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Lattice Uniformities Generated by Filters
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
209, 507]528 Ž1997.
AY965291
Lattice Uniformities Generated by Filters Anna Avallone* Dipartimento di Matematica, Uni¨ ersita ` della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy
and Hans Weber † Dipartimento di Matematica, Uni¨ ersita ` di Udine, Via delle Scienze, 206, 33100 Udine, Italy Received May 30, 1995
INTRODUCTION In wW2 x, uniform lattices are studied as a common generalization of topological Boolean rings and topological Riesz spaces. Lattice uniformities play a similar role in the study of modular functions as FN-topologies; see e.g., wD; W1 x for the use of FN-topologies in measure theory and wW4 x for the use of lattice uniformities. The main aim of this paper is to give a contribution to the examination of the lattice structure of the space L Ue Ž L. of all exhaustive lattice uniformities on a lattice L Žsee Section 1 for definitions.. In particular, in the case where L is a modular sectionally complemented lattice, we get a satisfactory result; in this case L Ue Ž L. is a complete Boolean algebra Žsee Corollary 5.11.. Such a result can be used as the main tool in obtaining decomposition theorems for modular functions as well as information about the structure of uniform completions of L Žcf. wW5 x.. In the special case that L is a topological Boolean ring, the result 5.11 mentioned above as well as the connection with decomposition theorems in measure theory is given in wW1 x; for related results about locally solid l-groups, see wB-Tx. *E-mail: [email protected]. † E-mail: [email protected]. 507 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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AVALLONE AND WEBER
Important for our investigations is the fact that a lattice uniformity on a sectionally complemented lattice L is uniquely determined by its 0neighbourhood system; moreover the filters which are 0-neighbourhood filters for lattice uniformities are precisely the ‘‘distributive’’ filters, the space of which we indicate by FN D Ž L. Žsee Section 2.. In Section 4 we study this space FN D Ž L. also for an arbitrary lattice L. Every filter of FN D Ž L. induced on L a lattice uniformity in a similar way that a distributive ideal in L induces a congruence relation. In fact, the concept of ‘‘distributive’’ filters introduced here and some statements about them were suggested by the concept and some properties of distributive ideals Žsee wG, Chap. III.3x.. The main result of Section 5 essentially says that, for a Hausdorff exhaustive lattice uniformity w on L generated by some F0 g FN D Ž L., the space of all lattice uniformities generated by filters of FN D Ž L. coarser than F0 is isomorphic to the space of all distributive elements of the completion of Ž L, w .. As a consequence, we get the result 5.11 mentioned above and the fact proved in wW3 x that the space of all exhaustive lattice uniformities on an orthomodular lattice is a complete Boolean algebra.
1. PRELIMINARIES Throughout the paper, let L be a lattice and D [ Ž x, x . : x g L4 the diagonal in L = L. We denote by N the set of natural numbers. A lattice uniformity on L is a uniformity on L which makes the lattice operations k and n uniformly continuous. For any lattice uniformity u on L, we denote by N Ž u. the intersection of all members of u, i.e., the closure of D in Ž L, u. 2 ; N Ž u. is a congruence relation on L. If N is a congruence ˆ uˆ. [ Ž L, u.rN is a relation contained in N Ž u., then the quotient Ž L, lattice endowed with a lattice uniformity. If u is Hausdorff, then the ˜ u˜. of Ž L, u. is a lattice endowed with a lattice uniform completion Ž L, uniformity. u is called exhausti¨ e if every monotone sequence is Cauchy in Ž L, u. Žsee wW2 x, Sect. 1, 6x.. We denote, respectively, by L U Ž L. and L Ue Ž L. the sets of all lattice uniformities and of all exhaustive lattice uniformities; these are complete lattices, where the partial ordering is inclusion. If a lattice has a smallest or a greatest element, we denote these elements with 0 or 1, respectively. L is called sectionally complemented if L has a 0, and, for every a g L, the interval w0, ax is complemented. An element a of L is called distributi¨ e Žin L. if a k Ž x n y. s Ž a k x. n Ž a k y.
LATTICE UNIFORMITIES GENERATED BY FILTERS
509
for all x, y g L, i.e., if x ª a k x defines a lattice homomorphism. The set DŽ L. of all distributive elements of L is closed with respect to k ŽwG, Theorem III, 2.9 Ža.x.. It follows, since a lattice satisfying one the distributive laws satisfies both ŽwG, Lemma I.4.10x.: 1.1. If a n b g DŽ L. for any a, b g DŽ L., then DŽ L. is a distributi¨ e sublattice of L. Dualizing the definition of distributive elements, one gets the definition of dually distributive elements. 1.2 wG, Theorem III.2.4x. equi¨ alent:
If a g L, the following three conditions are
Ž1. a is distributi¨ e, a is dually distributi¨ e, and a k x s a k y, a n x s a n y imply x s y for any x, y g L. Ž2. There is an embedding f of L into a direct product A = B of lattices, where A has a 1, B has a 0, and f Ž a. s Ž1, 0.. Ž3. For any x, y g L, the sublattice generated by a, x, and y is distributi¨ e. An element satisfying the equivalent conditions of 1.2 is called neutral. The set N Ž L. of all neutral elements of L is a distributive sublattice of L. 1.3 wG, Theorem III.2.6x. then DŽ L. s N Ž L..
If L is modular or relati¨ ely complemented,
1.4. There are finite sectionally complemented lattices with DŽ L. / N Ž L., see Fig. 1, where a g DŽ L. _ N Ž L..
FIGURE 1
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AVALLONE AND WEBER
If L has 0 and 1, then the centre C Ž L. of L is the set of all complemented neutral elements of L; any element of C Ž L. has a unique complement in L. 1.5 wM-M, Theorem 4.15x. subalgebra of L.
If L has 0 and 1, then C Ž L. is a Boolean
2. LATTICE UNIFORMITIES ON SECTIONALLY COMPLEMENTED LATTICES The study of the structure of L U Ž L. and L Ue Ž L. for a sectionally complemented lattice L is based on the fact that its lattice uniformities are generated by certain filters on L. 2.1. PROPOSITION. Let u be a lattice uniformity on a sectionally complemented lattice L and F the 0-neighbourhood system in Ž L, u.. Then the sets F* [ Ž x, y . g L2 : ' a g F : Ž x n y . k a s x k y 4 ,
F g F,
form a base of u, as do the sets F=[ Ž x, y . g L2 : ' a g F : x k a s y k a4 ,
F g F.
Proof. The first assertion is proved in wW2 , 6.10x. It follows that F=g u for F g F , since F* : F=. We now show that any U g u contains a set F= for some F g F. Choose a symmetric vicinity V g u with Ž D k V .( Ž D k V . : U and set F [ V Ž0.. Then F=: U: In fact, if Ž x, y . g F= and a g F with x k a s y k a, then
Ž x, y k a. s Ž x, x . k Ž 0, a. g D k V and
Ž y k a, y . s Ž y, y . k Ž a, 0 . g D k V ; hence Ž x, y . g Ž D k V .(Ž D k V . : U. Proposition 2.1 generalizes a known fact about congruence relations. If N is a congruence relation on L, then the principal filter u N [ U : N : U : L = L4 generated by N is a lattice uniformity. For principal filters of L U Ž L., the first and second assertion of Proposition 2.1 exactly mean that each congruence relation on a sectionally complemented lattice is standard and distributive, respectively Žsee wG, Theorem III.3.10x..
LATTICE UNIFORMITIES GENERATED BY FILTERS
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The pentagon has congruence relations which are not distributive, namely 04 , c4 , 14 , a, b44 by the notation of Fig. 4; therefore the conclusion of Proposition 2.1 does not hold in general for complemented lattices. The correspondence F =ª F of Proposition 2.1 yields, for sectionally complemented lattices, an order isomorphism between L U Ž L. and FN D Ž L.; see Corollary 4.8 and Definition 4.3. This motivates the examination}in the next sections}of the spaces of filters FN D Ž L. and FN Ž L., which contains FN D Ž L.; see Definition 3.1. The principal filters of L U Ž L., FN Ž L., and FN D Ž L. correspond, respectively, to congruence relations, ideals, and distributive ideals; cf. Proposition 4.9.
3. THE LATTICE OF FILTERS FN Ž L. We call a subset S of L solid if x g S whenever y G x g L for some y g S. For A : L, the set s Ž A . [ x g L : there exists y g A with x F y 4 is the smallest solid subset of L containing A, i.e., the solid hull of A. 3.1 DEFINITION. We denote by FN Ž L. the set of all filters F on L satisfying one of the following two equivalent conditions: Ž1. ;F g F, ' G g F : ; x, y g G, ; z g L, z F x k y « z g F. Ž2. Ži. F has a base of solid sets and Žii. ;F g F , ' G g F : G k G [ x k y : x, y g G4 : F. To prove Ž1. « Ž2.Ži., one verifies that, under assumption Ž1., sŽ F . : F g F 4 is a filter base of F. The notation FN Ž L. is motivated by the fact that, for a Boolean ring L, F g FN Ž L. iff F is the 0-neighbourhood system of an FN-topology on L Žfor FN-topologies, cf. wDx or wW1 x.. 3.2. PROPOSITION. If u is a lattice uniformity on a lattice L with 0 and F is the 0-neighbourhood system in Ž L, u., then F g FN Ž L.. The proof follows from the fact that, by wW2 , 1.1.6x, every point x g L has a neighbourhood base of convex sets and every convex subset of L containing 0 is solid. On the other hand, there are filters belonging to FN Ž L. on lattices with 0, which are not 0-neighbourhood systems with respect to lattice uniformities: The principal filter on the pentagon generated by 0, c4 Žsee Fig. 4. is an example of this.
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AVALLONE AND WEBER
FN Ž L. is, with respect to the inclusion, a partially ordered set. It is easy to see and follows from Theorem 3.4 that Ž FN Ž L., :. is a lattice and that, for F1 , F2 g FN Ž L., the sets F1 l F2 and sŽ F1 k F2 ., where F1 g F1 and F2 g F2 , form a base of F1 k F2 and F1 n F2 , respectively. The fact that herein F1 l F2 / B follows from the next proposition, the proof of which is obvious. If F1 , F2 are solid subsets of L, then
3.3. PROPOSITION.
F1 l F2 s F1 n F2 [ x 1 n x 2 : x 1 g F1 , x 2 g F2 4 . 3.4. THEOREM. Ž FN Ž L., :. is a complete lattice. Moreo¨ er, if Ž Fa .a g A is a family in FN Ž L., then the sets F a g I Fa , where I is a finite subset of A and Fa g Fa , form a base of supa g A Fa ; and the sets s
ž
`
n
D E D
ns1 ks1 a gA
Fa , k
/
s x g L : ' n g N , a k g A, x k g Fa k , k : x F sup x k ,
½
1FkFn
5
where Fa , k g Fa for a g A and k g N , form a base of inf a g A Fa . Proof. Ži. Let A / B and Fa g FN Ž L. for a g A. For any finite subset I of A and Fa g Fa Ž a g I ., the sets F a g I Fa are by Proposition 3.3 non-empty; now it is obvious that these sets form a filter base of a filter of FN Ž L. and that this filter is the supremum of Fa : a g A4 . Since FN Ž L. has a smallest element, namely L4 , it follows that Ž FN Ž L., :. is a complete lattice. Žii. We prove the description of inf Fa given in Theorem 2.3. Obviously `
n
½ žD E D / s
ns1 ks1 a gA
Fa , k : Fa , k g Fa
5
is a filter base. To prove that the filter F generated by this filter base belongs to FN Ž L., we show that, for every F g F , there is a G g F with n G k G : F. Let F g F and Fa , k g Fa with sŽj `ns1 k ks1 j a g A Fa , k . : F. Put G[s
`
ž
n
D E D
ns1 ks1 a gA
2k
/
Ga , k ,
where Ga , k s
F Fa , i . is1
LATTICE UNIFORMITIES GENERATED BY FILTERS
513
Let x, y g G. Choose a 1 , . . . , a n , b 1 , . . . , bm g A and x k g Fa k , k , y k g Fb k , k with n
x F sup x k ks1
and
m
y F sup y k ; ks1
we may assume that m F n. Put
Ž g 1 , . . . , gnqm . s Ž a 1 , b 1 , . . . , a m , bm , a mq1 , . . . , a n . and
Ž z1 , . . . , z nqm . s Ž x 1 , y 1 , . . . , x m , ym , x mq1 , . . . , x n . . nq m Then z k g Fg k , k for k s 1, . . . , n q m and x k y F sup ks1 z k ; hence x k y g F.
We prove that F s inf Fa . Obviously, F F Fa for a g A. If F 9 g F N Ž L. with F 9 F Fa for all a g A, then F 9 F F : Let F0 g F 9. We may assume that F0 is solid. Choose Fk g F 9 with Fk k Fk : Fky1 for k g N . Then Fa , k [ Fk g Fa for all a g A and k g N , and n sŽD `ns 1 E ks1 D a g A Fa , k . : F0 ; hence F0 g F. In the case of L being a Boolean ring, Theorem 3.4 describes the supremum and the infimum of a family of FN-topologies by means of the 0-neighborhood systems. The description of the infimum is suggested by Waelbroecks’ description of the infimum of linear topologies, see wWx, which also holds true for group topologies.
4. THE LATTICE OF FILTERS FN D Ž L. In this section we introduce the space FN D Ž L. of ‘‘distributive’’ filters, which generate lattice uniformities on L in a similar way as distributive ideals generate the congruence relation; cf. wG, Theorem III.3.4x. The precise relationship between the filters of FN D Ž L. and distributive ideals in L is given in Proposition 4.9. 4.1. Notation. We put F=[ Ž x, y . g L2 : ' a g F : x k a s y k a4 for F : L and F =[ G : L2 : ' F g F : F=: G 4 for a filter F on L.
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AVALLONE AND WEBER
4.2. LEMMA. Let F g FN Ž L.. Then F = is the finest uniformity, which makes k uniformly continuous and which is contained in the filter generated by F = F : F g F 4 . Moreo¨ er, F has a base of closed sets as well as a base of open sets with respect to the topology induced by F =. Proof. Obviously, F = is a uniformity, which makes k uniformly continuous, since D k F=: F=. Moreover, F = is contained in the filter F 2 generated by F = F : F g F 4 , since F = F : Ž F k F .=. Let now u be a uniformity on L, which makes k uniformly continuous, and u F F 2 . We show that u F F =. Let U g u. Choose V, W g u such that V is symmetric, V (V : U, D k W : V and a solid set F g F with F = F : W. Then F=: U: Let Ž x, y . g F= and a g F with x k a s y k a. Then
Ž x, y k a. s Ž x, x k a. s Ž x, x . k Ž x n a, a. g D k Ž F = F . :DkW:V and
Ž y k a, y . s Ž y, y . k Ž a, y n a. g D k Ž F = F . : V ; hence Ž x, y . g V (V : U. To prove the last assertion, we show that, for every F g F , there is a ˚ Let F, G g F with sŽ G k G. : F and G g F with G : F and G : F. = ˚ since U [ G . Then G : UŽ G . : sŽ G k G . : F and therefore G : F, UŽ G . : F. 4.3. DEFINITION. We denote by FN D Ž L. the set of all filters F g FN Ž L. for which F = is a lattice uniformity. 4.4. PROPOSITION. ¨ alent:
For F g FN Ž L., the following conditions are equi-
Ž1. F g FN D Ž L.. Ž2. ;F g F , ' G g F : D n G=: F=. Ž3. ;F g F , ' G g F : ; x, y g L and ;a g G, ' b g F : Ž x k a. n y F Ž x n y . k b. Ž4. ;F g F, 'G g F : ;F1 , F2 : L, sŽ F1 k G . l sŽ F2 k G . : sŽŽ F1 n F2 . k F .. Proof. Ž1. m Ž2. follows from Lemma 4.2, since condition Ž2. is equivalent to the uniform continuity of n. Ž2. « Ž3. Let F, G be given according to Ž2., a g G and x, y g L. Since Ž x, x k a. g G=, we have
Ž x n y, Ž x k a. n y . g G=n D : F= ;
LATTICE UNIFORMITIES GENERATED BY FILTERS
515
hence
Ž x n y . k b s Ž Ž x k a. n y . k b G Ž x k a. n y for some b g F. Ž3. « Ž4. Let F0 g F. Choose F g F with F k F : F0 and G for F according to Ž3.. We show that s Ž F1 k G . l s Ž F2 k G . : s Ž Ž F1 n F2 . k F0 .
for any F1 , F2 : L:
Let z g sŽ F1 k G . l sŽ F2 k G .; moreover x i g Fi and a i g G with z F Ž x 1 k a1 . n Ž x 2 k a2 .. Applying Ž3. twice, we get b1 , b 2 g F with
Ž x 1 k a1 . n Ž x 2 k a2 . F Ž x 1 n Ž x 2 k a2 . . k b1 F Ž Ž x 1 n x 2 . k b 2 . k b1 ; hence z F Ž x 1 n x 2 . k b, where b [ b1 k b 2 g F k F : F0 . It follows that z g sŽŽ F1 n F2 . k F0 .. Ž4. m Ž2. Let F, G be given according to Ž4.. We show that D n G=: Ž F k F .=. Let x g L and Ž y 1 , y 2 . g G=; moreover a g G with y 1 k a s y 2 k a. Applying Ž4. for F1 s x 4 and F2 s yi 4 , where i g 1, 24 , we get x n Ž yi n a . g s Ž F1 n G . l s Ž F2 k G . : s Ž Ž F1 n F2 . k F . . Therefore there exists bi g F with x n Ž yi k a. F Ž x n yi . k bi . Then, with b [ b1 k b 2 , we have
Ž x n y1 . k b s Ž x n Ž y1 k a. . k b s Ž x n Ž y 2 k a. . k b s Ž x n y 2 . k b; hence Ž x n y 1 , x n y 2 . g Ž F k F .=. 4.5. THEOREM. Ža. Ž FN D Ž L., :. is a Dedikind complete lattice. Žb. If Fa g FN D Ž L. for a g A, then F [ inf a g A Fa g FN D Ž L. and F =s inf a g A Fa=. Ž Here, inf Fa and inf Fa= are built in FN Ž L. and L U Ž L., respecti¨ ely.. Žc. If F g FN D Ž L. and Fa g FN Ž L. for a g A, then supa g A Ž F n Fa . s F n sup a g A Fa . Proof. Ža. follows from Theorem 3.4 and Žb.. Žb. For the proof of F g FN D Ž L., we verify condition Ž3. of Proposition 4.4. Let F g F. By Theorem 3.4, there are sets Fa , k g Fa Ž a g A, k g N . with s
ž
`
n
D E D
ns1 ks1 a gA
Fa , k : F.
/
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AVALLONE AND WEBER
Choose Ga , k g Fa for Fa , k according to Proposition 4.4 Ž3. and G [ n sŽj `ns 1 k ks1 j a g A Ga , k .. Let x, y g L and g g G. We have to show that Ž x k g . n y F Ž x n y . k f for some f g F. Let n g N and a k g A, g k g Ga k , k for k s 1, . . . , n with g F E nks1 g k . Using Proposition 4.4 Ž3. for Ga , k and Fa , k , we inductively get f k g Fa k , k with n
ž
xk
E gi isk
n
/
nyF
žž
xk
E
/ /
gi n y k fk
iskq1
for k s 1, . . . , n.
It follows that Ž x k g . n y F Ž x n y . k f, where f [ E nks 1 f k g F. Žb. Obviously F =F Fa= for every a g A and therefore F =F u [ inf Fa= . We now prove that u F F =. Let U0 g u and Uk g u with Uk (Uk : Uky 1 for k g N . Choose Fa , k g Fa with Fa=, k : Uk for k g N and a g A. Put F[s
ž
`
n
D E D
ns1 ks1 a gA
/
Fa , k .
We show that F=: U0 (U0 . Let Ž x, y . g F=. Then there are a 1 , . . . , a n g A n and z k g Fa k , k for k s 1, . . . , n such that x k E ks1 z k s y k E nks1 z k . ky 1 k = Since Ž x k E is1 z i , x k E is1 z i . g Fa k , k : Uk for k s 1, . . . , n, we get n
ž
x, x k
E zi is1
/
g U1 ( ??? (Un : U0 ;
n analogously Ž y k E is1 z i , y . g U0 . Hence Ž x, y . g U0 (U0 . Žc. supa Ž F n Fa . F F n supa Fa obviously holds. Since every member of supa g A Fa belongs to supa g B Fa for some finite B : A, it is enough to prove the other inequality G for a finite index set A. Inductively, one can reduce the proof to the case of A containing two elements; i.e., we have to prove that
Ž F n F1 . k Ž F n F2 . G F n Ž F1 k F2 . for F g FN D Ž L. and F1 , F2 g FN Ž L.. But that immediately follows from Proposition 4.4 Ž1. « Ž4. and the characterization of the infimum and supremum of two filters of FN Ž L.. We now clarify the relationship between the filters of FN D Ž L. and the induced lattice uniformities. 4.6. Notation. 4.7. LEMMA.
L U D Ž L. [ F = : F g FN D Ž L.4 . F : sŽ F=Ž a.. and F=Ž a. : sŽ a k F . for a g L and F : L.
LATTICE UNIFORMITIES GENERATED BY FILTERS
517
Proof. If x g F, then
Ž x k a. k x s a k x and therefore Ž x k a, a. g F=. Hence x k a g F=Ž a. and x g sŽ F=Ž a... If x g F=Ž a., then x k z s a k z for some z g F; hence x F a k z g a k F and x g sŽ a k F .. 4.8. COROLLARY. Ža. F ª F = defines a lattice isomorphism from FN D Ž L. onto L U D Ž L.. Žb. If L is a lattice with 0 and F g FN D Ž L., then F is the 0neighbourhood system in Ž L, F =. . Žc. If ua g L U D Ž L. for a g A, then inf a g A ua g L U D Ž L.. Proof. Ža. We have only to show that F1 , F2 g FN D Ž L. and F1= F F2= imply F1 F F2 . Let G g F1 and F1 g F1 with sŽ F1 k F1 . : G. Choose F2 g F2 with F2=: F1= and a g F1. Then we have by Lemma 4.7. F2 : s Ž F2= Ž a . . : s Ž F1= Ž a . . : s Ž a k F1 . : G; hence G g F2 . Žb. follows from the fact that, if F g F is solid, then F=Ž 0. s = sŽ F Ž 0.. s F. Žc. follows from Theorem 4.5 Žb.. A bijection f : L1 ª L2 from a lattice L1 onto a lattice L2 such that x F y iff f Ž x . G f Ž y . is called a dual isomorphism. 4.9. PROPOSITION. Let I Ž L., I D Ž L., C Ž L., and C D Ž L. be the spaces of all ideals, distributi¨ e ideals wG, Def. III.3.1x, congruence relations, and distributi¨ e congruence relations in L wG, p. 149x, respecti¨ ely. Denote by FI Ž or u N . the principal filter on L Ž or on L = L. generated by a non-empty set I : L Ž or N : L = L, respecti¨ ely .. Let Fa s Fxy`, ax and ua s wy`, ax=s Ž x, y . g L2 : x k a s y k a4 . Ža.
Then for N : L = L, I : L, a g L, we ha¨ e
Ži. u N g L U Ž L. iff N g C Ž L.; Žii. FI g FN Ž L. iff I g I Ž L.; Žiii. FI g FN DŽ L. iff I g I DŽ L.; Živ. u N g L U D Ž L. iff N g C D Ž L.; Žv. Fa g FN D Ž L. iff ua g C D Ž L. iff ua g C Ž L. iff a g DŽ L.. Ž b . FL is the smallest element of FN D Ž L.. If L is a lattice with 0, then F04 is the greatest element of FN D Ž L.. Ž c . N ª u N Ž I ª FI , a ª Fa . defines dual isomorphisms from C Ž L. Ž I Ž L., DŽ L.. onto a sublattice of L U Ž L. Ž FN Ž L., FN D Ž L., respecti¨ ely ..
518
AVALLONE AND WEBER
Ž d . If L is finite, then C Ž L., C D Ž L., L, and DŽ L. are dual isomorphic, respecti¨ ely, to L U Ž L., L U D Ž L., FN Ž L., and FN D Ž L. by the dual isomorphism of Žc.. Proof. Ža. Ži. and Žii. obviously hold. Žiii. One verifies successively the equivalence of the following conditions, using u I=s Ž FI .=: FI g FN D Ž L . ,
u I=g L U Ž L . ,
I=g C Ž L . ,
I g I D Ž L. .
Živ. If N g C D Ž L., then N s I= for some I g I D Ž L. and FI g FN D Ž L. by Žiii.; hence u N s Ž FI .=g L U D Ž L.. Let now u N g L U D Ž L. and F g FN D Ž L. with F =s u N . Let I g F with I=s N. We show that F s FI : Let F, G g F with sŽ G k G . : F and a g G; then, by Lemma 4.7, I : s Ž I= Ž a . . s s Ž N Ž a . . : s Ž G= Ž a . . : s Ž a k G . : F. By Žiii., FI s F g FN D Ž L. implies I g I D Ž L. and N s I=g C D Ž L.. Žv. By Žiii., Fa g FN D Ž L. iff x y `, ax g I D Ž L.. By wG, Corollary III.3.3 and Theorem III.2.2x, x y `, ax g I D Ž L. iff a g DŽ L. iff ua g C D Ž L. iff ua g C Ž L.. Žb. and Žc. are obvious. Žd. follows from Ža. and Žc.. It follows from Proposition 4.9 Ža. Žv. and Žc. that FN D Ž L. s FN Ž L. iff L is distributive iff FN Ž L. is distributive. If L is a lattice such F1 k F2 g FN D Ž L. whenever F1 , F2 g FN D Ž L., then FN D Ž L. is, by Theorem 4.5, a distributive sublattice of FN Ž L.. In Fig. 2 is given a lattice L, for which the lattice FN D Ž L. is not distributive; this follows from the fact that FN D Ž L. and DŽ L. are dually isomorphic by Proposition 4.9 Žd. and that the lattice DŽ L. }given in Fig. 5}is not distributive. In particular, in this example, FN D Ž L. is not a sublattice of FN Ž L.. In general, L U DŽ L. is not a sublattice of L U Ž L.. An example of this is the lattice of Fig. 6; this follows from Proposition 4.9 Žd. and the fact that a, d g DŽ L.; hence ua , ud g C D Ž L., but ua l ud g C Ž L. _ C D Ž L.. There are lattice uniformities on lattices with 0, the 0-neighbourhood system of which does not belong to FN D Ž L.; an example of this is the lattice of Fig. 3 endowed with the uniformity u N , where N is the congruence N [ 0, b4 , a1 , a2 , a3 4 , c, 144 . If L is a sectionally complemented lattice, then L U Ž L. s L U D Ž L.. The last equality is an essential assumption in Corollary 5.10. There are also lattices L which are not sectionally complemented, for which however L U Ž L. s L U D Ž L. holds. An example is the lattice of Fig. 7; the only non-trivial congruence of this lattice is uc and c g DŽ L..
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FIGURE 2
FIGURE 3
FIGURE 4
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FIGURE 5
FIGURE 6
FIGURE 7
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5. THE LATTICE STRUCTURE OF L U D Ž L, w . FOR w BEING EXHAUSTIVE For w g L U Ž L., we denote by L U Ž L, w . the space of all lattice uniformities on L coarser than w. Moreover, we put L U D Ž L, w . s L U Ž L, w . l L U D Ž L.. As a main result of this section, we prove that, under additional assumptions, L U D Ž L, w . is a Boolean algebra. Let f : X ª Y be a map and u a uniformity on Y. Then the collection of sets
Ž f=f.
y1
Ž U . [ Ž a, b . g X 2 : Ž f Ž a. , f Ž b . . g U 4 ,
U g u,
is a base for the coarsest uniformity uŽ f . on X, which makes f uniformly continuous. uŽ f . is called the in¨ erse image of u under f. 5.1. PROPOSITION. Let L1 and L2 be lattices, f : L1 ª L2 a lattice homomorphism, and u g L U Ž L2 .. Then the in¨ erse image uŽ f . of u under f is a lattice uniformity on L1 and N Ž uŽ f .. s Ž x, y . g L1 = L1: Ž f Ž x ., f Ž y .. g N Ž u.4 . Proof. Let D i be the diagonal of L2i . If U, V g u with D 1 k V : U, then D 2 k Ž f = f .y1 Ž V . : Ž f = f .y1 ŽU .. Hence k is uniformly continuous on L1 = L1. Anagously one sees that n is uniformly continuous on L1 = L1. The description of N Ž uŽ f .. obviously holds true. The following proposition serves to reduce the proof of Corollary 5.10 to the case of w being Hausdorff. 5.2. PROPOSITION. Let w g L U Ž L.. For u g L U Ž L, w ., denote by u ˆ the ˆ [ LrNŽ w . induced by u and by p : L ª L ˆ the quotient uniformity on L quotient map. Ža. Then u ª u ˆ defines a lattice isomorphism from L U Ž L, w . onto ˆ w . L U Ž L, . Here u is exhausti¨ e iff u ˆ ˆ is so. ˆ w Žb. If u g L U D Ž L, w ., then u ˆ g L U DŽ L, ˆ .. If w g L U DŽ L., then ˆ wˆ .. u g L U D Ž L, w . iff u ˆ g L U DŽ L, Proof. Ža. follows from wW2 , Sect. 1.2, 5.1x. Žb. Put ˆ x s p Ž x ., Fˆ s p Ž F ., and Uˆ s Žp = p .ŽU . for x g L, F : L, and U : L = L. Let u g L U D Ž L. and F g FN D Ž L. with F =s u. Using the fact that, for x, y g L with ˆ x Fˆ y, there exists x9 g ˆ x with x9 F y, one immediately sees that Fˆ[ p Ž F . [ Fˆ : F g F 4 g FN Ž L..
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ˆ ., we show that uˆ s Fˆ=. We have Fˆ=F u, since To prove u ˆ g L U DŽ L = Uˆ : Fˆ for F g F and U [ F=. To prove u ˆ F Fˆ=, let U g u and F, G g F = ˆ=: U. ˆ If Ž ˆx, ˆy . g Gˆ=, then with F : U and G k G : F. We show that G ˆx k aˆ s ˆy k aˆ for some a g G; hence Ž x k a, y k a. : N Ž w . : G= and therefore x k a k b s y k a k b for some b g G. Since a k b g F, it ˆ Let w g follows that Ž x, y . g F=: U and therefore Ž ˆ x, ˆ y . g U. ˆ .. We prove that u g L U DŽ L.. L U D Ž L., u g L U Ž L., and u ˆ g L U DŽ L ˆ . with Fˆ=s u, Let Fˆg FN D Ž L ˆ F0 g FN DŽ L. with F0= s w, and Fˆ0 [ = p Ž F0 .. Then Fˆ s u ˆF w ˆ s Fˆ0 =; hence FˆF Fˆ0 by Corollary 4.8 Ža.. y1 Obviously p Ž Fˆ. : Fˆ g Fˆ 4 is a base for a filter F g FN Ž L.. We show that u s F =. Let U g u. Choose Vˆ g u ˆ with Žp = p .y1 Ž Vˆ . : U and = y1 ˆ Then F s p Ž Fˆ. g F and F =: U; hence U g F =. Fˆ g Fˆ with Fˆ : V. ˆ g Fˆ and G s py1 Ž Gˆ. with G k G : F Conversely, let F g F ; choose G = ˆ . We show that U : F=, which implies F=g u. Let and U g u with Uˆ : G ˆ= and ˆx k aˆ s ˆy k aˆ for some a g G. Since Ž x, y . g U. Then Ž ˆ x, ˆ y. g G G g F : F0 , we obtain Ž x k a, y k a. g N Ž w . : G=; hence x k a k b s y k a k b for some b g G. It follows that Ž x, y . g F=, since a k b g F. 5.3. PROPOSITION. Let w be a Hausdorff lattice uniformity on L and ˜ w Ž L, ˜ . the completion of Ž L, w .. Ža. Then u ˜ ª u˜< L Ž:s uniformity induced by u˜ on L. defines a lattice ˜ w isomorphism from L U Ž L, ˜ . onto L U Ž L, w .. Here u˜ is exhausti¨ e iff u˜< L is so. ˜ w Žb. If w g L U D Ž L., and u ˜ g L U DŽ L, ˜ ., then u˜< L g L U DŽ L, w .. Proof. Ža. follows from wW4 , 3.8 Ža.x. Žb. Let w g L U D Ž L ., F0 g FN D Ž L . with F 0= s w, u ˜g ˜ w ˜ . with F˜=s u, and u [ u˜< L . Obviously, F [ F˜ L U D Ž L, ˜ ., F˜g FN DŽ L l L : F˜ g F˜4 g FN Ž L.. We will prove that F =s u. We first show that F =F w: Let F˜ g F˜ and F s F˜ l L. We may assume that F˜ is solid and, ˜ u˜.. Then F is open in Ž L, u. and by Lemma 4.2, that F˜ is open in Ž L, therefore in Ž L, w .. Consequently, if a g F, then G=Ž a. : F for some G g F0 . It follows with Lemma 4.7 that G : F; hence F g F0 and F=g w. ˜ G˜ g F˜ with G˜ k G˜ : F, ˜ F s F˜ l L, G s G˜ l L, and F =F u: Let F, ˜=lŽ L = L.. By Lemma 4.2, we may assume that G˜ is open in UsG ˜ u˜. and therefore in Ž L, ˜ w .. We show that U : F=. Since F =F w, there Ž L, ˜gw is a W ˜ with W˜ l Ž L = L. : G=. Let Ž x, y . g U and z g G˜ with x k z s y k z. Let Ž za .a g A be a net in L converging to z with respect to ˜ we may assume that za g G for all a g A. w. ˜ Since G˜ is open and z g G, Since ŽŽ x k za , y k za ..a g A converges to a diagonal element with respect to w, ˜ we have Ž x k zg , y k zg . g W˜ l Ž L = L. : G= for some g g A. Therefore there exists a z9 g G with x k zg k z9 s y k zg k z9. It follows that Ž x, y . g F=, since zg k z9 g G k G : F.
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It remains to show that u F F =: Let U g u and F˜ g F˜ with F˜=l Ž L = L. : U. Then F=: U for F [ F˜ l L.
˜ w More complicated is the proof that u ˜ g L U Ž L, ˜ . and u s u˜< L g ˜ w˜ .. We can prove it only in the case L U D Ž L, w . implies u ˜ g L U DŽ L, that w is exhaustive; see Proposition 5.6. The proof of Proposition 5.6 is based on the following propositions. Proposition 5.4 puts together some results of wW2 , 6.3, 6.7, 6.14 Žb., 7.1.7x. 5.4. PROPOSITION. Let w be a Hausdorff exhausti¨ e lattice uniformity and ˜ w Ž L, ˜ . the uniform completion of Ž L, w ..
˜ . is a complete lattice and w Ža. Then Ž L,F ˜ is order-continuous Ž i.e., order con¨ ergence implies topological con¨ ergence in Ž L, w ... ˜ . is meet-continuous Ž i.e., xa x implies xa n y x n y . and Žb. Ž L,F Ž dually . join-continuous. ˜ w ˜a Žc. For u, ˜ ¨ g L U Ž L, ˜ ., we ha¨ e N Ž u˜. = N Ž ¨˜. iff u˜ induces on L coarser topology than ¨˜. 5.5. PROPOSITION. A complete lattice is join-continuous iff lim supŽ xa k ˜ ya . s lim sup xa k lim sup ya for all nets Ž xa .a g A , Ž ya .a g A in L. Proof. ¥ is obvious. « Let x s lim sup xa , y s lim sup ya , z s lim sup Ž xa k ya ., xa s supg G a xg , ya s supg G a yg . Then, for a, b g A, we have z F xb k ya ; hence z F inf a Ž xb k ya . s xb k inf a ya s xb k y and z F inf b Ž xb k y . ˜ is join-continuous. The other inequality, z G x k y, holds s x k y, since L in any complete lattice. 5.6. PROPOSITION. Let w be an exhausti¨ e Hausdorff lattice uniformity on ˜ w ˜ w L, Ž L, ˜ . the completion of Ž L, w ., u g L U DŽ L, uw ., and u˜ g L U Ž L, ˜. ˜ ˜ ˜ Ž . Ž . Ž . 4 with u s u . Then u g L U D L, w , a [ sup 0 g D L , and N u s ˜< L ˜ ˜ ˜ ˜ = L: ˜ x k a s y k a4. Moreo¨ er, if ¨˜ g L U Ž L, ˜ w ua [ Ž x, y . g L ˜ ., ¨ s ¨˜< L , and b s sup 04¨ , then u F ¨ iff a G b. Proof. We use in the following Proposition 5.4 Ža., Žb.. From Proposi˜ w tion 5.4 Ža. follows that every closed ideal in Ž L, ˜ . is order closed, hence a ˜ is complete. principal ideal, since L Ži. Let F g FN D Ž L. with F =s u. Then the filter F˜ generated by ˜ F g F 4 obviously belongs to FN Ž L ˜ .; here and in the following A F: ˜ ˜ Ž . denotes the closure of A in L, w ˜ for any A : L. ˜ F g F , and W g w. Žii. Let x g L, ˜ We show that there are s g L and c g F with x k c s s k c and Ž s, x . g W.
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Let F0 [ F and Fn g F such that Fn k Fn : Fny1 and x F y g Fn , with x g L, implies x g Fny 1 for n g N . Then I [ F `ns1 Fn is a closed ideal ˜ hence I s w0, c0 x for some c0 g L. ˜ We use in the following that, for in L; any sequence Ž z n . with z n g Fn , we have sup`isn z i g Fny1 : Fm for n, m g N with n ) m and therefore lim n sup z n F c 0 . Choose symmetric vicinities Un , Vn g u ˜ with Un l Ž L = L. : Fn= and Vn (Vn : Un . By wW2 , Lemma 5.3x, there are Wa , n g w ˜ for a g A, n g N and some upward directed set A such that x s lim supa lim sup n xa , n whenever Ž xa , n , x . g Wa , n for a g A and n g N . ŽWe repeat the idea of the proof: Let ŽWa , 0 .a g A be a base of w ˜ and A upward directed; take Wa , n g w ˜ with Wa , n k Wa , n : Wa , ny1 and Wa , n n Wa , n : Wa , ny1 for a g A and n g N .. We may assume that Wa , n : W l Vn for a g A and n g N }choose xa , n g Wa , nŽ x . l L for a g A and n g N and yn [ xb , n for some fixed b g A. Let a g A. Since Ž xa , n , x . g Wa , n : Vn , in particular Ž yn , x . g Vn , we have
Ž xa , n , x . g Ž Vn (Vn . l Ž L = L . : Un l Ž L = L . : Fn= ; hence xa , n k za , n s yn k za , n for some za , n g Fn . Put xa s lim sup xa , n , n
za s lim sup za , n ,
y s lim sup yn .
n
n
Then xa k za s y k za by Proposition 5.5 and therefore xa k c 0 s y k c 0 , since za F c 0 . Since x s lim supa xa , we get}again by Proposition 5.5} x k c0 s y k c0 . We now prove that y k c1 s y 2 k c1 for some c1 g F1. Since
Ž yn , x . g Vn
and
Ž x, ynq1 . g Vnq1 : Vn ,
we have Ž yn , ynq1 . g Vn (Vn : Un ; hence Ž yn , ynq1 . g Un l Ž L = L. : Fn= and therefore yn k z n s ynq1 k z n for some z n g Fn . Then `
c1 [ sup z i g F1
and
yn k c1 s ynq1 k c1
for n G 2.
is2
Inductively one gets y 2 k c1 s yn k c1 for all n G 2 and therefore y 2 k c1 s y k c1 by Proposition 5.5. Finally we obtain c [ c 0 k c1 g F1 k F1 : F, s [ y 2 g L, x k c s y k c s s k c, and Ž s, x . g Wb , 2 : W. Žiii. F˜=F u: ˜ Let F g F. Choose F1 g F with F1 k F1 k F1 : F, symmetric vicinities U, V g u ˜ with U l Ž L = L. : F1= and V 3 : U. We = show that V : F . Let Ž x 1 , x 2 . g V. By Žii., there are si g L and c i g Fi such that x i k c i s si k c i
and
Ž x i , si . g V
Ž i s 1, 2 . .
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Then Ž s1 , s2 . g V 3 : U; hence Ž s1 , s2 . g U l Ž L = L. : F1= and therefore s1 k c 3 s s2 k c 3 for some c 3 g F1. It follows that c [ c1 k c 2 k c 3 g F
and
x 1 k c s s1 k c s s2 k c s x 2 k c;
hence Ž x 1 , x 2 . g F=. Živ. F˜=G u ˜ immediately follows from Lemma 4.2, since any U g u˜ contains F = F for some F g F as u s F =; hence F = F : U if U is ˜ .. closed. With Žiii. we get F˜=s u ˜ and therefore u˜ g L U DŽ L u ˜ Žv. We show that N Ž u ˜. s ua , where a s sup 0 4 . This implies by ˜ ., since N Ž u˜. is Proposition 4.9 Ža. Žv. or wG, Theorem III.2.2x that a g DŽ L a congruence relation. u ˜ ˜ w 0 4 u is a closed ideal in Ž L, ˜ . since u˜ : w; ˜ hence a g 0 4 and ˜ and x k a s y k a, then Ž x, y . g Ž a, 0 . g N Ž u˜.. Therefore, if x, y g L NŽ u ˜., since N Ž u˜. is a congruence relation. Vice versa, if Ž x, y . g N Ž u˜., then Ž x, y . g F= for any F g F , since u ˜ s F˜=; hence x k z F s y k z F for some z F g F. Then x k z s y k z, where z s inf F g F z F . Since z g F F g F F s w0, ax, we get z F a and x k a s y k a, since x k z s y k z. Žvi. By ŽProposition 5.3 Ža., u : ¨ iff u ˜ : ¨˜. If u˜ : ¨˜, then obviously a G b. Conversely, if a G b, then ua = u b ; hence N Ž u ˜. = N Ž ¨˜. by Žv.. It ˜ ¨˜. Ž . follows with Proposition 5.4 c that the 0-neighbourhood system in Ž L, ˜ Ž . Ž . Ž . is finer than that in L, u ˜ . Consequently, by Corollary 4.8 a , b , we get u ˜ : ¨˜. 5.7. THEOREM. Let w g L U D Ž L. and exhausti¨ e. For u g L U D Ž L, w ., ˆ uˆ. s Ž L, u.rNŽ w ., by Ž L, ˜ w denote by u ˆ the quotient uniformity of Ž L, ˜ . the ˆ w ˜ w uniform completion of Ž L, ˆ ., and by u˜ the unique uniformity of L U Ž L, ˜u. ˜ extending u ˆ Ž see Propositions 5.2 Ža. and 5.3 Ža... Then u ª f Ž u. [ sup 0 4 ˜ .. defines a dual isomorphism f : L U D Ž L, w . ª DŽ L Proof. By Propositions 5.2, 5.3, and 5.6, u ª u ˆ and uˆ ª u˜ define ˆ w ˆ wˆ . . isomorphisms from L U D Ž L, w . onto L U D Ž L, and from L U D Ž L, ˆ u ˜ ˜ w . 4 onto L U D Ž L, ; moreover, u ª sup 0 defines a dual isomorphism ˜ ˜ ˜ w ˜ .. It remains to prove that from L U D Ž L, ˜ . onto a subspace of DŽ L ˜. is the range of the last map. DŽ L ˜ . and F˜a the principal filter on L ˜ generated by w0, ax. Then Let a g DŽ L ˜ . by Proposition 4.9 Ža., Žv.. Since w g L U DŽ L., there is, by F˜a g FN D Ž L ˜. with F˜0 s w. Proposition 5.6, a filter F˜0 g FN D Ž L ˜ By Theorem 4.5 Žb., ˜ . and u˜ [ F˜=g L U DŽ L ˜ .. Since the sets sŽw0, ax F˜[ F˜0 n F˜a g FN D Ž L ˜ u˜., we k F ., F g F˜, form a base of the 0-neighbourhood system in Ž L, u ˜ have x g 0 4 iff, for every F g F˜, there is x F g F with x F a k x F . Since ˜ w here Ž x F . converges to 0 in Ž L, Hausdorff, it follows that x ˜ . and w ˜ is u u ˜ ˜ g 0 4 iff x F a and therefore a s sup 0 4 .
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˜ .. By Theorem 5.7, L U D Ž L, w . can be studied by studying DŽ L 5.8. LEMMA. Under the assumption of Proposition 5.4, we ha¨ e:
˜ . for any a, b g DŽ L ˜ ., then DŽ L ˜ . is a Browerian Ža. If a n b g DŽ L ˜ and dually Browerian complete lattice and an order closed sublattice of L. ˜. s NŽ L ˜ . and therefore DŽ L ˜ . is a Žb. If L is modular, then DŽ L ˜ sublattice of L. Proof. Ža. Obviously, the set of distributive elements is a closed set in any Hausdorff topological lattice. Therefore, by Proposition 5.4 Ža. and 1.1, ˜ . is an order closed sublattice of L. ˜ Since L ˜ is complete, meet-conDŽ L ˜ . is so. Hence DŽ L ˜ . is Browerian tinuous, and join-continuous, also DŽ L and dually Browerian by wB 2 , Theorem 24, p. 128x. ˆ and L ˜ are so. Hence DŽ L ˜. s NŽ L ˜. Žb. If L is modular, then L by 1.3. From Theorem 5.7 and Lemma 5.8 follows immediately: 5.9. COROLLARY. Let L be modular, w g L U D Ž L., and exhausti¨ e. Then L U D Ž L, w . is a complete Browerian and dually Browerian lattice. 5.10. COROLLARY. Let w g L Ue Ž L.. Suppose that L U Ž L, w . : L U D Ž L.. Ža. Then L U Ž L, w . is a complete Browerian and dually Browerian lattice. Žb. By the notation of Theorem 5.7, we ha¨ e
a g D Ž L˜ . : a has a complement in D Ž L˜ . 4 s N Ž L˜ . s C Ž L˜ . . ˜. s NŽ L ˜., in particular if L is modular, then L U Ž L, w . is a Žc. If DŽ L complete Boolean algebra. Proof. By Propositions 5.2, 5.3, and 5.6 we may assume that Ž L, w . is ˜ w Hausdorff and complete; hence Ž L, ˜ . s Ž L, w .. Put wa s fy1 Ž a. for Ž . a g D L , where f is defined as in Theorem 5.7. Ža. By Theorem 5.7 and Lemma 5.8, we have only to prove that a n b g DŽ L. for a, b g DŽ L.. By Proposition 5.6, and Theorem 5.7 we have
04
w a kw b
s 04
wa
l 04
wb
s w 0, a x l w 0, b x s w 0, a n b x
and therefore a n b s f Ž wa k w b . g DŽ L.. Žb. Ži. Let a g N Ž L.. We show that a has a complement in DŽ L..
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527
Since f Ž x . [ x n a defines a lattice homomorphism on L, the inverse image w a [ w Ž f . belongs by Proposition 5.1 to L U Ž L, w .. Then, by Theorem 5.7, b [ f Ž w a . g DŽ L.. Since Ž0, a. g N Ž wa . and Ž a, 1. g N Ž w a ., we have Ž0, 1. g N Ž wa n w a . and therefore a k b s f Ž wa n w a . s 1. By 1.2 and Proposition 5.1, we have N Ž wa k w a . s N Ž wa . l N Ž w a . s Ž x, y . g L2 : x k a s y k a and x n a s y n a4 s D ; hence a n b s f Ž w 1 k w a . s 0. Therefore b is a complement of a. Žii. Let a, b g DŽ L. and b a complement of a. We show that Ž a g C L.. Denote by pa and p b the quotient maps from L onto Lrua and Lru b , respectively, where ua and u b are defined as in Proposition 4.9. We now show that
p [ Ž pa , p b . : L ª Lrua = Lru b is injective. Here we use that wa k w b s w, since a n b s 0 and f is a dual isomorphism. Let p Ž x . s p Ž y .. Then, by Theorem 5.7, Ž x, y . g ua l u b s N Ž wa k w b . s N Ž w . s D, since w is Hausdoff; hence x s y. Moreover
p Ž a . s Ž 0, p b Ž a . . s Ž 0, p b Ž a . k p b Ž b . . s Ž 0, p b Ž a k b . . s Ž 0, p b Ž 1 . . s Ž 0, 1 . . We have shown that condition Ž2. of 1.2 is satisfied. Therefore a is neutral, hence central, since complemented. Žc. follows from Ža., Žb., and Lemma 5.8 Žb., since the centre of a lattice is always a Boolean algebra Žsee 1.5.. If L is a sectionally complemented lattice, then L U Ž L. s L U D Ž L. by Proposition 2.1. Therefore we obtain from Corollary 5.10Žc.: 5.11. COROLLARY. If L is a sectionally complemented modular lattice, then L Ue Ž L. is a complete Boolean algebra. By 1.4, the assumption in Corollary 5.11 that L is modular cannot be omitted. If L is an orthomodular lattice ŽSee wB 1 x. and w g L U Ž L., then, by ˜ of Ž L, w . is an orthomodular wW3 , 1.5x, also the uniform completion L lattice. Therefore we obtain from Corollary 5.10 Žc. and 1.3: 5.12. COROLLARY wW3 , Corollary 2.5x. If L is an orthomodular lattice, then L Ue Ž L. is a complete Boolean algebra.
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REFERENCES wB-Tx A. Basile and T. Traynor, Monotonely Cauchy locally solid topologies, Order 7 ŽŽ1991., 407]416. wB1 x L. Beran, Orthomodular lattices, in ‘‘Mathematics and Its Applications,’’ Reidel, Dordrecht, 1985. wB 2 x G. Birkhoff, ‘‘Lattice Theory,’’ 3rd ed., Am. Math. Soc., Providence, RI, 1967. wDx L. Drewnowski, Topological rings of sets, continuous set functions, integration I, II, and III, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 20 Ž1972., 269]276, 277]286, 439]445. wGx G. Gratzer, ‘‘General Lattice Theory,’’ Pure and Applied Mathematics Series, Aca¨ demic Press, San Diego, 1978. wM-Mx F. Maeda and S. Maeda, ‘‘Theory of Symmetric Lattices,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1970. wWx Waelbroecks, Topological vector spaces and algebras, in ‘‘Lecture Notes in Mathematics No. 230,’’ Springer-Verlag, BerlinrNew York, 1971. wW1 x H. Weber, Group- and vector-valued s-bounded contents, in ‘‘Measure Theory’’ ŽOberwolfach 1983., Lecture Notes in Mathematics, Vol. 1089, pp. 181]198, Springer-Verlag, BerlinrNew York, 1984. wW2 x H. Weber, Uniform Lattice: A generalization of topological Riesz spaces and topological Boolean rings; Uniform Lattice II: Order continuity and exhaustivity, Ann. Mat. Pura Appl. 160 Ž1991. 347]370; 165 Ž1993. 133]158. wW3 x H. Weber, Lattice uniformities and modular functions on orthomodular lattices, Order 12 Ž1995., 295]305. wW4 x H. Weber, On modular functions, Funct. Approx., to appear. wW5 x H. Weber, manuscript.
209, 507]528 Ž1997.
AY965291
Lattice Uniformities Generated by Filters Anna Avallone* Dipartimento di Matematica, Uni¨ ersita ` della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy
and Hans Weber † Dipartimento di Matematica, Uni¨ ersita ` di Udine, Via delle Scienze, 206, 33100 Udine, Italy Received May 30, 1995
INTRODUCTION In wW2 x, uniform lattices are studied as a common generalization of topological Boolean rings and topological Riesz spaces. Lattice uniformities play a similar role in the study of modular functions as FN-topologies; see e.g., wD; W1 x for the use of FN-topologies in measure theory and wW4 x for the use of lattice uniformities. The main aim of this paper is to give a contribution to the examination of the lattice structure of the space L Ue Ž L. of all exhaustive lattice uniformities on a lattice L Žsee Section 1 for definitions.. In particular, in the case where L is a modular sectionally complemented lattice, we get a satisfactory result; in this case L Ue Ž L. is a complete Boolean algebra Žsee Corollary 5.11.. Such a result can be used as the main tool in obtaining decomposition theorems for modular functions as well as information about the structure of uniform completions of L Žcf. wW5 x.. In the special case that L is a topological Boolean ring, the result 5.11 mentioned above as well as the connection with decomposition theorems in measure theory is given in wW1 x; for related results about locally solid l-groups, see wB-Tx. *E-mail: [email protected]. † E-mail: [email protected]. 507 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Important for our investigations is the fact that a lattice uniformity on a sectionally complemented lattice L is uniquely determined by its 0neighbourhood system; moreover the filters which are 0-neighbourhood filters for lattice uniformities are precisely the ‘‘distributive’’ filters, the space of which we indicate by FN D Ž L. Žsee Section 2.. In Section 4 we study this space FN D Ž L. also for an arbitrary lattice L. Every filter of FN D Ž L. induced on L a lattice uniformity in a similar way that a distributive ideal in L induces a congruence relation. In fact, the concept of ‘‘distributive’’ filters introduced here and some statements about them were suggested by the concept and some properties of distributive ideals Žsee wG, Chap. III.3x.. The main result of Section 5 essentially says that, for a Hausdorff exhaustive lattice uniformity w on L generated by some F0 g FN D Ž L., the space of all lattice uniformities generated by filters of FN D Ž L. coarser than F0 is isomorphic to the space of all distributive elements of the completion of Ž L, w .. As a consequence, we get the result 5.11 mentioned above and the fact proved in wW3 x that the space of all exhaustive lattice uniformities on an orthomodular lattice is a complete Boolean algebra.
1. PRELIMINARIES Throughout the paper, let L be a lattice and D [ Ž x, x . : x g L4 the diagonal in L = L. We denote by N the set of natural numbers. A lattice uniformity on L is a uniformity on L which makes the lattice operations k and n uniformly continuous. For any lattice uniformity u on L, we denote by N Ž u. the intersection of all members of u, i.e., the closure of D in Ž L, u. 2 ; N Ž u. is a congruence relation on L. If N is a congruence ˆ uˆ. [ Ž L, u.rN is a relation contained in N Ž u., then the quotient Ž L, lattice endowed with a lattice uniformity. If u is Hausdorff, then the ˜ u˜. of Ž L, u. is a lattice endowed with a lattice uniform completion Ž L, uniformity. u is called exhausti¨ e if every monotone sequence is Cauchy in Ž L, u. Žsee wW2 x, Sect. 1, 6x.. We denote, respectively, by L U Ž L. and L Ue Ž L. the sets of all lattice uniformities and of all exhaustive lattice uniformities; these are complete lattices, where the partial ordering is inclusion. If a lattice has a smallest or a greatest element, we denote these elements with 0 or 1, respectively. L is called sectionally complemented if L has a 0, and, for every a g L, the interval w0, ax is complemented. An element a of L is called distributi¨ e Žin L. if a k Ž x n y. s Ž a k x. n Ž a k y.
LATTICE UNIFORMITIES GENERATED BY FILTERS
509
for all x, y g L, i.e., if x ª a k x defines a lattice homomorphism. The set DŽ L. of all distributive elements of L is closed with respect to k ŽwG, Theorem III, 2.9 Ža.x.. It follows, since a lattice satisfying one the distributive laws satisfies both ŽwG, Lemma I.4.10x.: 1.1. If a n b g DŽ L. for any a, b g DŽ L., then DŽ L. is a distributi¨ e sublattice of L. Dualizing the definition of distributive elements, one gets the definition of dually distributive elements. 1.2 wG, Theorem III.2.4x. equi¨ alent:
If a g L, the following three conditions are
Ž1. a is distributi¨ e, a is dually distributi¨ e, and a k x s a k y, a n x s a n y imply x s y for any x, y g L. Ž2. There is an embedding f of L into a direct product A = B of lattices, where A has a 1, B has a 0, and f Ž a. s Ž1, 0.. Ž3. For any x, y g L, the sublattice generated by a, x, and y is distributi¨ e. An element satisfying the equivalent conditions of 1.2 is called neutral. The set N Ž L. of all neutral elements of L is a distributive sublattice of L. 1.3 wG, Theorem III.2.6x. then DŽ L. s N Ž L..
If L is modular or relati¨ ely complemented,
1.4. There are finite sectionally complemented lattices with DŽ L. / N Ž L., see Fig. 1, where a g DŽ L. _ N Ž L..
FIGURE 1
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If L has 0 and 1, then the centre C Ž L. of L is the set of all complemented neutral elements of L; any element of C Ž L. has a unique complement in L. 1.5 wM-M, Theorem 4.15x. subalgebra of L.
If L has 0 and 1, then C Ž L. is a Boolean
2. LATTICE UNIFORMITIES ON SECTIONALLY COMPLEMENTED LATTICES The study of the structure of L U Ž L. and L Ue Ž L. for a sectionally complemented lattice L is based on the fact that its lattice uniformities are generated by certain filters on L. 2.1. PROPOSITION. Let u be a lattice uniformity on a sectionally complemented lattice L and F the 0-neighbourhood system in Ž L, u.. Then the sets F* [ Ž x, y . g L2 : ' a g F : Ž x n y . k a s x k y 4 ,
F g F,
form a base of u, as do the sets F=[ Ž x, y . g L2 : ' a g F : x k a s y k a4 ,
F g F.
Proof. The first assertion is proved in wW2 , 6.10x. It follows that F=g u for F g F , since F* : F=. We now show that any U g u contains a set F= for some F g F. Choose a symmetric vicinity V g u with Ž D k V .( Ž D k V . : U and set F [ V Ž0.. Then F=: U: In fact, if Ž x, y . g F= and a g F with x k a s y k a, then
Ž x, y k a. s Ž x, x . k Ž 0, a. g D k V and
Ž y k a, y . s Ž y, y . k Ž a, 0 . g D k V ; hence Ž x, y . g Ž D k V .(Ž D k V . : U. Proposition 2.1 generalizes a known fact about congruence relations. If N is a congruence relation on L, then the principal filter u N [ U : N : U : L = L4 generated by N is a lattice uniformity. For principal filters of L U Ž L., the first and second assertion of Proposition 2.1 exactly mean that each congruence relation on a sectionally complemented lattice is standard and distributive, respectively Žsee wG, Theorem III.3.10x..
LATTICE UNIFORMITIES GENERATED BY FILTERS
511
The pentagon has congruence relations which are not distributive, namely 04 , c4 , 14 , a, b44 by the notation of Fig. 4; therefore the conclusion of Proposition 2.1 does not hold in general for complemented lattices. The correspondence F =ª F of Proposition 2.1 yields, for sectionally complemented lattices, an order isomorphism between L U Ž L. and FN D Ž L.; see Corollary 4.8 and Definition 4.3. This motivates the examination}in the next sections}of the spaces of filters FN D Ž L. and FN Ž L., which contains FN D Ž L.; see Definition 3.1. The principal filters of L U Ž L., FN Ž L., and FN D Ž L. correspond, respectively, to congruence relations, ideals, and distributive ideals; cf. Proposition 4.9.
3. THE LATTICE OF FILTERS FN Ž L. We call a subset S of L solid if x g S whenever y G x g L for some y g S. For A : L, the set s Ž A . [ x g L : there exists y g A with x F y 4 is the smallest solid subset of L containing A, i.e., the solid hull of A. 3.1 DEFINITION. We denote by FN Ž L. the set of all filters F on L satisfying one of the following two equivalent conditions: Ž1. ;F g F, ' G g F : ; x, y g G, ; z g L, z F x k y « z g F. Ž2. Ži. F has a base of solid sets and Žii. ;F g F , ' G g F : G k G [ x k y : x, y g G4 : F. To prove Ž1. « Ž2.Ži., one verifies that, under assumption Ž1., sŽ F . : F g F 4 is a filter base of F. The notation FN Ž L. is motivated by the fact that, for a Boolean ring L, F g FN Ž L. iff F is the 0-neighbourhood system of an FN-topology on L Žfor FN-topologies, cf. wDx or wW1 x.. 3.2. PROPOSITION. If u is a lattice uniformity on a lattice L with 0 and F is the 0-neighbourhood system in Ž L, u., then F g FN Ž L.. The proof follows from the fact that, by wW2 , 1.1.6x, every point x g L has a neighbourhood base of convex sets and every convex subset of L containing 0 is solid. On the other hand, there are filters belonging to FN Ž L. on lattices with 0, which are not 0-neighbourhood systems with respect to lattice uniformities: The principal filter on the pentagon generated by 0, c4 Žsee Fig. 4. is an example of this.
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AVALLONE AND WEBER
FN Ž L. is, with respect to the inclusion, a partially ordered set. It is easy to see and follows from Theorem 3.4 that Ž FN Ž L., :. is a lattice and that, for F1 , F2 g FN Ž L., the sets F1 l F2 and sŽ F1 k F2 ., where F1 g F1 and F2 g F2 , form a base of F1 k F2 and F1 n F2 , respectively. The fact that herein F1 l F2 / B follows from the next proposition, the proof of which is obvious. If F1 , F2 are solid subsets of L, then
3.3. PROPOSITION.
F1 l F2 s F1 n F2 [ x 1 n x 2 : x 1 g F1 , x 2 g F2 4 . 3.4. THEOREM. Ž FN Ž L., :. is a complete lattice. Moreo¨ er, if Ž Fa .a g A is a family in FN Ž L., then the sets F a g I Fa , where I is a finite subset of A and Fa g Fa , form a base of supa g A Fa ; and the sets s
ž
`
n
D E D
ns1 ks1 a gA
Fa , k
/
s x g L : ' n g N , a k g A, x k g Fa k , k : x F sup x k ,
½
1FkFn
5
where Fa , k g Fa for a g A and k g N , form a base of inf a g A Fa . Proof. Ži. Let A / B and Fa g FN Ž L. for a g A. For any finite subset I of A and Fa g Fa Ž a g I ., the sets F a g I Fa are by Proposition 3.3 non-empty; now it is obvious that these sets form a filter base of a filter of FN Ž L. and that this filter is the supremum of Fa : a g A4 . Since FN Ž L. has a smallest element, namely L4 , it follows that Ž FN Ž L., :. is a complete lattice. Žii. We prove the description of inf Fa given in Theorem 2.3. Obviously `
n
½ žD E D / s
ns1 ks1 a gA
Fa , k : Fa , k g Fa
5
is a filter base. To prove that the filter F generated by this filter base belongs to FN Ž L., we show that, for every F g F , there is a G g F with n G k G : F. Let F g F and Fa , k g Fa with sŽj `ns1 k ks1 j a g A Fa , k . : F. Put G[s
`
ž
n
D E D
ns1 ks1 a gA
2k
/
Ga , k ,
where Ga , k s
F Fa , i . is1
LATTICE UNIFORMITIES GENERATED BY FILTERS
513
Let x, y g G. Choose a 1 , . . . , a n , b 1 , . . . , bm g A and x k g Fa k , k , y k g Fb k , k with n
x F sup x k ks1
and
m
y F sup y k ; ks1
we may assume that m F n. Put
Ž g 1 , . . . , gnqm . s Ž a 1 , b 1 , . . . , a m , bm , a mq1 , . . . , a n . and
Ž z1 , . . . , z nqm . s Ž x 1 , y 1 , . . . , x m , ym , x mq1 , . . . , x n . . nq m Then z k g Fg k , k for k s 1, . . . , n q m and x k y F sup ks1 z k ; hence x k y g F.
We prove that F s inf Fa . Obviously, F F Fa for a g A. If F 9 g F N Ž L. with F 9 F Fa for all a g A, then F 9 F F : Let F0 g F 9. We may assume that F0 is solid. Choose Fk g F 9 with Fk k Fk : Fky1 for k g N . Then Fa , k [ Fk g Fa for all a g A and k g N , and n sŽD `ns 1 E ks1 D a g A Fa , k . : F0 ; hence F0 g F. In the case of L being a Boolean ring, Theorem 3.4 describes the supremum and the infimum of a family of FN-topologies by means of the 0-neighborhood systems. The description of the infimum is suggested by Waelbroecks’ description of the infimum of linear topologies, see wWx, which also holds true for group topologies.
4. THE LATTICE OF FILTERS FN D Ž L. In this section we introduce the space FN D Ž L. of ‘‘distributive’’ filters, which generate lattice uniformities on L in a similar way as distributive ideals generate the congruence relation; cf. wG, Theorem III.3.4x. The precise relationship between the filters of FN D Ž L. and distributive ideals in L is given in Proposition 4.9. 4.1. Notation. We put F=[ Ž x, y . g L2 : ' a g F : x k a s y k a4 for F : L and F =[ G : L2 : ' F g F : F=: G 4 for a filter F on L.
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AVALLONE AND WEBER
4.2. LEMMA. Let F g FN Ž L.. Then F = is the finest uniformity, which makes k uniformly continuous and which is contained in the filter generated by F = F : F g F 4 . Moreo¨ er, F has a base of closed sets as well as a base of open sets with respect to the topology induced by F =. Proof. Obviously, F = is a uniformity, which makes k uniformly continuous, since D k F=: F=. Moreover, F = is contained in the filter F 2 generated by F = F : F g F 4 , since F = F : Ž F k F .=. Let now u be a uniformity on L, which makes k uniformly continuous, and u F F 2 . We show that u F F =. Let U g u. Choose V, W g u such that V is symmetric, V (V : U, D k W : V and a solid set F g F with F = F : W. Then F=: U: Let Ž x, y . g F= and a g F with x k a s y k a. Then
Ž x, y k a. s Ž x, x k a. s Ž x, x . k Ž x n a, a. g D k Ž F = F . :DkW:V and
Ž y k a, y . s Ž y, y . k Ž a, y n a. g D k Ž F = F . : V ; hence Ž x, y . g V (V : U. To prove the last assertion, we show that, for every F g F , there is a ˚ Let F, G g F with sŽ G k G. : F and G g F with G : F and G : F. = ˚ since U [ G . Then G : UŽ G . : sŽ G k G . : F and therefore G : F, UŽ G . : F. 4.3. DEFINITION. We denote by FN D Ž L. the set of all filters F g FN Ž L. for which F = is a lattice uniformity. 4.4. PROPOSITION. ¨ alent:
For F g FN Ž L., the following conditions are equi-
Ž1. F g FN D Ž L.. Ž2. ;F g F , ' G g F : D n G=: F=. Ž3. ;F g F , ' G g F : ; x, y g L and ;a g G, ' b g F : Ž x k a. n y F Ž x n y . k b. Ž4. ;F g F, 'G g F : ;F1 , F2 : L, sŽ F1 k G . l sŽ F2 k G . : sŽŽ F1 n F2 . k F .. Proof. Ž1. m Ž2. follows from Lemma 4.2, since condition Ž2. is equivalent to the uniform continuity of n. Ž2. « Ž3. Let F, G be given according to Ž2., a g G and x, y g L. Since Ž x, x k a. g G=, we have
Ž x n y, Ž x k a. n y . g G=n D : F= ;
LATTICE UNIFORMITIES GENERATED BY FILTERS
515
hence
Ž x n y . k b s Ž Ž x k a. n y . k b G Ž x k a. n y for some b g F. Ž3. « Ž4. Let F0 g F. Choose F g F with F k F : F0 and G for F according to Ž3.. We show that s Ž F1 k G . l s Ž F2 k G . : s Ž Ž F1 n F2 . k F0 .
for any F1 , F2 : L:
Let z g sŽ F1 k G . l sŽ F2 k G .; moreover x i g Fi and a i g G with z F Ž x 1 k a1 . n Ž x 2 k a2 .. Applying Ž3. twice, we get b1 , b 2 g F with
Ž x 1 k a1 . n Ž x 2 k a2 . F Ž x 1 n Ž x 2 k a2 . . k b1 F Ž Ž x 1 n x 2 . k b 2 . k b1 ; hence z F Ž x 1 n x 2 . k b, where b [ b1 k b 2 g F k F : F0 . It follows that z g sŽŽ F1 n F2 . k F0 .. Ž4. m Ž2. Let F, G be given according to Ž4.. We show that D n G=: Ž F k F .=. Let x g L and Ž y 1 , y 2 . g G=; moreover a g G with y 1 k a s y 2 k a. Applying Ž4. for F1 s x 4 and F2 s yi 4 , where i g 1, 24 , we get x n Ž yi n a . g s Ž F1 n G . l s Ž F2 k G . : s Ž Ž F1 n F2 . k F . . Therefore there exists bi g F with x n Ž yi k a. F Ž x n yi . k bi . Then, with b [ b1 k b 2 , we have
Ž x n y1 . k b s Ž x n Ž y1 k a. . k b s Ž x n Ž y 2 k a. . k b s Ž x n y 2 . k b; hence Ž x n y 1 , x n y 2 . g Ž F k F .=. 4.5. THEOREM. Ža. Ž FN D Ž L., :. is a Dedikind complete lattice. Žb. If Fa g FN D Ž L. for a g A, then F [ inf a g A Fa g FN D Ž L. and F =s inf a g A Fa=. Ž Here, inf Fa and inf Fa= are built in FN Ž L. and L U Ž L., respecti¨ ely.. Žc. If F g FN D Ž L. and Fa g FN Ž L. for a g A, then supa g A Ž F n Fa . s F n sup a g A Fa . Proof. Ža. follows from Theorem 3.4 and Žb.. Žb. For the proof of F g FN D Ž L., we verify condition Ž3. of Proposition 4.4. Let F g F. By Theorem 3.4, there are sets Fa , k g Fa Ž a g A, k g N . with s
ž
`
n
D E D
ns1 ks1 a gA
Fa , k : F.
/
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AVALLONE AND WEBER
Choose Ga , k g Fa for Fa , k according to Proposition 4.4 Ž3. and G [ n sŽj `ns 1 k ks1 j a g A Ga , k .. Let x, y g L and g g G. We have to show that Ž x k g . n y F Ž x n y . k f for some f g F. Let n g N and a k g A, g k g Ga k , k for k s 1, . . . , n with g F E nks1 g k . Using Proposition 4.4 Ž3. for Ga , k and Fa , k , we inductively get f k g Fa k , k with n
ž
xk
E gi isk
n
/
nyF
žž
xk
E
/ /
gi n y k fk
iskq1
for k s 1, . . . , n.
It follows that Ž x k g . n y F Ž x n y . k f, where f [ E nks 1 f k g F. Žb. Obviously F =F Fa= for every a g A and therefore F =F u [ inf Fa= . We now prove that u F F =. Let U0 g u and Uk g u with Uk (Uk : Uky 1 for k g N . Choose Fa , k g Fa with Fa=, k : Uk for k g N and a g A. Put F[s
ž
`
n
D E D
ns1 ks1 a gA
/
Fa , k .
We show that F=: U0 (U0 . Let Ž x, y . g F=. Then there are a 1 , . . . , a n g A n and z k g Fa k , k for k s 1, . . . , n such that x k E ks1 z k s y k E nks1 z k . ky 1 k = Since Ž x k E is1 z i , x k E is1 z i . g Fa k , k : Uk for k s 1, . . . , n, we get n
ž
x, x k
E zi is1
/
g U1 ( ??? (Un : U0 ;
n analogously Ž y k E is1 z i , y . g U0 . Hence Ž x, y . g U0 (U0 . Žc. supa Ž F n Fa . F F n supa Fa obviously holds. Since every member of supa g A Fa belongs to supa g B Fa for some finite B : A, it is enough to prove the other inequality G for a finite index set A. Inductively, one can reduce the proof to the case of A containing two elements; i.e., we have to prove that
Ž F n F1 . k Ž F n F2 . G F n Ž F1 k F2 . for F g FN D Ž L. and F1 , F2 g FN Ž L.. But that immediately follows from Proposition 4.4 Ž1. « Ž4. and the characterization of the infimum and supremum of two filters of FN Ž L.. We now clarify the relationship between the filters of FN D Ž L. and the induced lattice uniformities. 4.6. Notation. 4.7. LEMMA.
L U D Ž L. [ F = : F g FN D Ž L.4 . F : sŽ F=Ž a.. and F=Ž a. : sŽ a k F . for a g L and F : L.
LATTICE UNIFORMITIES GENERATED BY FILTERS
517
Proof. If x g F, then
Ž x k a. k x s a k x and therefore Ž x k a, a. g F=. Hence x k a g F=Ž a. and x g sŽ F=Ž a... If x g F=Ž a., then x k z s a k z for some z g F; hence x F a k z g a k F and x g sŽ a k F .. 4.8. COROLLARY. Ža. F ª F = defines a lattice isomorphism from FN D Ž L. onto L U D Ž L.. Žb. If L is a lattice with 0 and F g FN D Ž L., then F is the 0neighbourhood system in Ž L, F =. . Žc. If ua g L U D Ž L. for a g A, then inf a g A ua g L U D Ž L.. Proof. Ža. We have only to show that F1 , F2 g FN D Ž L. and F1= F F2= imply F1 F F2 . Let G g F1 and F1 g F1 with sŽ F1 k F1 . : G. Choose F2 g F2 with F2=: F1= and a g F1. Then we have by Lemma 4.7. F2 : s Ž F2= Ž a . . : s Ž F1= Ž a . . : s Ž a k F1 . : G; hence G g F2 . Žb. follows from the fact that, if F g F is solid, then F=Ž 0. s = sŽ F Ž 0.. s F. Žc. follows from Theorem 4.5 Žb.. A bijection f : L1 ª L2 from a lattice L1 onto a lattice L2 such that x F y iff f Ž x . G f Ž y . is called a dual isomorphism. 4.9. PROPOSITION. Let I Ž L., I D Ž L., C Ž L., and C D Ž L. be the spaces of all ideals, distributi¨ e ideals wG, Def. III.3.1x, congruence relations, and distributi¨ e congruence relations in L wG, p. 149x, respecti¨ ely. Denote by FI Ž or u N . the principal filter on L Ž or on L = L. generated by a non-empty set I : L Ž or N : L = L, respecti¨ ely .. Let Fa s Fxy`, ax and ua s wy`, ax=s Ž x, y . g L2 : x k a s y k a4 . Ža.
Then for N : L = L, I : L, a g L, we ha¨ e
Ži. u N g L U Ž L. iff N g C Ž L.; Žii. FI g FN Ž L. iff I g I Ž L.; Žiii. FI g FN DŽ L. iff I g I DŽ L.; Živ. u N g L U D Ž L. iff N g C D Ž L.; Žv. Fa g FN D Ž L. iff ua g C D Ž L. iff ua g C Ž L. iff a g DŽ L.. Ž b . FL is the smallest element of FN D Ž L.. If L is a lattice with 0, then F04 is the greatest element of FN D Ž L.. Ž c . N ª u N Ž I ª FI , a ª Fa . defines dual isomorphisms from C Ž L. Ž I Ž L., DŽ L.. onto a sublattice of L U Ž L. Ž FN Ž L., FN D Ž L., respecti¨ ely ..
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AVALLONE AND WEBER
Ž d . If L is finite, then C Ž L., C D Ž L., L, and DŽ L. are dual isomorphic, respecti¨ ely, to L U Ž L., L U D Ž L., FN Ž L., and FN D Ž L. by the dual isomorphism of Žc.. Proof. Ža. Ži. and Žii. obviously hold. Žiii. One verifies successively the equivalence of the following conditions, using u I=s Ž FI .=: FI g FN D Ž L . ,
u I=g L U Ž L . ,
I=g C Ž L . ,
I g I D Ž L. .
Živ. If N g C D Ž L., then N s I= for some I g I D Ž L. and FI g FN D Ž L. by Žiii.; hence u N s Ž FI .=g L U D Ž L.. Let now u N g L U D Ž L. and F g FN D Ž L. with F =s u N . Let I g F with I=s N. We show that F s FI : Let F, G g F with sŽ G k G . : F and a g G; then, by Lemma 4.7, I : s Ž I= Ž a . . s s Ž N Ž a . . : s Ž G= Ž a . . : s Ž a k G . : F. By Žiii., FI s F g FN D Ž L. implies I g I D Ž L. and N s I=g C D Ž L.. Žv. By Žiii., Fa g FN D Ž L. iff x y `, ax g I D Ž L.. By wG, Corollary III.3.3 and Theorem III.2.2x, x y `, ax g I D Ž L. iff a g DŽ L. iff ua g C D Ž L. iff ua g C Ž L.. Žb. and Žc. are obvious. Žd. follows from Ža. and Žc.. It follows from Proposition 4.9 Ža. Žv. and Žc. that FN D Ž L. s FN Ž L. iff L is distributive iff FN Ž L. is distributive. If L is a lattice such F1 k F2 g FN D Ž L. whenever F1 , F2 g FN D Ž L., then FN D Ž L. is, by Theorem 4.5, a distributive sublattice of FN Ž L.. In Fig. 2 is given a lattice L, for which the lattice FN D Ž L. is not distributive; this follows from the fact that FN D Ž L. and DŽ L. are dually isomorphic by Proposition 4.9 Žd. and that the lattice DŽ L. }given in Fig. 5}is not distributive. In particular, in this example, FN D Ž L. is not a sublattice of FN Ž L.. In general, L U DŽ L. is not a sublattice of L U Ž L.. An example of this is the lattice of Fig. 6; this follows from Proposition 4.9 Žd. and the fact that a, d g DŽ L.; hence ua , ud g C D Ž L., but ua l ud g C Ž L. _ C D Ž L.. There are lattice uniformities on lattices with 0, the 0-neighbourhood system of which does not belong to FN D Ž L.; an example of this is the lattice of Fig. 3 endowed with the uniformity u N , where N is the congruence N [ 0, b4 , a1 , a2 , a3 4 , c, 144 . If L is a sectionally complemented lattice, then L U Ž L. s L U D Ž L.. The last equality is an essential assumption in Corollary 5.10. There are also lattices L which are not sectionally complemented, for which however L U Ž L. s L U D Ž L. holds. An example is the lattice of Fig. 7; the only non-trivial congruence of this lattice is uc and c g DŽ L..
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FIGURE 2
FIGURE 3
FIGURE 4
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FIGURE 5
FIGURE 6
FIGURE 7
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5. THE LATTICE STRUCTURE OF L U D Ž L, w . FOR w BEING EXHAUSTIVE For w g L U Ž L., we denote by L U Ž L, w . the space of all lattice uniformities on L coarser than w. Moreover, we put L U D Ž L, w . s L U Ž L, w . l L U D Ž L.. As a main result of this section, we prove that, under additional assumptions, L U D Ž L, w . is a Boolean algebra. Let f : X ª Y be a map and u a uniformity on Y. Then the collection of sets
Ž f=f.
y1
Ž U . [ Ž a, b . g X 2 : Ž f Ž a. , f Ž b . . g U 4 ,
U g u,
is a base for the coarsest uniformity uŽ f . on X, which makes f uniformly continuous. uŽ f . is called the in¨ erse image of u under f. 5.1. PROPOSITION. Let L1 and L2 be lattices, f : L1 ª L2 a lattice homomorphism, and u g L U Ž L2 .. Then the in¨ erse image uŽ f . of u under f is a lattice uniformity on L1 and N Ž uŽ f .. s Ž x, y . g L1 = L1: Ž f Ž x ., f Ž y .. g N Ž u.4 . Proof. Let D i be the diagonal of L2i . If U, V g u with D 1 k V : U, then D 2 k Ž f = f .y1 Ž V . : Ž f = f .y1 ŽU .. Hence k is uniformly continuous on L1 = L1. Anagously one sees that n is uniformly continuous on L1 = L1. The description of N Ž uŽ f .. obviously holds true. The following proposition serves to reduce the proof of Corollary 5.10 to the case of w being Hausdorff. 5.2. PROPOSITION. Let w g L U Ž L.. For u g L U Ž L, w ., denote by u ˆ the ˆ [ LrNŽ w . induced by u and by p : L ª L ˆ the quotient uniformity on L quotient map. Ža. Then u ª u ˆ defines a lattice isomorphism from L U Ž L, w . onto ˆ w . L U Ž L, . Here u is exhausti¨ e iff u ˆ ˆ is so. ˆ w Žb. If u g L U D Ž L, w ., then u ˆ g L U DŽ L, ˆ .. If w g L U DŽ L., then ˆ wˆ .. u g L U D Ž L, w . iff u ˆ g L U DŽ L, Proof. Ža. follows from wW2 , Sect. 1.2, 5.1x. Žb. Put ˆ x s p Ž x ., Fˆ s p Ž F ., and Uˆ s Žp = p .ŽU . for x g L, F : L, and U : L = L. Let u g L U D Ž L. and F g FN D Ž L. with F =s u. Using the fact that, for x, y g L with ˆ x Fˆ y, there exists x9 g ˆ x with x9 F y, one immediately sees that Fˆ[ p Ž F . [ Fˆ : F g F 4 g FN Ž L..
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ˆ ., we show that uˆ s Fˆ=. We have Fˆ=F u, since To prove u ˆ g L U DŽ L = Uˆ : Fˆ for F g F and U [ F=. To prove u ˆ F Fˆ=, let U g u and F, G g F = ˆ=: U. ˆ If Ž ˆx, ˆy . g Gˆ=, then with F : U and G k G : F. We show that G ˆx k aˆ s ˆy k aˆ for some a g G; hence Ž x k a, y k a. : N Ž w . : G= and therefore x k a k b s y k a k b for some b g G. Since a k b g F, it ˆ Let w g follows that Ž x, y . g F=: U and therefore Ž ˆ x, ˆ y . g U. ˆ .. We prove that u g L U DŽ L.. L U D Ž L., u g L U Ž L., and u ˆ g L U DŽ L ˆ . with Fˆ=s u, Let Fˆg FN D Ž L ˆ F0 g FN DŽ L. with F0= s w, and Fˆ0 [ = p Ž F0 .. Then Fˆ s u ˆF w ˆ s Fˆ0 =; hence FˆF Fˆ0 by Corollary 4.8 Ža.. y1 Obviously p Ž Fˆ. : Fˆ g Fˆ 4 is a base for a filter F g FN Ž L.. We show that u s F =. Let U g u. Choose Vˆ g u ˆ with Žp = p .y1 Ž Vˆ . : U and = y1 ˆ Then F s p Ž Fˆ. g F and F =: U; hence U g F =. Fˆ g Fˆ with Fˆ : V. ˆ g Fˆ and G s py1 Ž Gˆ. with G k G : F Conversely, let F g F ; choose G = ˆ . We show that U : F=, which implies F=g u. Let and U g u with Uˆ : G ˆ= and ˆx k aˆ s ˆy k aˆ for some a g G. Since Ž x, y . g U. Then Ž ˆ x, ˆ y. g G G g F : F0 , we obtain Ž x k a, y k a. g N Ž w . : G=; hence x k a k b s y k a k b for some b g G. It follows that Ž x, y . g F=, since a k b g F. 5.3. PROPOSITION. Let w be a Hausdorff lattice uniformity on L and ˜ w Ž L, ˜ . the completion of Ž L, w .. Ža. Then u ˜ ª u˜< L Ž:s uniformity induced by u˜ on L. defines a lattice ˜ w isomorphism from L U Ž L, ˜ . onto L U Ž L, w .. Here u˜ is exhausti¨ e iff u˜< L is so. ˜ w Žb. If w g L U D Ž L., and u ˜ g L U DŽ L, ˜ ., then u˜< L g L U DŽ L, w .. Proof. Ža. follows from wW4 , 3.8 Ža.x. Žb. Let w g L U D Ž L ., F0 g FN D Ž L . with F 0= s w, u ˜g ˜ w ˜ . with F˜=s u, and u [ u˜< L . Obviously, F [ F˜ L U D Ž L, ˜ ., F˜g FN DŽ L l L : F˜ g F˜4 g FN Ž L.. We will prove that F =s u. We first show that F =F w: Let F˜ g F˜ and F s F˜ l L. We may assume that F˜ is solid and, ˜ u˜.. Then F is open in Ž L, u. and by Lemma 4.2, that F˜ is open in Ž L, therefore in Ž L, w .. Consequently, if a g F, then G=Ž a. : F for some G g F0 . It follows with Lemma 4.7 that G : F; hence F g F0 and F=g w. ˜ G˜ g F˜ with G˜ k G˜ : F, ˜ F s F˜ l L, G s G˜ l L, and F =F u: Let F, ˜=lŽ L = L.. By Lemma 4.2, we may assume that G˜ is open in UsG ˜ u˜. and therefore in Ž L, ˜ w .. We show that U : F=. Since F =F w, there Ž L, ˜gw is a W ˜ with W˜ l Ž L = L. : G=. Let Ž x, y . g U and z g G˜ with x k z s y k z. Let Ž za .a g A be a net in L converging to z with respect to ˜ we may assume that za g G for all a g A. w. ˜ Since G˜ is open and z g G, Since ŽŽ x k za , y k za ..a g A converges to a diagonal element with respect to w, ˜ we have Ž x k zg , y k zg . g W˜ l Ž L = L. : G= for some g g A. Therefore there exists a z9 g G with x k zg k z9 s y k zg k z9. It follows that Ž x, y . g F=, since zg k z9 g G k G : F.
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It remains to show that u F F =: Let U g u and F˜ g F˜ with F˜=l Ž L = L. : U. Then F=: U for F [ F˜ l L.
˜ w More complicated is the proof that u ˜ g L U Ž L, ˜ . and u s u˜< L g ˜ w˜ .. We can prove it only in the case L U D Ž L, w . implies u ˜ g L U DŽ L, that w is exhaustive; see Proposition 5.6. The proof of Proposition 5.6 is based on the following propositions. Proposition 5.4 puts together some results of wW2 , 6.3, 6.7, 6.14 Žb., 7.1.7x. 5.4. PROPOSITION. Let w be a Hausdorff exhausti¨ e lattice uniformity and ˜ w Ž L, ˜ . the uniform completion of Ž L, w ..
˜ . is a complete lattice and w Ža. Then Ž L,F ˜ is order-continuous Ž i.e., order con¨ ergence implies topological con¨ ergence in Ž L, w ... ˜ . is meet-continuous Ž i.e., xa x implies xa n y x n y . and Žb. Ž L,F Ž dually . join-continuous. ˜ w ˜a Žc. For u, ˜ ¨ g L U Ž L, ˜ ., we ha¨ e N Ž u˜. = N Ž ¨˜. iff u˜ induces on L coarser topology than ¨˜. 5.5. PROPOSITION. A complete lattice is join-continuous iff lim supŽ xa k ˜ ya . s lim sup xa k lim sup ya for all nets Ž xa .a g A , Ž ya .a g A in L. Proof. ¥ is obvious. « Let x s lim sup xa , y s lim sup ya , z s lim sup Ž xa k ya ., xa s supg G a xg , ya s supg G a yg . Then, for a, b g A, we have z F xb k ya ; hence z F inf a Ž xb k ya . s xb k inf a ya s xb k y and z F inf b Ž xb k y . ˜ is join-continuous. The other inequality, z G x k y, holds s x k y, since L in any complete lattice. 5.6. PROPOSITION. Let w be an exhausti¨ e Hausdorff lattice uniformity on ˜ w ˜ w L, Ž L, ˜ . the completion of Ž L, w ., u g L U DŽ L, uw ., and u˜ g L U Ž L, ˜. ˜ ˜ ˜ Ž . Ž . Ž . 4 with u s u . Then u g L U D L, w , a [ sup 0 g D L , and N u s ˜< L ˜ ˜ ˜ ˜ = L: ˜ x k a s y k a4. Moreo¨ er, if ¨˜ g L U Ž L, ˜ w ua [ Ž x, y . g L ˜ ., ¨ s ¨˜< L , and b s sup 04¨ , then u F ¨ iff a G b. Proof. We use in the following Proposition 5.4 Ža., Žb.. From Proposi˜ w tion 5.4 Ža. follows that every closed ideal in Ž L, ˜ . is order closed, hence a ˜ is complete. principal ideal, since L Ži. Let F g FN D Ž L. with F =s u. Then the filter F˜ generated by ˜ F g F 4 obviously belongs to FN Ž L ˜ .; here and in the following A F: ˜ ˜ Ž . denotes the closure of A in L, w ˜ for any A : L. ˜ F g F , and W g w. Žii. Let x g L, ˜ We show that there are s g L and c g F with x k c s s k c and Ž s, x . g W.
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Let F0 [ F and Fn g F such that Fn k Fn : Fny1 and x F y g Fn , with x g L, implies x g Fny 1 for n g N . Then I [ F `ns1 Fn is a closed ideal ˜ hence I s w0, c0 x for some c0 g L. ˜ We use in the following that, for in L; any sequence Ž z n . with z n g Fn , we have sup`isn z i g Fny1 : Fm for n, m g N with n ) m and therefore lim n sup z n F c 0 . Choose symmetric vicinities Un , Vn g u ˜ with Un l Ž L = L. : Fn= and Vn (Vn : Un . By wW2 , Lemma 5.3x, there are Wa , n g w ˜ for a g A, n g N and some upward directed set A such that x s lim supa lim sup n xa , n whenever Ž xa , n , x . g Wa , n for a g A and n g N . ŽWe repeat the idea of the proof: Let ŽWa , 0 .a g A be a base of w ˜ and A upward directed; take Wa , n g w ˜ with Wa , n k Wa , n : Wa , ny1 and Wa , n n Wa , n : Wa , ny1 for a g A and n g N .. We may assume that Wa , n : W l Vn for a g A and n g N }choose xa , n g Wa , nŽ x . l L for a g A and n g N and yn [ xb , n for some fixed b g A. Let a g A. Since Ž xa , n , x . g Wa , n : Vn , in particular Ž yn , x . g Vn , we have
Ž xa , n , x . g Ž Vn (Vn . l Ž L = L . : Un l Ž L = L . : Fn= ; hence xa , n k za , n s yn k za , n for some za , n g Fn . Put xa s lim sup xa , n , n
za s lim sup za , n ,
y s lim sup yn .
n
n
Then xa k za s y k za by Proposition 5.5 and therefore xa k c 0 s y k c 0 , since za F c 0 . Since x s lim supa xa , we get}again by Proposition 5.5} x k c0 s y k c0 . We now prove that y k c1 s y 2 k c1 for some c1 g F1. Since
Ž yn , x . g Vn
and
Ž x, ynq1 . g Vnq1 : Vn ,
we have Ž yn , ynq1 . g Vn (Vn : Un ; hence Ž yn , ynq1 . g Un l Ž L = L. : Fn= and therefore yn k z n s ynq1 k z n for some z n g Fn . Then `
c1 [ sup z i g F1
and
yn k c1 s ynq1 k c1
for n G 2.
is2
Inductively one gets y 2 k c1 s yn k c1 for all n G 2 and therefore y 2 k c1 s y k c1 by Proposition 5.5. Finally we obtain c [ c 0 k c1 g F1 k F1 : F, s [ y 2 g L, x k c s y k c s s k c, and Ž s, x . g Wb , 2 : W. Žiii. F˜=F u: ˜ Let F g F. Choose F1 g F with F1 k F1 k F1 : F, symmetric vicinities U, V g u ˜ with U l Ž L = L. : F1= and V 3 : U. We = show that V : F . Let Ž x 1 , x 2 . g V. By Žii., there are si g L and c i g Fi such that x i k c i s si k c i
and
Ž x i , si . g V
Ž i s 1, 2 . .
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Then Ž s1 , s2 . g V 3 : U; hence Ž s1 , s2 . g U l Ž L = L. : F1= and therefore s1 k c 3 s s2 k c 3 for some c 3 g F1. It follows that c [ c1 k c 2 k c 3 g F
and
x 1 k c s s1 k c s s2 k c s x 2 k c;
hence Ž x 1 , x 2 . g F=. Živ. F˜=G u ˜ immediately follows from Lemma 4.2, since any U g u˜ contains F = F for some F g F as u s F =; hence F = F : U if U is ˜ .. closed. With Žiii. we get F˜=s u ˜ and therefore u˜ g L U DŽ L u ˜ Žv. We show that N Ž u ˜. s ua , where a s sup 0 4 . This implies by ˜ ., since N Ž u˜. is Proposition 4.9 Ža. Žv. or wG, Theorem III.2.2x that a g DŽ L a congruence relation. u ˜ ˜ w 0 4 u is a closed ideal in Ž L, ˜ . since u˜ : w; ˜ hence a g 0 4 and ˜ and x k a s y k a, then Ž x, y . g Ž a, 0 . g N Ž u˜.. Therefore, if x, y g L NŽ u ˜., since N Ž u˜. is a congruence relation. Vice versa, if Ž x, y . g N Ž u˜., then Ž x, y . g F= for any F g F , since u ˜ s F˜=; hence x k z F s y k z F for some z F g F. Then x k z s y k z, where z s inf F g F z F . Since z g F F g F F s w0, ax, we get z F a and x k a s y k a, since x k z s y k z. Žvi. By ŽProposition 5.3 Ža., u : ¨ iff u ˜ : ¨˜. If u˜ : ¨˜, then obviously a G b. Conversely, if a G b, then ua = u b ; hence N Ž u ˜. = N Ž ¨˜. by Žv.. It ˜ ¨˜. Ž . follows with Proposition 5.4 c that the 0-neighbourhood system in Ž L, ˜ Ž . Ž . Ž . is finer than that in L, u ˜ . Consequently, by Corollary 4.8 a , b , we get u ˜ : ¨˜. 5.7. THEOREM. Let w g L U D Ž L. and exhausti¨ e. For u g L U D Ž L, w ., ˆ uˆ. s Ž L, u.rNŽ w ., by Ž L, ˜ w denote by u ˆ the quotient uniformity of Ž L, ˜ . the ˆ w ˜ w uniform completion of Ž L, ˆ ., and by u˜ the unique uniformity of L U Ž L, ˜u. ˜ extending u ˆ Ž see Propositions 5.2 Ža. and 5.3 Ža... Then u ª f Ž u. [ sup 0 4 ˜ .. defines a dual isomorphism f : L U D Ž L, w . ª DŽ L Proof. By Propositions 5.2, 5.3, and 5.6, u ª u ˆ and uˆ ª u˜ define ˆ w ˆ wˆ . . isomorphisms from L U D Ž L, w . onto L U D Ž L, and from L U D Ž L, ˆ u ˜ ˜ w . 4 onto L U D Ž L, ; moreover, u ª sup 0 defines a dual isomorphism ˜ ˜ ˜ w ˜ .. It remains to prove that from L U D Ž L, ˜ . onto a subspace of DŽ L ˜. is the range of the last map. DŽ L ˜ . and F˜a the principal filter on L ˜ generated by w0, ax. Then Let a g DŽ L ˜ . by Proposition 4.9 Ža., Žv.. Since w g L U DŽ L., there is, by F˜a g FN D Ž L ˜. with F˜0 s w. Proposition 5.6, a filter F˜0 g FN D Ž L ˜ By Theorem 4.5 Žb., ˜ . and u˜ [ F˜=g L U DŽ L ˜ .. Since the sets sŽw0, ax F˜[ F˜0 n F˜a g FN D Ž L ˜ u˜., we k F ., F g F˜, form a base of the 0-neighbourhood system in Ž L, u ˜ have x g 0 4 iff, for every F g F˜, there is x F g F with x F a k x F . Since ˜ w here Ž x F . converges to 0 in Ž L, Hausdorff, it follows that x ˜ . and w ˜ is u u ˜ ˜ g 0 4 iff x F a and therefore a s sup 0 4 .
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˜ .. By Theorem 5.7, L U D Ž L, w . can be studied by studying DŽ L 5.8. LEMMA. Under the assumption of Proposition 5.4, we ha¨ e:
˜ . for any a, b g DŽ L ˜ ., then DŽ L ˜ . is a Browerian Ža. If a n b g DŽ L ˜ and dually Browerian complete lattice and an order closed sublattice of L. ˜. s NŽ L ˜ . and therefore DŽ L ˜ . is a Žb. If L is modular, then DŽ L ˜ sublattice of L. Proof. Ža. Obviously, the set of distributive elements is a closed set in any Hausdorff topological lattice. Therefore, by Proposition 5.4 Ža. and 1.1, ˜ . is an order closed sublattice of L. ˜ Since L ˜ is complete, meet-conDŽ L ˜ . is so. Hence DŽ L ˜ . is Browerian tinuous, and join-continuous, also DŽ L and dually Browerian by wB 2 , Theorem 24, p. 128x. ˆ and L ˜ are so. Hence DŽ L ˜. s NŽ L ˜. Žb. If L is modular, then L by 1.3. From Theorem 5.7 and Lemma 5.8 follows immediately: 5.9. COROLLARY. Let L be modular, w g L U D Ž L., and exhausti¨ e. Then L U D Ž L, w . is a complete Browerian and dually Browerian lattice. 5.10. COROLLARY. Let w g L Ue Ž L.. Suppose that L U Ž L, w . : L U D Ž L.. Ža. Then L U Ž L, w . is a complete Browerian and dually Browerian lattice. Žb. By the notation of Theorem 5.7, we ha¨ e
a g D Ž L˜ . : a has a complement in D Ž L˜ . 4 s N Ž L˜ . s C Ž L˜ . . ˜. s NŽ L ˜., in particular if L is modular, then L U Ž L, w . is a Žc. If DŽ L complete Boolean algebra. Proof. By Propositions 5.2, 5.3, and 5.6 we may assume that Ž L, w . is ˜ w Hausdorff and complete; hence Ž L, ˜ . s Ž L, w .. Put wa s fy1 Ž a. for Ž . a g D L , where f is defined as in Theorem 5.7. Ža. By Theorem 5.7 and Lemma 5.8, we have only to prove that a n b g DŽ L. for a, b g DŽ L.. By Proposition 5.6, and Theorem 5.7 we have
04
w a kw b
s 04
wa
l 04
wb
s w 0, a x l w 0, b x s w 0, a n b x
and therefore a n b s f Ž wa k w b . g DŽ L.. Žb. Ži. Let a g N Ž L.. We show that a has a complement in DŽ L..
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527
Since f Ž x . [ x n a defines a lattice homomorphism on L, the inverse image w a [ w Ž f . belongs by Proposition 5.1 to L U Ž L, w .. Then, by Theorem 5.7, b [ f Ž w a . g DŽ L.. Since Ž0, a. g N Ž wa . and Ž a, 1. g N Ž w a ., we have Ž0, 1. g N Ž wa n w a . and therefore a k b s f Ž wa n w a . s 1. By 1.2 and Proposition 5.1, we have N Ž wa k w a . s N Ž wa . l N Ž w a . s Ž x, y . g L2 : x k a s y k a and x n a s y n a4 s D ; hence a n b s f Ž w 1 k w a . s 0. Therefore b is a complement of a. Žii. Let a, b g DŽ L. and b a complement of a. We show that Ž a g C L.. Denote by pa and p b the quotient maps from L onto Lrua and Lru b , respectively, where ua and u b are defined as in Proposition 4.9. We now show that
p [ Ž pa , p b . : L ª Lrua = Lru b is injective. Here we use that wa k w b s w, since a n b s 0 and f is a dual isomorphism. Let p Ž x . s p Ž y .. Then, by Theorem 5.7, Ž x, y . g ua l u b s N Ž wa k w b . s N Ž w . s D, since w is Hausdoff; hence x s y. Moreover
p Ž a . s Ž 0, p b Ž a . . s Ž 0, p b Ž a . k p b Ž b . . s Ž 0, p b Ž a k b . . s Ž 0, p b Ž 1 . . s Ž 0, 1 . . We have shown that condition Ž2. of 1.2 is satisfied. Therefore a is neutral, hence central, since complemented. Žc. follows from Ža., Žb., and Lemma 5.8 Žb., since the centre of a lattice is always a Boolean algebra Žsee 1.5.. If L is a sectionally complemented lattice, then L U Ž L. s L U D Ž L. by Proposition 2.1. Therefore we obtain from Corollary 5.10Žc.: 5.11. COROLLARY. If L is a sectionally complemented modular lattice, then L Ue Ž L. is a complete Boolean algebra. By 1.4, the assumption in Corollary 5.11 that L is modular cannot be omitted. If L is an orthomodular lattice ŽSee wB 1 x. and w g L U Ž L., then, by ˜ of Ž L, w . is an orthomodular wW3 , 1.5x, also the uniform completion L lattice. Therefore we obtain from Corollary 5.10 Žc. and 1.3: 5.12. COROLLARY wW3 , Corollary 2.5x. If L is an orthomodular lattice, then L Ue Ž L. is a complete Boolean algebra.
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