Lattice Uniformities on Orthomodular Structures

Mathematica Slovaca Anna Avallone Lattice uniformities on orthomodular structures Mathematica Slovaca, Vol. 51 (2001), No. 4, 403--419 Persistent URL: http://dml.cz/dmlcz/131921 Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 2001 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Mathematica Slovaca » A ^L 4-1 r-» /0/-.01 \ IM A >»,-.-» A i rv Math. SlOVaCa, 5 1 (2001), NO. 4, 4 0 3 - 4 1 9 ©2001 Mathematical Institute Slovák Aeademy of Sciences LATTICE UNIFORMITIES ON ORTHOMODULAR S T R U C T U R E S A N N A AVALLONE (Communicated by Anatolij Dvurecenskij ) ABSTRACT. We prove that every lattice uniformity on an orthomodular lattice is generated by a family of weakly subadditive functions and that every modular measure on a difference-lattice generates a topological structure as modular functions on orthomodular lattices. Introduction Starting from the seventies, many authors, a s L . D r e w n o w s k i , Z. L i p e c k i , H. W e b e r and others (see for example [D-J, [L], [W-J), introduced, in classical measure theory, topological methods based on the theory of FrechetNikodym topologies, which gave many contributions t o the study of measures on Boolean algebras. In the last years, similar topological methods have been developed for the study of modular functions on orthomodular lattices in non-commutative measure theory (see for example [A-J, [A 2 ], [A-B-C], [A-D], [A-L], [W 4 ], [W 5 ], [W 6 ], [W8]) and for the study of measures on fuzzy structures in fuzzy measure theory (see [B-W], [B-L-W], [G]). In this context, the theory of Frechet-Nikodym topologies is replaced by the theory of lattice uniformities — i.e. uniformities which makes the lattice operations uniformly continuous — developed in [W 2 ], [W 3 ], [W 4 ], [W 7 ], [A-W], starting from the fact that every modular function on an orthomodular lattice generates a lattice uniformity which makes the orthocomplementation uniformly continuous ([W 4 ; 1.1]) and every measure on a A -^-semigroup or on a Vitali space generates a lattice uniformity which makes uniformly continuous the operations of these structure ([B-W; 3.1.2], and [G; 5.3]). It is known (see, for example, [D]) that, in a Boolean algebra, every FrechetNikodym topology is generated by a family of subadditive functions. In the first 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Primary 28B05, 06C15. Key w o r d s : orthomodular lattice, difference-lattice, modular function, lattice uniformities. 403 ANNA AVALLONE part of the present paper, we prove that a similar result also holds for lattice uniformities on orthomodular lattices: we introduce a class of weakly subadditive functions — the k-submeasures — and we prove that, for every k-submeasure rj, there exists the weakest lattice uniformity which makes 77 uniformly continuous (see 2.6) and, conversely, every lattice uniformity is generated by a family of fc-submeasures (see 2.8). In particular, every lattice uniformity with a countable base coincides with the uniformity generated by a fc-submeasure. In the second part, we prove that it is possible to use topological methods also in the study of modular measures on difference-lattices (D-lattices), since every modular measure on a D-lattice L generates a lattice uniformity which makes the difference operation of L uniformly continuous (see 3.2.2). As example of consequence of this result, we derive by standard topological methods the equivalence in any D-lattice between Vitali-Hahn-Saks and BrooksJewett theorems for modular measures (see 3.6). In particular, as consequence of [D-P] — in which the Brooks-Jewett theorem has been proved for measures on quasi-(j-complete D-posets — we obtain the Vitali-Hahn-Saks theorem for modular measures on quasi- a -complete D-lattices (see 3.7). We recall that D-posets and D-lattices have been introduced in [C-KJ as a generalization of many structures as orthomodular lattices, MV-algebras, orthoalgebras, weakly complemented posets and others. For a study, see for example [B-F], [C-KJ, [C-K 2 ], [D-D-P], [F-G-P], [P 2 ], [P 3 ], [R]. 1. Preliminaries Let L be a lattice. If L has a smallest or a greatest element, we denote these elements by 0 and 1, respectively. We set A = {(a, b) G Lx L : a = b} . If {an} is an increasing sequence and a = s u p a n in L (respectively, {an} is decreasing n and a = inf an in L), we write an t a (respectively, an I a). If a < 6, we set [a, b] = {c G L : a < c < b}. A lattice uniformity U on L is a uniformity on L which makes the lattice operations of L uniformly continuous (for a study, see [ W J ) . U is called exhaustive if every monotone sequence in L is Cauchy in (L,U) and a-order continuous (cr-o.c.) if a n t a or a n 4- a imply an —•> a in (L,U). If (G, +) is an Abelian group, a function /1: L —> G is called modular if, for every a, b G L, fi(a V b) + fi(a A b) = 11(a) + 11(b). If G is a topological group and \i: L —> G is a modular function, by [W 5 ; (3.1)] there exists the weakest lattice uniformity U(fi) which makes fi uniformly continuous and a base of U(\i) is the family consisting of the sets {(a, b) G L x L : fi(c) - 11(d) G W for all c, d G [a A b, a V b]} , 404 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES where W is a 0-neighbourhood in G. L is called orthomodular lattice if it has 0 and 1 and there exists a map /: a G L —>> a' G L, with the following properties: (1) (2) (3) (4) a V a ' = 1 and a A a' = 0 . a<b ==> a' >b'. (a')' = a. a < b => b = aV(bAa'). For a study, we refer to [K] or [P-P]. A difference-poset (or D-poset) is a non-empty partially ordered set (L, <) with a greatest element 1 and a binary partial operation 0 , called difference^ such that a © b is defined if and only if b < a and the following properties hold: (1) (2) (3) (4) bQa< b. be (bQ a) = a. If a < b < c, then c © b < c G a. If a < b < c, then (c 0 a) © (c© b) = b 0 a. If (L, <) is a lattice, a D-poset is called D-lattice. For every a,6 G L, we set aAb = (a V 6) 0 (a A 6) and a x = 1 © a. It is easy to see that (a- 1 )" 1 = a for every a E L and a < b implies a -1 > 5 1 . If a,6 G L, we say that a _L b if a < b1. If a _L 6, we set a © 6 = (a1 © 6)- 1 . It is easy to see that © is commutative and, if b © c and a® (bee) are defined, then a © b and (a © b) © c are defined, too, and (a © b) © c = a © (6 © c). More in general, for n > 3 , we inductively define ax © • • • © an = (a x © • • • © an_l) © a n if ax © • • • © a n and (ax © • • • © a n _ 1 ) © a n are defined, and the definition is independent on any permutation of the elements. We say that a family { a x , . . . , an} in L is orthogonal if 0 ai = ax © • • • © a n is defined. We say that {an} is orthogonal if, for every finite M C j V , {a n : n G M } is orthogonal. We use the following properties of D-lattices. PROPOSITION (1.1). ([R; 1.3, 1.4, 1.7, 2.2, 2.4, 2.6]) Let L be a D-lattice. Then: (1) If c<a and c<b, then (oV b) © c = (aec)\/ (be c) and ( a A b ) © c = (a © c) A (b © c). (2) If a<b, then b = a © (b © a ) . (3) If a <b < c, then be a < c 0 a. (4) If a ±.b, then a < a © b and (a © 6) © a = 6. (5) If a<b< c, then (c© a) 0 (b © a) = c 0 6. (6) If c> a and c>b, then c 0 ( a V6) = ( c © a ) A (c© 6) ana1 c 0 ( a A b) = (c 0 a) V (c © 6). (7) If a <b< c, then (ceb)®>a exists and (c © 6) © a = c © (b 0 a). 405 ANNA AVALLONE If G is an Abelian group, a function \i: L —> G is called a measure if, every a, 6 G L with a JL 6, /i(a © 6) = /i(a) + /z(6). By (1.1) (2) and (4), it easy to see that \i is a measure if and only if, for every a, b G L, with b < /x(a © 6) = /z(a) — /i(b). Moreover, by induction, we obtain that, if { a 1 ? . . . , for is a, an} is orthogonal, then /if 0 a i J = ]T} M a i ) • Many structures are examples of D-lattices (see [P 2 ; Chapter 12]). In particular, every orthomodular lattice is a D-lattice if we define, for b < a, a©b = aAb'. In this case, a1- = a' and, if a JL b, then a © 6 - - a V 5 . In the following, we denote by M the set of the positive integer numbers. = |x - y|, where oo — co = 0 and Moreover, for x,y G [0,oo], we set d^x^y) oo— x = x — co = oo for every x G [0, +oo[. 2. Lattice uniformities on orthomodular lattices In this section, L is an orthomodular lattice, and rj: L —> [0, +oo]. DEFINITION (2.1). If k > 1, we say that rj is k-subadditive a, 6 G L, rj(a V 6) < k7y(a) + 77(6). If rj is 1-subadditive, we say that rj is subadditive. if, for every DEFINITION (2.2). We say that 77 is a k-submeasure if 77(0) = 0, 77 is monotone and k-subadditive and, for every a,b £ L, rj[(a V b) A 6') < k77(a). A 1-submeasure is called submeasure. Every k-submeasure is k-triangular and null-additive in the sense of [P 2 ]. If L is a Boolean algebra, every monotone and k-subadditive function 77, with rj(0) = 0, is a k-submeasure. (2.3). (1) Every positive real-valued modular function \i with \i(0) = 0, is a submeasure, since /i is monotone and subadditive and, for every a, b G L, a V 6 = b V ((a V b) A V), where 6 _L (a V 6) A b', from which EXAMPLES //((a V b) A 6') = //(a V 6) - /i(6). (*) (2) A positive real-valued measure \i is a k-submeasure if and only if \i is k-subadditive, because (*) of (1) holds. (3) Let [i be a positive real-valued modular function with \i(0) = 0, k > 1, and 0: [0,+oo[ —> [0,+oo[ an increasing function such that 0(0) = 0 and, for every x,y G [0,+oo[, |</>(x) — <Ky)| -^ k<j)(\x — y\). Then the function A: L —> [0, +oo[ defined by A(a) = </>(/L(a)) for a G L is a k-submeasure. The following result generalizes the equivalence for a real-valued measure on L between modularity and subadditivity (see [R]). 406 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES PROPOSITION (2.4). Let ji: L -> [0,+oo[ be a measure and k > 1. Then \i is a k-submeasure if and only if, for every a, b G L, \i(a V b) + k/x(a A b) < kfi(a) + fj,(b) < kfi(a V b) + ji(a A b). P r o o f . The proof of « = kfi(a A b) < kfx(a) + /x(b). ==>: Let a,b E L. Since is trivial by (2.3) (2), since/i(aVb) < / i ( a V b ) + a V b = (a A b) V (aAb) with a A b _L a A b , aAb = [a A (a A b)'] V [b A (a A b)'] , a = (a A b) V (a A (a A b)') and b = (a A b) V (b A (a A b)') , we get fi(a V b) = /x(a A b) + /i(aAb) < /x(a A b) + kfx(a) —fc/x(aA b) + /i(b) — jx(a A b), from which /i(a V b) + k/x(a A b) < fc^(a) + /i(b). Moreover, since a V b = a V [(a V b) A a'] = b V [(a V b) A b'] , aAb < [aA(a V b)] V [(a V b) Ab] = [(a V b) A a1] V [(a V b) A b'] , we get /i(a V b) = \i(a A b) + \x(aAb) < \x(a A b) + fc//(a V b) - k/i(a) + \i(a V b) - //(b), from which k\x(a V b) + //(a A b) > k/i(a) + /x(b). D We use the following result of [W 3 ; (1.1)]. THEOREM (2.5). Let T be a filter on L with the following properties: (1) For every F G T, there exists G G T such that a G L, b, c G G and a < b V c zmp/y « G F . (2) For ei>en/ F e T, there is G E T such that a e G implies (aVb) Ab' G F1 / o r eac/i b G F. Tlten /liere exists a unique lattice uniformity U on L which has T as base of 0 -neighbourhoods and a base for U is the family consisting of the sets {(a, b) G L x L : aAb G F } wztt F e T. PROPOSITION (2.6). If r) is a k-submeasure, there exists the weakest lattice uniformity U(rj) which makes rj uniformly continuous. P r o o f . By (2.5), the family consisting of the sets {(a, b) G L x L : r](aAb) < e], where e > 0, is a base for a lattice uniformity U(r)) on L. (i) We prove that 77 is uniformly continuous with respect to U(r}). 407 ANNA AVALLONE Let £ > 0 and a,6 G L such that r](aAb) < ejk. Since oV6 = (oA6)V(aA6), then 77(0 V 6) < 77(0 A 6) + krj(aAb) < 77(0 A 6) + e. Then, if 77(0 V 6) = + 0 0 , we get 77(0 A 6) = + 0 0 , from which 77(0) = 77(6) = + o o . If 77(0 V 6) < + 0 0 , then 77(a) < +00 and 77(6) < -f-oo, from which 77(0) - 77(6) < 77(0 V 6) - 77(0 A 6) < e and 77(6) —77(a) < e. In both the cases, dOG(r](a),r](b)) < e. (ii) Let V be a lattice uniformity which makes 77 uniformly continuous. We prove that ^(77) < V. Let U G U(rj) and e > 0 such that U£ = {(a, 6) G L x L : rj(aAb) < e} CU. Since 77 is V-uniformly continuous, we can choose V G V such that {a,b)eV ==» doo(r](a),r](b))<e. By [W x ; 1.1.2, 1.1.3], we can choose V, (*) 1 V" G V such that V" C V C V, V'AACFand (a, 6) G V " = [ > [0 A 6, 0 V 6] x [a A 6, 0 V 6] C V 7 . (**) We prove that V" <ZU. Let (a, 6) G V". By (**), (0 A 6, a V 6) G V . Then (0, 0A6) = ((a A 6) A (0 A 6) ; , (a V 6) A (a A 6)') = (a A 6, a V 6) A ((a A 6)', (a A 6)7) G V ' A A C V . By (*), we get r](aAb) < e, from which (a, 6) G U£ C U. D COROLLARY (2.7). Le^ rj be a k-submeasure and U a lattice uniformity. 77 is U -uniformly continuous if and only if U(rj) < U. Then We say that a fc-submeasure 77 is a uniform k-submeasure if the following conditions hold: (1) There exists M > 0 such that, for every a, 6, c G L with b A c = 0, 77((aV6) Ac) < Mkr](a). (2) There exist M 1 } M 2 > 0 such that, for every o, 6, cG F, 77((aVc)A(6Vc)) < Mxkr](aAb) and ^((o A c)A(6 A c)) < M2krj(aAb). We want to prove the following result. THEOREM (2.8). Let U be a lattice uniformity on L. (1) For every k > 1, there exists a family {fja} such that U = suipU(r]a). Then: of uniform k-submeasures a (2) If U has a countable base, for every k > 1 there exists a uniform k -submeasure fj such that U = ^(77). (3) If U is generated by a modular function ]i\ L —> G where G is a topological Abelian group, then there exists a family {fia} of uniform submeasures such that U = s\iipU(ft,a). a To prove (2.8), essential tools are the following two results, which hold in any lattice. 408 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES THEOREM (2.9). ([W 3 ; 1.4]) Let k > 1 and U a lattice uniformity. Then U is generated by a family {da : a € A} of pseudometrics with the following properties: (i) For every a,b,c G L, da(aV c, b V c) < da(a, b). (ii) For every a,b,c G L, da (a A c, b A c) < kda (a, b). Moreover, if U has a countable base, we can choose \A\ = 1. THEOREM (2.10). ([F-T 2 ; Theorem 3]) Let \x be a modular function with values in a topological Abelian group G. Then U(\x) is generated by a family of pseudometrics da defined by d a (a, b) = snp{p a (/L(c)—/i(d)) : c, d G [aAb, aVb], c < d} , cYG^4, a,beL, where {pa : a E A} is a family of group seminorms which generate the topology of G. and da have the properties (i) and (ii) of (2.9) with k = 1. LEMMA (2.11). Let k > 1 and d a pseudometric with the following (i) d(a V c,b\/ c) < d(a, b) for every a,b,c G L. (ii) d(a A c, b A c) < kd(a, b) for every a,b,c G L. Then d has the following properties: (1) I/ c < a < b < d ; then d(a, b) < kd(c, d). (2) d(a A b, a) < kd(b, a V b). (3) d(b, a V b) < d(a A b, a ) . (4) d(b, a V b) < d(a, b) < 2kd(a A b, a V b). (5) d ( a A b , a V b ) < 2kd(a,b). (6) d(aAb,0) < kd(aAb,aVb). (7) d ( a A b , a V b ) < d ( a A b , 0 ) . (8) IfbAc= 0, then d ( ( a V b ) Ac,0) < kd(aAb,a) < properties: fc2d(a,0). P r o o f . The proof of (1) - (5) can be obtained in similar way as in [W 3 ; 1.7]. (6): By (ii) and (5), we get d(aAb, 0) = d((a V b) A (a A b)', (a A b) A (a A b)') < kd(aAb,aVb). (7): By (i), d(a A b, a V b) = d((a A b) V 0, (a A b) V (aAb)) < d(aAb, 0 ) . (8): By (ii), (1), (3) and (7), we get d((a V b) A c, 0) = d((a V b) A c, b A c) < kd(a V b, b) < kd(a A b, a) = kd(a A (a A b), a V (a A b)) < kd(aA(a A b), 0) 2 = kd(a A (a A 6)', 0) < k d(a, 0 ) . D 409 ANNA AVALLONE PROPOSITION (2.12). Let k>\ and d be as in (2.11). For a G L, let 77(a) = sup{d(6,0) : be [0,o]}. Then f) is a uniform generated by d. k2 -submeasure and U(f)) coincides with the uniformity Proof. (i) First we prove that, for every a e L, 77(a) = sup {d(6, c) : 6, c G [0, a ] , 6 < c} . (*) Let a e L and denote by 77(a) the right side of (*). The inequality 77(a) < 77(a) is trivial. Let 6, c G [0,o] with 6 < c, and set d = c A 6'. Since c = 6 V d and 6 A d = 0, by (3) of (2.11) we get d(6, c) = d(6,6 V d) < d(b A d, d) = d(0, d) < 77(a), since d < c < a. Hence 77(a) < 77(a). (ii) We prove that 77 is a uniform fc-submeasure. Trivially 77 is monotone and 7/(0) = 0 . L e t ce [0,o V 6], By (1) and by (1), (2), (3) of (2.11), we get d(c, 0) < d(c, c A a) -f d(c A a, 0) < kd(a, a V c) + 77(a) < k2d(a, a V 6) + f)(a) < k2d(a A 6,6) + 77(a) < k2f)(b) + 77(a), from which 77(0 V 6) < k2f)(b) + 77(a). Now let a, 6, c G L with 6 A c = 0 and choose e, / G [0, (a V 6) A c] with e < / • By (ii), (1) and (3) of (2.11), we get d(e, / ) < fcd(0, (a V 6) A c) = fcd(6 A c, (a V 6) A c) < fc2d(6, a V 6) < k2d(a A 6, a) < fc277(a). Hence fj((a V 6) A c) < fc277(a). Let a,6,c G L and d < ( o V c ) A ( 6 V c ) . By (i), (1), (4), (5), (6) and (7) of (2.11), we get d(0, d) < fcd(0, (a V c) A(6 V c)) < 2fc3d(a V c, 6 V c) < 2fc3d(a, 6) < 4fc4d(a A 6, a V 6) < 4fc 4 d(aA6,0) < 4k4f)(aAb) , from which 77((a V c) A(6 V c)) < 4fc477(aA6). Now let d < ( a A c ) A ( 6 A c ) . By (ii), (1), (4), (5), (6) and (7) of (2.11), we get d(0, d) < fcd(0, (o A c) A(6 A c)) < 2fc3d(a A c, 6 A c) < 2fc4d(a, 6) < 4fc5d(a A 6, a V 6) < 4fc 5 d(aA6,0) < 410 4k5f)(aAb), LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES from which f)((a A c) A(b A c)) < 4k5fj(aAb). (iii) Denote by V the uniformity generated by d. Since, for every a G L, d(0,a) < 77(a) and, by (1) of (2.11), d(Oya) < £ implies 77(a) < ke, then U(fj) and V have the same base of 0-neighbourhoods. Hence, by (2.5), U(fj) = V. • P r o o f o f (2.8) . By (2.12), if U is generated by a family {da} of pseudometrics with the properties (i) and (ii) of (2.9) with k > 1, then there exists a family {fja} of uniform k2-submeasures such that U = s\npU(f]a). Then (1) a and (2) follow from (2.9), and (3) follows from (2.10). • R e m a r k . By (2.10) and (2.12), we get that the submeasures \xa in (2.8) (3) are defined by ^ « ( a ) = r f a ( M ) = s u p { p a ( / i ( c ) - / i ( d ) ) : c , d G [ 0 , a ] , c < d] = sui{pa(fi(b) - fi(0)) : 6 E [ 0 , a ] } for every a £ L. R e m a r k . In [W2] it is proved that, in general, the conclusions of (2.9) fail if k = l. A consequence of (2.8) is a characterization of lattice uniformities with (a) by means of the family of k-submeasures which generate them. Property (<r) has been introduced in [W 2 ; (3.1)] for arbitrary lattices and it is an essential tool for many results, for example to obtain that a uniform lattice is a Baire space ([W 2 ; 3.15)], or to obtain extension theorems (see [W 2 ; 8.2.1] and its applications in [A-D]). We say that a lattice uniformity U has (a) if, for every U GM, there exists a sequence {Un} in U with the following property: if an f a or an I a and (avaj) e Un for z, j > n , then (a1,a) e U. By [W 2 ; 3.3 ], if U has a countable base, then U has (a) if and only if every monotone Cauchy sequence {an} in (L,U), with an t a or an 4- a, converges to a in (L,U). Property (a) is connected with the a-order continuity by the following result of [W 2 ; (8.1.2)], which holds in any lattice. PROPOSITION (2.13). (1) If U is exhaustive and has (a), then U is a-o.c. (2) If U is a-o.c, then U has (a). (3) 1/ (L, <) is a-complete, then U is a-o.c. if and only if U is exhaustive and has (a). If L is a Boolean algebra, the uniformity generated by a submeasure 77 has (a) if and only if 77 is a -subadditive. Moreover the uniformity generated by a 411 ANNA AVALLONE Frechet-Nikodym topology r has (<J) if and only if r is generated by a family of a -subadditive submeasures (see [W 2 ; 3.17]). We prove that a similar characterization holds for lattice uniformities on orthomodular lattices. We need the following definitions. a We say that a function rj: L —r [0,-f-oo] has (a) if a n t or an \. a and limry(a n Aa m ) = 0 imply limry(a n Aa) = 0. n,m n,m Then, if rj is a fc-submeasure, rj has (a) if and only if U(rj) has (a). If fc > 1, we say that a function rj: L -> [0,+co] is a-k-subadditive if, for every sequence {an} in L such that V a ex n s i ^ s m < k ]T) L, rj\\J an) n ^n ' rj(an). n=l A a-fc-subadditive fc-submeasure is called a -k -submeasure. The following result has been proved in [A-D; (2.2)] for a-subadditive submeasures and the proof is the same for cr-fc-submeasures. PROPOSITION (2.14). Every a-k-submeasure has (a). LEMMA (2.15). Let fc, d and fj be as in (2.12). Then, if r) has (a), r) is a 2 a-k -submeasure. P r o o f . By (2.12), U(fj) coincides with the uniformity generated by d. Since U(rj) has (<T), by [W 2 ; 3.3] we get that, for every sequence {a n } in L such that a = \Jan exists in L, d(ax,a) a < k Y, ^ n ^ n + i ) - Let { n } n C L be such that n=l n oo a = V a n exists in L. Set bn = V a i ? where a 0 = 0. Then a = V bn. If b < a, n i=0 n=0 by (2.11) we get d(6,0) <fcd(a,b 0 ) ОО п=0 ОО ОО / п п п=0 ^г=0 г=0 ОО <к^<1(0,ап+1)<к^гЦап+1), n=0 n=0 oo 2 from which 77(a) < fc E r?(a n ). D n=l By (2.8) (3) and (2.15), we get: COROLLARY (2.16). Let fc > 1. Then every k-submeasure with (a) is equiv­ alent (i.e. generates the same uniformity) to a a -fc-submeasure. Now we can give a characterization of lattice uniformities with 412 (a). LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES COROLLARY (2.17). Let k > 1 and U be a lattice uniformity. lowing conditions are equivalent: (1) U has (a). (2) There exists a family of a -k-submeasures Then the fol- which generates U. Proof. (1) => (2): By [W 2 ; 3.2], there exists a family {Ua : a e A) of pseudometrizable lattice uniformities with (cr) such that U = sup Ua. By (2.8) a£A and (2.15), for each a e A we can choose a cr-fc-submeasure A a such that ua=u(AJ. (2) =>* (1): By (2.14), U is the supremum of a family of lattice uniformities with (a). Hence U has (a) by [W 2 ; 3.2]. • 3. Uniform D-lattices In this section, L is a D-lattice and G is a topological Abelian group. A measure \i\ L —> G is called modular measure if it is a modular function. We prove that the lattice uniformity generated by every modular measure on L makes uniformly continuous the difference operation of L. As an example of consequence of this result, we obtain the equivalence between Vitali-Hahn-Saks and Brooks-Jewett theorems for modular measures on D-lattices. Following the terminology of [P-J, a lattice uniformity on L is called D-lattice uniformity if © is uniformly continuous. By the definition of 0 , it is clear that a D-lattice uniformity makes 0 uniformly continuous, too. For [/, V C L x L, we set UeV = {(a 0 c, bed) : c<a, UeV = { ( a 0 c , b 0 d ) : a _ L c , bid, d<b, (a,b)eU, (c,d)eV), (a,b)eU, (c,d)eV). Then 0 is uniformly continuous if and only if, for every U eU, there exists V e U such that V eV CU, and 0 is uniformly continuous if and only if, for every U GW, there exists V eU such that V 0 V C U. PROPOSITION (3.1). Let U be a lattice uniformity. Then U is a D-lattice uniformity if and only if, for every U eU, there exists V eU such that V © A C U and AeV CU. P r o o f . It is clear that the condition is necessary. We prove that it is sufficient, too. Let U eU and choose V,V1,V2 eU such that VoVoV cu, V;©ACV, A e vi c v, V2AV2CV1. 413 ANNA AVALLONE We prove that V2 © V2 C U. Let (a, 6), (c, d) G V2 such that c < a and d < 6. By (c, c A d) E Vj, we get (a 0 c, a 0 (c A d)) G A 0 Vi C V . (1) By (c A d, d) G Vj, we get ( 6 © ( c A d ) , 6 © d ) G A e V i CV. (2) Moreover, since (a, 6) G V2 C V x , we have ( a © ( c A d ) , 6 © ( c A d ) ) G Vx © A C V . (3) By (1), (2) and (3), we get (a © c, 6 0 d) G V o V o 17 C U. In similar way, we can prove that, if U is a lattice uniformity, then 0 is uniformly continuous if and only if, for every U G W, there exists V G W such that V0ACU. • THEOREM (3.2). If \i\ L -+ G is a modular measure, then U(\i) is a D-lattice uniformity and a base of U(\i) is the family consisting of the sets Aw = {(a, 6) G L x L : fi(c) G W for all c < aA6} , where W is a 0-neighbourhood in G. P r o o f . For every 0-neighbourhood W in G, set [/^ = { ( a , b) e Lx L: fi(c) - /i(d) G W for all c, d G [a A 6, a V 6], c > d] . (i) First we prove that Aw = Uw. Let (a, 6) G Aw and c,rfG [a A 6, a V 6], with c > d. By the definition of © and (1.1)(3), we get cG d < (a V 6) © (a A 6) = aA6, from which \i(c © d) G W. Since, by (1.1)(2), c = d 0 (c © d), we get ji(c) - /x(d) = \x(c © d) G TV, from which (a, 6) G U^ . Now let (a, 6) G Uw and c < aA6. By (1.1) (2), we can find d G L such that aA6 = c 0 d and therefore a V 6 = (a A 6) 0 (aA6) = (a A 6) 0 c 0 d. By (1.1) (4), we get c = (a V 6) © ((a A 6) 0 d ) . Then (a A 6) 0 d G [a A 6, a V 6] and lx(c) = \i(aVb) - \i((aNb) 0 d) G W , from which (a, 6) G A ^ . (ii) Now we prove that aA6 = (a © c) A(6 © c) for every a, 6 > c. Set d = a © c and e = 6 © c. By (1.1)(1), we get d V e = (a V 6) © c and d A e = (a A 6) © c. Then (a © c) A(6 © c) = dAe = ((a V 6) © c) © ((a A 6) © c) . Since c < aA6 < aV6, by (1.1) (5), we get (a©c)A(6©c) = (aV6)©(aA6) = aA6. (iii) We prove that aA6 = (c © a)A(c © 6) for every a, 6 < c. By (1.1) (1),(6) and by the definition of ©, we get ( c © a ) A ( c © 6 ) = ((cQa) V ( c © 6 ) ) © ( ( c © a ) A ( c © 6 ) ) = (c©(aA6)) © (c©(aV6)) = (aV6)©(aA6) = aA6. 414 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES (iv) By (ii) and (iii), we get Aw 0 A C Aw and A 0 Aw C Aw. Then, by (i), U(\i) is a D-lattice uniformity and {Aw : W is a 0-neighbourhood in G} • is a base of U(\x). As consequence of (3.2), we obtain — in a similar way as in [A-L] for modular functions on orthomodular lattices — the equivalence in any D-lattice between Brooks-Jewett and Vitali-Hahn-Saks theorems for modular measures. First we recall the definitions which we need. We say that a family K of G-valued measures on L is uniformly exhaustive if, for every orthogonal sequence {an} in L, /i(a n ) —> 0 in G uniformly for \x £ K. If G' is another topological Abelian group and A: L -> G! is a measure, we say that K is X-equicontinuous if, for every 0-neighbourhood W in (7, there exists a 0-neighbourhood W! in G! such that, if a E L and X(b) E W! for every b < a, then /i(a) E TV for every /z E if. In particular, if /L: L -» (7 is a measure, we say that lz is exhaustive or A-continuous if K = {/z} is exhaustive or A-equicontinuous, respectively. If A is a modular measure, by (3.2), a base of 0-neighbourhoods in U(X) is the family consisting of the sets {a E L : X(b) E W for all b < a} , where W is a 0-neighbourhood in G. Then, in this case, if we denote by T ^ the topology of the uniform convergence in GK, we have that K is A-equicontinuous if and only if the function v = ( / i ) ^ ^ - : (L,U(X)) -> (GK,TQO) is continuous and K is is exhaustive. uniformly exhaustive if and only if v: L —r (GKJroo) The notion of exhaustive measure given here is a particular case of the notion of x0 -exhaustive measure given in [D-P] and we need it in the proof of (3.7). The following result allows to prove in a standard way (see (3.4)) that this notion is equivalent to that of H. W e b e r in [W 5 ], which we need in the proof of (3.6). LEMMA (3.3). Let a0ialy... , a n be in L such that aQ < ax < • • • < a n and set bi = a{ Oai_l for every i E { 1 , . . . , n } . Then {bli... , 6 n } is orthogonal and bx 9 • • • © bn = an 0 a 0 . P r o o f . Since axOa0 < ax < a 2 , by (1.1)(9), 6X ©6 2 = (ax Ga0)^(a2Qa1) exists and it is equal to a 2 0 (a x 0 (ax 0 a 0 )) = a2 Q a0. Then the assertion is true for n = 2. Now suppose that the assertion is true for n — 1. Since b10---0bri_1 = a n _ i e a 0 < an_l < a n , by (1.1)(7), we get that 6 1 © - - - © 6 n = (bx 0 • • • 0 6 n _i) © (a n 0 « n _ i ) exists and it is equal to an 0 (an_1 0 ( o n _ 1 0 D K i©ao)) =«n©aoPROPOSITION (3.4). Let /x: L —r G 6e a measure. Then the following tions are equivalent: (1) /i is exhaustive. (2) For e-venl monotone sequence {an} in G. condi- in L, {n>(an)} is a Cauchy sequence 415 ANNA AVALLONE (3) For every increasing sequence {an} in L, {/i(aj} in G. is a Cauchy sequence Proof. bn (1) => (2): (i) Let {an} be an increasing sequence in L and, for each n G JV, set = an 9 an__1, where a0 = 0. By (3.3), {b n } is orthogonal. Then /x(aj - MK~I) = M U - > ° (ii) Now let {aj be a decreasing sequence in L and set bn = an . {bn} is an increasing sequence. By (i), /x(aj -/x(an_x) = /x(bn_x) -/Ahn) (2) -=> (3) is trivial. (3) => (1): Let {an} be an orthogonal sequence in L and set bx ^ n = ® ai f° r e v e r Y rc > 2. Then {b n } is increasing and an © bn = Then ~> °= 0, bn+1. i_n — 1 Therefore M(an) =-- /i(b n + 1 ) - M b J -> 0. D PROPOSITION (3.5). Let /x: L -» G be a measure and U a D-lattice uniformity on L. Then /x is continuous in 0 if and only if /x is uniformly continuous. P r o o f . Let TV, W be 0-neighbourhoods in G with W - W C VV, and choose U eU such that ( a , 0 ) G U = * L<a)GlV'. (*) Let V,V GU such that V e A C U and, for every (a, b) G V, [a A b, a V b] x [aAb,aVb] C V (see[W 2 ; 1.1.3]). Let (a,b) G V'. We prove that /x(a)-/x(b) G TV. Set c = a 0 (a A b) and d = bQ(aAb). Then (c, 0) = (a, a A b) 0 (a A b, a A b) G V 0 A C [/. By (*), we get /i(c) G W . In similar way we obtain /x(d) G TV7. Then /x(a) - /x(b) = /x(c) - /x(d) eW. • We say that L has the Vitali-Hahn-Saks property ( VHS-property) if, for every topological Abelian group G', for every G"-valued modular measure A on L and for every sequence {/xn : n G N} of exhaustive A-continuous G -valued modular measures on L which is pointwise convergent on L to a function fi0, {/xn : n e Af U {0}} is A-equicontinuous. We say that L has the Brooks-Jewett property (BJ-property) if, for every sequence {/xn : n G JV} of exhaustive G-valued modular measures on L which is pointwise convergent on L to a function /x0 , {/xn : n G AfU {0}} is uniformly exhaustive. Then, if we denote by c(G) the space of all convergent sequences in G, by A the topology of the pointwise convergence in c(G) and by A^ the topology of the uniform convergence in c(G), it is clear (see (3.5)) that L has the VHS-property if and only if, for every topological Abelian group G' and for every modular measure X: L —> G', every exhaustive A-continuous modular 416 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES measure ii: L —> (c(G), A ) is A-continuous with respect to A ^ , and L has the BJ-property if and only if every exhaustive modular measure \x: L —> (c(C?), A ) is A^-exhaustive. Using (3.4) and (3.5), the equivalence between VHS and BJ properties can be proved in a similar way as in [A-L; (1.2.15)]. THEOREM (3.6). L has the VHS-property if and only if L has the BJ-property. Proof. <£=: The VHS-property follows from the BJ-property as consequence of the following result of [W 5 ; (6.2)], which holds in an arbitrary lattice: if U is a lattice uniformity and K is a uniformly exhaustive family of ZY-continuous modular functions, then K is ZY-equicontinuous. = > : Let \i: L -> (c(G),A ) be an exhaustive modular measure and set U — U(\i). By [W 5 ; 3.6], U is exhaustive since ii is exhaustive. Since ii: (L,U) —r (c(G),A ) is uniformly continuous, by the VHS-property, ii: is continuous, too. By (3.5), \x: (L,U) -> ^(G),^) (L,U) -> {c(G)^oo) is uniformly continuous. Then, since U is exhaustive, by (3.4) we obtain that \i: L —> ( ^ G ^ A ^ ) is exhaustive, too. Therefore L has the BJ-property. • In [D-P; (12.4)], the Brooks-Jewett theorem has been proved for measures on quasi- a -complete D-posets (i.e. on D-posets L such that, for every orthogonal sequence {an} in L, there exists a subsequence {an : n £ M} such that 0 a{ iei exists for every I C M). Then, by [D-P] and (3.6), we obtain the Vitali-HahnSaks theorem for modular measures on quasi-a-complete D-lattices. COROLLARY (3.7). If L is a quasi-a -complete D-lattice, then L has BJ and VHS properties. We conclude remarking that we can derive by [W 5 ; (4.1)] a characterization of modular measures with weakly relatively compact range. PROPOSITION (3.8). Let G be a complete locally convex linear space and /i: L -> G a modular measure. Then li(L) is weakly relatively compact if and only if \i is exhaustive. In particidar, ifG^W1, \x is exhaustive if and only if ft is bounded. P r o o f . In [W 5 ; (4.1)], it is proved that, if L0 is an arbitrary lattice, the equivalence between exhaustivity and relative weak compactness of the range holds for any modular function \x: L0 —> G which satisfies the following condition: for every a0 < • • • < an in L0 and for every I C { 1 , . . . , n } , ]CN a i)~M a i-i)] eM^o)- (*) i£l 417 ANNA AVALLONE Therefore we have only to observe that (*) holds for any measure on L. For every i < n , set bi = at Q a>i-\- By (3.3), for every i < n , {b1,...,bi} is a orthogonal and bx 0 • • • ® b{ = ai 0 a0. Let I C { 1 , . . . , n}. Then £ [M ;) "" ІЄ/ мK-i)] = ENв. o0)-Moi-i ao)] = E H Ф O - / * ! ІЄ/ EÍĽ/^jiЄІ 3<i iЄ/ E / ^ ) 1 = E M M = M( j<i-l iЄ1 V iЄ1 L V j<i У 0Є/І(L). S<i-1 Ь-)1 / J D 7 REFERENCES [A! [A 2 [A-B-C [A-D [A-L; [A-W; [в-w; [B-L-W [B-F [C-K г [C-K 2 [D-D-P AVALLONE, A . : Liapunov theorem for modular functions, Internat. J. Theoгet. Phys. 3 4 (1995), 1197-1204. AVALLONE, A . : Nonatomic vector-valued modular functions, Annal. Soc. Matli. Polon. Ser. I: C o m m e n t . Math. 39 (1999), 37-50. AVALLONE, A — B A R B I E R I , G.—CILIA, R.: Control and separating points of modular functions, Math. Slovaca 4 9 (1999), 155-182. AVALLONE, A . — D E S I M O N E , A . : Extensions of modular functions on orthomodular lattices, Ital. J. P u r e Appl. M a t h . 9 (2001), 109-122. Uniform boundedness AVALLONE, A . — L E P E L L E R E , M. A . : Modular functions: and compactness, Rend. Circ. M a t . Palermo (2) 4 7 (1998), 221-264. generated by filters, J. M a t h . AVALLONE, A . — W E B E R , H . : Lattice uniformities Anal. Appl. 2 0 9 (1997), 507-528. B A R B I E R I , G . — W E B E R , H . : A topological approach to the study of fuzzy measures. In: Functional Analysis and Economic Theory, Springer, Samos, 1996, p p . 17-46. B A R B I E R I , G . — L E P E L L E R E , M. A — W E B E R , H . : The Hahn decomposition theorem for fuzzy measures and applications, Fuzzy Sets and Systems (To appear). B E N N E T T , M. K . — F O U L I S , D. J . : Effect algebras and unsharp quantum logics, Found. Phys. 2 4 (1994), 1331-1352. C H O V A N E C , F . — K Ô P K A , F . : D-posets, M a t h . Slovaca 4 4 (1994), 21-34. C H O V A N E C , F . — K Ô P K A , F . : D-lattices, Intern. J. Theoret. Phys. 3 4 (1995), 1297-1302. D E LUCIA, P . — D V U R E Č E N S K I , A . — P A P , E . : On a decomposition theorem and its applications, M a t h . Japonica 4 4 (1996), 145-164. D E LUCIA, P . — P A P , E . : Nikodym convergence theorem for uniform space-valued [D-P functions defined on D-poset, M a t h . Slovaca 4 5 (1995), 367-376. P: D R E W N O W S K I , L.: Topological rings of sets, continuous set functions, integration I, II, Bull. Polish. Acad. Sci. M a t h . 2 0 (1972), 269-286. F L E I S C H E R , I . — T R A Y N O R , T . : Equivalence o] group-valued measures on an ab[F-T stract lattice, Bull. Polish. Acad. Sci. M a t h . 28 (1980), 549-556. [F-T. F L E I S C H E R , L — T R A Y N O R , T . : Group-valued modular functions, Algebra Univeгsalis 1 4 (1982), 287-291. [F-G-P F O U L I S , D. J . — G R E E C H I E , R. J . — P U L M A N N O V Á , S.: The center of an effect algebra, Order 12 (1995), 91-106. G R A Z I A N O , M. G . : Uniformities of Fréchet-Nikodym type on Vitali spaces, Semi[G group Forum (To appear). 418 LATTICE UNIFORMITIES ON ORTHOMODULAR STRUCTURES [K] K A L M B A C H , G . : Orthomodular Lattices. London M a t h . Soc. Monographs 18, Academic Press Inc., London, 1983. [L] L I P E C K I , Z . : Extensions of additive set functions with values in a topological group, Bull. Polish. Acad. Sci. M a t h . 2 2 (1974), 19-27. [ P j P A L K O , V . : Topological difference posets, M a t h . Slovaca 4 7 (1997), 211-220. [P 2 ] P A P , E . : Null-Additive Set Functions, Kluwer Academic Publ, Dordrecht, 1995. [P-P] P T K K , P . — P U L M A N N O V A , S.: Orthomodular Structures as Quantum Logics, Kluwer Acad. Publ., Dordrecht, 1991. [P 3 ] P U L M A N N O V A , S.: On connection among some orthomodular structures, Demonstratio M a t h . 3 0 (1997), 313-328. [R-N] R I E C A N , B — N E U B R U N N , T . : Integral, Measure and Ordering, Dordrecht, Kluwer Acad. Publ., 1997. [R] R I E C A N O V A , Z.: A topology on a quantum logic induced by a measure. In: Proceed. Conf. Topology and Measure V, Wissensch Beitr, Greifswald, 1988, p p . 126-130. [W-J W E B E R , H.: Group-and vector-valued-s-bounded contents. In: Lecture Notes in M a t h . 1089, Springer, New York, 1984, p p . 181-198. [W 2 ] W E B E R , H . : Uniform lattices I: A generalization of topological Riesz space and topological Boolean rings; Uniform lattices II, A n n . M a t . P u r a Appl. 1 6 0 ; 1 6 5 (1991; 1993), 347-370; 133-158. [W 3 ] W E B E R , H . : Metrization of uniform lattices, Czechoslovak M a t h . J. 4 3 (1993), 271-280. [W 4 ] W E B E R , H . : Lattice uniformities and modular functions on orthomodular lattices, Order 1 2 (1995), 295-305. [W 5 ] W E B E R , H.: On modular functions, Funct. Approx. C o m m e n t . M a t h . 2 4 (1996), 35-52. [W 6 ] W E B E R , H . : Valuations on complemented lattices, Intern. J. Theoret. Phys. 3 4 (1995), 1799-1806. [W 7 ] W E B E R , H . : Complemented uniform lattices, Topology Appl. 1 0 5 (2000), 47-64. [W 8 ] W E B E R , H . : Two extension theorems. Modular functions on complemented lattices Preprint. Received December 16, 1999 Revised May 30, 2000 Dipartimento di Matematica Universita della Basilicata via Nazario Sauro 85 1-85100 Potenza ITALY E-mail: [email protected] 419