proof is complete.

5. Compact and Dual Compact Generation Throughout this section we shall assume that 0 is an U L D E R on a partially ordered set P. Definition 5.1. In a lattice L an element c is compact if and only ifc _c U S for S = L implies that c ~ U So for a finite subset S O=cS. L is compactly generated if and only if L is complete and every element of L is the join of compact elements. Dually compact elements and dual compact generation of L are defined dually. A lattice is distributive if and only if it is isomorphic to a ring of sets. More particularly, a distributive lattice is both compactly generated and dually compactly generated if and only if L is isomorphic to a complete ring of sets, namely the lattice LM(P) of M-closed subsets of the partially ordered set P of c-join-irreducible elements of L. When 0 is an arbitrary ULDER, LMo(P) is in general neither compactly generated nor dually compactly generated. The following examples illustrate these facts. Example 5.1. Let P = {t 1, t 2. . . . ; xl, x2 .... } with t, < x, (n = 1, 2.... ) and x i Oxj for all i,j. LMo(P) is compactly generated. LMo(P) is not dually compactly generated, since Y = {tl, t2 .... } 3 ~i~:1 ( P - t [ ) = 0 but Y contains no finite (or any proper) submeet. Note even that both chain conditions hold in P. Example 5.2. Let P = {vl < v2 < --" C x; t < xo} with x OXo. The c-join~) r) / but is not contained in any irreducible x M is contained in x~w Ui-~ finite submeet. Thus LMo(P) is not compactly generated, though it is dually compactly generated. In view of the fact that the complete lattice LMo(P) is both smooth and dually smooth by Theorems 2.9 and 2.10, the determination of whether or not LMo(P) is compactly generated rests upon whether or not every c-join-irreducible x M is compact; and dually. In LM(P), where 0 is equality, E is compact if and only i r e is the M-closure of a finite crown: E = {xi,x2 ..... x,} M. The characterization of compact elements in LMo(P) is somewhat more complicated. Theorem 5.t. In LMo(P) C is compact if and only if C satisfies (A): (A) For each [~-selection of exactly one x, aOx, for each x, e C there exists

a finite subset {x~a, ..., x,,a} such that C~ ~)~=1 x~p. We may interpret (A) as follows: for each selection of one representative of each 0-equivalence class represented in C there exists a finite subset of elements that collectively bound from above representatives from every 0-equivalence class represented in C.

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Proof of Theorem 5.1. Let C be compact and consider an arbitrary selection of one x,aOx ~ for each x ~ e C . Since C * = { { G ~ } = { x ~ } I G e C } , C = U~ x ~ . By compactness C = U~'=l xM ,~, a finite subjoin, Conversely let (A) be satisfied by C e LMo(P) and consider C__CI,_)~,r Eo= ~ r E~. For each x~e C we have x~Ox~ e ~ r E~, so that x~a e E ~ ) for some 6(~)e T and x ~ -_ E aM , ~= E a , P By hypothesis (A) there exists a finite subset {x,~li = 1..... n} of the x~t~, s so selected tor which C __cUT= 1 ~,~ c= UT: E~,) with 6(cq) e T (i = 1, ..., n). Hence C is compact. Corollary 5.1.t. LMo(P) is compactly generated if and only if x M satisfies condition (A) .for every x e P. Corollary 5.t.2. I f x M is finite for all x e P, then LMo(P ) is compactly generated and atomic. Moreover C E LMo(P) is compact if and only if C is finite. Proof. Trivially the DCC holds in P, so that LMo(P ) is atomic by Theorem 3.4. Also x M satisfies condition (A) trivially, so t h a t LMo(P) is compactly generated. If C is finite, C = {x~ ..... x,} __c~ r E~ = ~ r Ea implies x~Ox~p *__ n e E~., 6~ e T, (i = 1. ., . . n). Whence C* -_- ~ = I E ~ - (~)i= 1 Ea)* and by Theorem 2.2 C__
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esis (B) there exists a finite subset {v, ..... v,} of the v,p's such that Y 3 (-]7=, Yt,~, where vi C x~. ¢ E~ N (i = 1..... n). (The x~, need not be distinct, and likewise for the E~,N). Now {Vi}'E,,N=O if and only if M = cp _ Ivan0 v,} = Y~. Thus Y3 ("]~'=, Y~, 3= (~}~'=~ E,,N~ O,~s {E,t,.lx~ E~N} 3=(~bETEo. F r o m all three cases we conclude Y is dually compact. Condition (B) can be interpreted as follows. For each x,, a ~ S, associated with the c-meet-irreducible Y and for each fl-selection of exactly one v,a C x, there exists a finite subset {v, ..... Vn} of the v~'s such that each x~ is bounded properly below either by one of the v~'s or by an element v~ 0 v~(1 < i < n). This interpretation follows from Y= P - Z {x~[c~e S} 3 (~7=~ Y,~,= 1-[7=~(P - Z, {v{,]v,~Ov,} being equivalent to Z{x~l~es}cZ~=,Z~{vJ~lvi~Ov~}, which in turn is equivalent to saying for each ~ e S that x, 3v~. for some v~rOv~ for some v~ (1 < i < n ) . Corollary 5.2.t. LMo(P) is dually compactly generated if and only if every c-meet-irreducible Y satisfies (B). Corollary 5.2.2. / f {x~} is a finite O-equivalence class then the associated

c-meet-irreducible Y is dually compact. I f all O-equivalence classes are finite, then LMo(P ) is dually compactly 9enerated. The converse of Corollary 5.2.2 is decidedly false, as shown by Example 5.2 presented earlier. Theorem 5.3. Let { Y~ l fl ~ R} = { Y~ ~ P' (LMo(P)) ] YIJ3=D} for an element D E LMo(P) that is not c-meet-irreducible. Let {x¢~]~ E S/~} be the O-equivalence

class associated with Ya. Then D = ~o~R Y~ is dually compact if and only if D satisfies condition (C): (C) For each fl ~ R and for each ~ ~ Sp make an arbitrary y-selection of exactly one vfl~r C=xt~~. Then there exists a finite subset {V1. . . . . Un} of this 7-selection such that D 3=0~= 1 Yo~ where Yu. is the c-meet-irreducible associated with the O-equivalence class {vi} (i= 1..... n). The proof follows the form of that for Theorem 5.2, but it is actually simplified by the fact that the inclusions va, 7 _cxt~"and D 3= (~= a Y~, are not strictly proper inclusions as are their counterparts in Theorem 5.2. We omit the details. (C) can be interpreted as fotlows. For each xa, (fl ~ R and ct 6 St0 and for each 7-selection of exactly one va,~ ~ xa, there exists a finite subset {v, ..... v,} of this selection such that each xa, is bounded below by (at least) one of the v~ or by vi~Ov i ( l < i < n ) . Detinition 5.2. An element p of a lattice L is perfect if and only if p ___[_) S for S ~ L implies p c_s 6 S. A dually perfect element is defined dually. A perfect element is necessarily both compact and c-join-irreducible, and dually. In a complete lattice L perfect elements and dually perfect elements occur in pairs (x,y) for which L splits simply: L = xS+ ym and x ~.yM= O. Conversely, any such splitting of an arbitrary lattice L implies x and y are respectively perfect and dually perfect.

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Theorem 5.4. C ~ LMo(P ) is perfect if and only' if C = x M for a one element O-equivalence class {x}. D ~ LMo(P ) is dually perfect if and only if D = Y - P - x J for a one element equivalence class. Proof. The upper splitting of LMo(P), determined by the c-meet-irreducible Y= P - ~ , ~ r x~ and the corresponding c-join-irreducibles X~ = x~, becomes a simple splitting if and only if the associated 0-equivalence class {x~ t c~~ T} is of order one. Corollary 5.4.t. All c-join-irreducible elements are peJfect if and only if all c-meet-irreducible elements are perfect if and only if 0 is equality.

6. TheFinite Case Four of the seven conditions discussed in Section 4 are trivially true in a finite lattice L, namely completeness, local atomicity, smoothness, and dual smoothness. The principal theorems of Section 2 and the solution to the converse problem, Theorem 4.4, are then encompassed in the following theorem for the finite case. Theorem 6.t. In a finite lattice L the conditions (USP) and (ULD) are equivalent and imply (USM). They also imply that L is isomorphic to LMo(P(L)) under the natural mapping d--*D = {x ~ P(L)]x c=d} where xl Ox2 in P(L) if and only if they correspond to the same c-meet-irreducible y: b/y is the common maximal prime upper transpose of the prime quotients xl/a 1 and x2/a2. Proof. (USP) implies (ULD) and (USM) by Lemmas 4.2 and 4.3 respectively. Now let L satisfy (ULD). For c ~ L let C' = {y e P'(L) Iy __3c}. In 1-7] DILWORTH proved that c > d implies D ' - C ' consists of just one c-meet-irreducible y. Since c > c n y = d, then by (USM) y < c w y = b, the unique element covering y. Hence c/dC__rb/y, and b/y is a maximal prime upper transpose by meet-irreducibility of y. If r/s c=rc/d, then r/s c=rb/y by transitivity and S ' - R ' = y. By induction c/d is projective with the prime quotient e/f implies F' - E' = y and e/f C=rb/y. For a minimal lower prime transpose x,/a, of b/y x~ is c-joinirreducible by Lemma 2.3. It remains to show that the partially ordered set Q of prime quotients projective with c/d, ordered by transposition and having greatest element b/y and minimal elements xJa,, determines an upper splitting of L. Now t _cy implies t ;~ x~ for xJa~ c_c_rb/y, since x~ ~ y. If t q: y, a minimal element x satisfying t = x q: y must cover just one element a, with a _--_y; for if x > a , ate=y, then x = a u a l C = y , a contradiction. Therefore tZ=x, a c-joinirreducible, with x/a c=rb/y. Hence L is upper splitting. 7. Dependence of Conditions on 0

In the solution of the converse problem in Section 4, to obtain the isomorphism of a complete, locally atomic, upper splitting lattice L with LMo(P) under the natural mapping d ~ D = {x eP(L) lx C=d} it was reasonable to require the conditions (F) and (G) on 0: (F) x ~ = x - ~ for all x ~ P . (G) x M - x = x - - g - x for all x ~ P .

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We then obtained the correspondence of c-join-irreducibles x~--~xu ~ LMo(P) and the correspondence a~-~(xM - x)~ LMo(P ) of the unique elements a < x and x u - x < x M. Smoothness of L: d = U D for all d ~ L then guaranteed the isomorphism. We summarize here the dependence of (F) and (G) with the following closely related conditions on the equivalence relation 0 defined over the partially ordered set P. (H) Xl OX2 :~ x I implies x2 ~ x 1 (and xl C x2). (D) XlOX2 4--xl implies (x~)* ¢ (x~)* (and (xM)* ~ (xM)*). (E) x 2 + x l and x M - - x 2 C x M - - x l imply x2Oxl. (K) x 2 ~ x 1 and xM--x2C=xM--xl imply x2Oxl. (J) xlOx 2 and XM--xlCxM--X~XM2--X2 imply xl,xzZ=X or XI~X2 ~'X.

(H0 x~ ~ ~--~-~-x. We shall omit all proofs, which merely involve application of Definition 2.1 of bar closure. Theorem 7.i. The following diagram gives the only simple implications among the conditions ~(H)~(H (O) ~ ( E ) ~

1)

~(K)

(F) ::~ ( J ) ~ Theorem 7.2, (H) and (K) imply (E). Theorem 7.3. (G) and (K) imply all the conditions. Theorem 7.4. (H) and (F) imply all the conditons Theorem %5. (H) and (J) do not imply (G), (F), (D). If P satisfies the DCC, then (H) and (J) imply all the conditions. The following four simple examples show that all essentially independent implications have been included in the five theorems. Example 7.1. P = {v2 Or1 > r < x > V2 ~ S}. LMo(P) satisfies (D), (E), (H), (K) only (and (H0). Example 7.2. P = {xl > r < XEOXt}. LMo(P) satisfies (G) and (H) only (and (Hi)). Example 7.3. P = {x0 < tl < t2 < "" cxlOxo} • LMo(P) satisfies (F), (J), (K) only. Example 7,4. P = { x l > x 2 > ' " ; w > v , > t , ( n = 1,2 .... );x, Ov,(n= 1,2 .... )}. LMo(P) satisfies (D), (E), (H), (K) only (and (H0). The results presented here suggest a study of the structure of LMo(P ) for more general equivalence relations on P than an ULDER.

Bibliography Dependence of finiteness conditions in distributive lattices. Math. Z. 85, 245--256 ( !964). 2. BALANCHANDRAN,V. K. : A characterizatic~nof Z--A ringsof subsets. Fund. Math. 4t, 38--41 (t954). 1. AVANN,S. P.:

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3. B1RKHOFF,G. : Lattice theory, rev. ed. Vol. 25. New York: Am. Math. Soc. Coll. Publ. 1948. 4. BRUNS,G.: Verbandstheorie Kennzeichnung vollstfindiger Mengenringe. Arch. Math. t0, 109--112 (1959). 5. BOCHe,J. R.: Representations of complete lattices by sets. Port. Math. t t , 151 167 (1952). 6. DILWORTn, R. P., and P. CRAWLEY: Decomposition theory for lattices without chain conditions. Trans. Am. Math. Soc. 96, 1--22 (1960). 7. - - Lattices with unique irreducible decompositions. Ann, Math. 41-2, 771--777 (1940). 8. MENZEL,W.: ~ber den Untergruppenverband einer abelschen Operatorgruppe. I. m-Verb~nde. Math. Z. 74, 3 9 - 5 1 (1960). 9. RANEY,G. N.: Completely distributive complete lattices. Proc. Am. Math. Soc. 3, 677--680 (1952)~ Prof. S. P. AVANN Department of Mathematics University of Washington Seattle, Washington 98105 USA

(Received August 3, 1966)