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Tensor Products and the Loomis–Sikorski Theorem for MV-Algebras
Advances in Applied Mathematics 22, 227]248 Ž1999. Article ID aama.1998.0631, available online at http:rrwww.idealibrary.com on
Tensor Products and the Loomis]Sikorski Theorem for MV-Algebras Daniele Mundici* Department of Computer Science, Uni¨ ersity of Milan, Via Comelico 39-41, 20135 Milan, Italy E-mail: [email protected] Received May 15, 1998; accepted September 19, 1998
MV-algebras are the models of the time-honored equational theory of magnitudes with unit. Introduced by Chang as a counterpart of the infinite-valued sentential calculus of Łukasiewicz, they are currently investigated for their relations with AF C*-algebras, toric desingularizations, and lattice-ordered abelian groups. Using tensor products, in this paper we shall characterize multiplicatively closed MV-algebras. Generalizing work of Loomis and Sikorski, we shall investigate the relationships between s-complete multiplicatively closed MV-algebras, Q 1999 Academic and pointwise s-complete MV-algebras of w0, 1x-valued functions. Press
1. INTRODUCTION Introduced by Chang in w2, 3x, MV-algebras are the models of the equational theory of magnitudes with a distinguished archimedean unit w1, pp. 106]107x, in a sense that will be made precise by Theorem 1.4 below. The monograph w5x, as well as the survey paper w4x, provide all the necessary background information. The prototypical MV-algebra is given by the real unit interval w0, 1x equipped with the operations ! x s 1 y x,
x [ y s min Ž 1, x q y . ,
x( y s max Ž 0, x q y y 1 . . Ž 1.
MV-algebras are the algebras satisfying precisely the same equations that are satisfied by w0, 1x. Equivalently, by Chang’s completeness theorem w2, *Partially supported by CNR-GNSAGA Project on Symbolic Computation, and by the European Cost Action 15 on Many-valued logics for computer science applications. 227 0196-8858r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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3x, an MV-algebra A s Ž A, 0, 1, !, [ , ( . is an abelian monoid Ž A, 0, [ . with an involution ! such that x [ 1 s 1 s !0, x( y s !Ž! x [ ! y ., and y [ !Ž y [ ! x . s x [ !Ž x [ ! y .. An MV-algebra is called tri¨ ial iff it only consists of the zero element. Unless otherwise specified, all MV-algebras considered in this paper shall be nontrivial. Every MV-algebra A s Ž A, 0, 1, !, [ , ( . is further equipped with the operations k and n given by a k b s !Ž! a [ b . [ b and a n b s !Ž! a k ! b .. The binary relation F on A given by a F b iff a( ! b s 0, is a partial order; as a matter of fact, Chang w2x proved that Ž A, k, n, 0, 1. is a distributive lattice with least element 0 and greatest element 1; this is called the underlying lattice of A. We say that MV-algebra A is s-complete iff its underlying lattice is s-complete, i.e., every nonempty countable subset of A has a supremum in A. Ideals are kernels of homomorphisms. Unless otherwise specified, every ideal I of an MV-algebra A considered in this paper shall be assumed to be proper, i.e., I / A. An ideal I is said to be maximal iff there is no ideal of A strictly containing I Žother than the improper ideal A.. Thus, maximal ideals are automatically proper. We let M Ž A. denote the set of all maximal ideals of A. Since I is maximal iff it is maximal for the property of not containing the unit element 1, we easily conclude that M Ž A. / B. The following result is an MV-algebraic variant of Holder’s theorem ¨ w1, 2.6, 12.2.1x. THEOREM 1.1. For any MV-algebra A and maximal ideal I of A, there is a unique isomorphism i I of the quotient MV-algebra ArI onto a subalgebra of w0, 1x. Let r I : A ª ArI be the quotient map. Then the map I ¬ i I ( r I is a one-to-one correspondence of the set of maximal ideals of A onto the set of homomorphisms of A into w0, 1x. The in¨ erse of this map sends any such homomorphism into its kernel. Proof. See, e.g., w5, Theorems 2.4.14, 2.5.7x. The set RadŽ A. s F K < K g M Ž A.4 is called the radical of A. A is said to be semisimple iff RadŽ A. s 04 . Equivalently, for each nonzero element x g A, there is a homomorphism h : A ª w0, 1x with h Ž x . / 0. In order to define the spectral Žalso called hull-kernel . topology on the set M Ž A. of all maximal ideals of A, for every ideal I of A, let OI s K g M Ž A. < K W I 4 . Then the collection of all sets of the form OI is a compact Hausdorff topology on M Ž A.. Any closed subset is of the form CI s K g M Ž A. < K = I 4 , for some ideal I of A. For any compact Hausdorff space X, let us agree to denote by C Ž X . the MV-algebra of all continuous w0, 1x-valued functions on X with pointwise
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229
operations as in Ž1.; let, as usual, w0, 1x X denote the MV-algebra of all w0, 1x-valued functions on X. In the light of Theorem 1.1, let the map a g A ¬ a* g w0, 1x X be defined by a* Ž I . s Ž i I ( x I . Ž a . g w 0, 1 x
for all I g M Ž A . .
Ž 2.
The following theorem gives a concrete representation of semisimple MV-algebras; its proof is a fundamental consequence of Chang’s completeness theorem. THEOREM 1.2. Let A be a semisimple MV-algebra. Then the map a ¬ a* is an isomorphism of A onto a separating subalgebra A* of C Ž M Ž A.., in the sense that whene¨ er I, J are distinct maximal ideals of A, there are elements a*, b* g A* with a*Ž I . / b*Ž J .. COROLLARY 1.3. An MV-algebra A is semisimple iff A is isomorphic to a separating MV-algebra of w0, 1x-¨ alued continuous functions defined o¨ er some compact Hausdorff space, iff A is isomorphic to an MV-algebra of w0, 1x¨ alued functions defined o¨ er some set. We assume familiarity with lattice-ordered abelian groups, for short
l-groups, for which we refer to w1x. For any l-group G, an element u g G
is said to be a strong unit of G iff for all g g G there is an integer n G 1 such that nu G g. By a morphism f : Ž G, u. ª Ž G9, u9. we mean a group homomorphism f : G ª G9 that also preserves the lattice structure and satisfies the condition f Ž u. s u9. THEOREM 1.4 w7x. For any l-group G with a strong unit u, let G Ž G, u. be the unit inter¨ al w0, u x s h g G < 0 F h F u4 , equipped with the operations ! g s u y g, g [ h s u n Ž g q h., and g(h s 0 k Ž g q h y u.. Then Ži. A s Žw0, u x, !, [ , ( . s G Ž G, u. is an MV-algebra. Žii. Letting, for any morphism f : Ž G, u. ª Ž G9, u9., G Ž f . be the restriction of f to w0, u x, then G is a categorical equi¨ alence Ž i.e., a full, faithful, dense functor . from l-groups with strong unit to MV-algebras. Žiii. The lattice operations on A agree with those of G. Živ. The map J ¬ J l w0, u x is an isomorphism between the lattice of l-ideals of G Ž equipped with set-theoretic inclusion. and the lattice of ideals of A Ž also equipped with inclusion.. All infinite suprema and infima are preser¨ ed by this correspondence. There is a natural isomorphism between G Ž GrJ, urJ . and G Ž G, u.rŽ J l w0, u x.. Thus, in particular, up to isomorphism, every MV-algebra A can be identified with the unit interval of a unique l-group G with strong unit u;
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in symbols, A s G Ž G, u . .
Ž 3.
We say that G is the l-group with strong unit u corresponding to A. As an application of the G functor we have the following. PROPOSITION 1.5. An MV-algebra is s-complete iff e¨ ery bounded sequence of elements in its corresponding l-group G, has a supremum in G. Any s-complete MV-algebra A is semisimple. Proof. The first statement follows from the preservation properties of the G functor w9, Proposition 9.4.3x. The second statement now immediately follows from its counterpart for l-groups Žsee, e.g., w1, Proposition 11.2.2x., in the light of Theorem 1.4Živ..
2. BIMORPHISMS OF MV-ALGEBRAS Let A and B be MV-algebras, and A = B be their cartesian product Žas a set.. DEFINITION 2.1. A bimorphism b of A = B into an MV-algebra C is a function b : A = B ª C such that b Ž1, 1. s 1 and for all a, a1 , a2 g A and b, b1 , b 2 g B, we have ŽI. b Ž a, 0. s 0 s b Ž0, b .; ŽII. b Ž a, b1 k b 2 . s b Ž a, b1 . k b Ž a, b 2 .; b Ž a1 k a2 , b . s b Ž a1 , b . k b Ž a2 , b .; ŽIII. b Ž a, b1 n b 2 . s b Ž a, b1 . n b Ž a, b 2 .; b Ž a1 n a2 , b . s b Ž a1 , b . n b Ž a2 , b .; ŽIV. if b1(b 2 s 0, then b Ž a, b1 . ( b Ž a, b 2 . s 0 and b Ž a, b1 [ b 2 . s Ž b a, b1 . [ b Ž a, b 2 .; symmetrically, if a1(a2 s 0, then b Ž a1 , b . ( b Ž a2 , b . s 0 and b Ž a1 [ a2 , b . s b Ž a1 , b . [ b Ž a2 , b .. We denote by bimŽ A, B, C . the set of all biomorphisms b : A = B ª C. If h : C ª C9 is a homomorphism of MV-algebras and b g bimŽ A, B, C ., then h ( b g bimŽ A, B, C9.. To increase readability, we shall adopt the usual convention that ! is more binding than ( , the latter being more binding than [. We also assume that the lattice operations k and n are less binding than any other operation. Following Chang w2x, in every MV-algebra A we define the distance function distŽ x, y . by dist Ž x, y . s x( ! y [ ! x( y.
Ž 4.
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
PROPOSITION 2.2. Ži. Žii. Žiii.
231
Let b g bimŽ A, B, C .. Then
if b1 F b 2 , then b Ž a, b1 . F b Ž a, b 2 .; b Ž a, ! b . s b Ž a, 1. ( ! b Ž a, b .; b Ž a, b1 [ b 2 . s b Ž a, 1. n Ž b Ž a, b1 . [ b Ž a, b 2 ...
Proof. Ži. This is an immediate consequence of ŽIII. in the above definition. To prove Žii., from b( ! b s 0 and b [ ! b s 1 we get by ŽIV. 0 s b Ž a, b . ( b Ž a, ! b .
whence b Ž a, ! b . F ! b Ž a, b .
Ž 5.
and
b Ž a, ! b . [ b Ž a, b . s b Ž a, 1 . .
Ž 6.
By Ži., b Ž a, b . F b Ž a, 1., i.e., ! b Ž a, 1 . ( b Ž a, b . s 0.
Ž 7.
Recalling w2, 3.15x, we get b Ž a, b . [ distŽ b Ž a, 1., b Ž a, b .. s b Ž a, 1., and hence, b Ž a, b . [ b Ž a, 1. ( ! b Ž a, b . [ ! b Ž a, 1. ( b Ž a, b . s b Ž a, 1., whence by Ž7.
b Ž a, 1 . ( ! b Ž a, b . [ b Ž a, b . s b Ž a, 1 . .
Ž 8.
From Ž6. and Ž8. together with the inequalities b Ž a, ! b . F ! b Ž a, b . and b Ž a, 1. ( ! b Ž a, b . F ! b Ž a, b ., it follows that cancellation can be applied Žsee w2, 1.13x., whence b Ž a, ! b . s b Ž a, 1. ( ! b Ž a, b ., as required. Finally, to prove Žiii. let us write
b Ž a, b1 [ b 2 . s b Ž a, b1 k Ž b1 [ b 2 . . s b Ž a, b1 [ Ž ! b1( Ž b1 [ b 2 . . . s b Ž a, b1 . [ b Ž a, ! b1( Ž b1 [ b 2 . . s b Ž a, b1 . [ b Ž a, ! b1 n b 2 . s b Ž a, b1 . [ Ž b Ž a, ! b1 . n b Ž a, b 2 . . s Ž b Ž a, b1 . [ b Ž a, ! b1 . . n Ž b Ž a, b1 . [ b Ž a, b 2 . . s b Ž a, 1 . n b Ž a, b1 . [ b Ž a, b 2 . , as required. For every element w in an MV-algebra A, the inter¨ al MV-algebra A w s Žw0, w x, 0, w, !w , [w , (w . is obtained by equipping the set w0, w x s x g A < 0 F x F w4 with the operations !w x s w( ! x, x [w y s w n Ž x [ y . , x(w y s !w Ž !w x [w !w y . .
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Note that A w is a trivial MV-algebra iff w s 0. For any b g bimŽ A, B, C ., a g A and b g B, we define the maps b a : B ª C and b b : A ª C by stipulating that for all x g A and y g B,
b a Ž y . s b Ž a, y .
and
b b Ž x . s b Ž x, b . .
Ž 9.
PROPOSITION 2.3. Let b g bimŽ A, B, C . and a g A. Then b a is a homomorphism of B into the inter¨ al MV-algebra w0, b Ž a, 1.x. Similarly, for each b g B, b b is a homomorphism of A into w0, b Ž1, b .x. Proof. This is essentially the content of Proposition 2.2Žii. ] Žiii., upon noting that, by Ži., rangeŽ b a . Žresp., rangeŽ b b .. is contained in w0, b Ž a, 1.x Žresp., in w0, b Ž1, b .x.. THEOREM 2.4. Let A and B be semisimple MV-algebras, respecti¨ ely identified with separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .. using the maps a ¬ a* and b ¬ b* of Theorem 1.2 Ž whence in particular a* s a and b* s b .. Then for each b g bimŽ A, B, w0, 1x., there is a unique pair of maximal ideals Ib g M Ž A. and Jb g M Ž B . such that, for all a g A and b g B,
b Ž a, b . s a Ž Ib . ? b Ž Jb . , where ? denotes natural pointwise multiplication. The function b ¬ Ž Ib , Jb . is a one-to-one correspondence of bimŽ A, B, w0, 1x. onto M Ž A. = M Ž B .. Proof. For any element a g A, let w s b aŽ1. g R, where b a : B ª w0, w x is the homomorphism of Proposition 2.3 given by Ž9.. In the light of Theorem 1.4, write B s G Ž G, u. as in Ž3.. Similarly, write w0, w x s G Ž L, w ., for a suitable l-subgroup L of the additive group R with natural order. Also write b a s G Žh . for a uniquely determined l-homomorphism h : G ª L such that h Ž u. s w. We shall now prove the existence of a maximal ideal Ja g M Ž A. and a real number m a G 0 such that, for all b g B, b aŽ b . s m a bŽ Ja .. In case w s 0 upon choosing m a s 0, any arbitrary maximal ideal in the nonempty set M Ž A. will do. In case w / 0, by Holder’s theorem there is a real number m a such that the map c : ¨ x ¬ xrm a is the only l-homomorphism of L into R sending w into 1. It follows that the composite map c (h : G ª L ª R is an l-homomorphism of G into R sending u to 1, and G Ž c (h . is a homomorphism of B into w0, 1x. By Theorem 1.1, there is a unique maximal ideal Ja g M Ž A. such that, for all b g B,
b a Ž b . s m a ? b Ž Ja . .
Ž 10 .
Dually, for each b g B, there is a maximal ideal Ib g M Ž B . and a real number n b G 0 such that, for all a g A,
b b Ž a . s n b ? a Ž Ib . .
Ž 11 .
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233
In particular, when a s 1 and b s 1, we get 1 s b 1Ž1. s m1 ? 1 s m1 and symmetrically, 1 s b 1 Ž1. s n1 , whence m1 s n1 s 1. By Ž10. and Ž11., we can write a Ž I1 . s n1 ? a Ž I1 . s b 1 Ž a . s b Ž a, 1 . s b a Ž 1 . s m a ,
Ž 12 .
and symmetrically, b Ž J1 . s m1 ? b Ž J1 . s b 1 Ž b . s b Ž 1, b . s b b Ž 1 . s n b ,
Ž 13 .
whence a Ž I1 . ? b Ž Ja . s m a ? b Ž Ja . s b Ž a, b . s n b ? a Ž Ib . s b Ž J1 . ? a Ž Ib . . Ž 14 . Claim: bŽ J1 ..
For all a g A and b g B, we have the identity b Ž a, b . s aŽ I1 . ?
Case 1. b Ž a, 1. / 0. Then by Ž12. and Ž14., there is an ideal Ja g M Ž B . such that, for all b g B,
b Ž a, b . s b Ž a, 1 . ? b Ž Ja . .
Ž 15 .
Assume Ja / J1 Žabsurdum hypothesis., and pick b g B such that bŽ Ja . ) 0 and bŽ J1 . s 0. The existence of such b is ensured by the fact that B is a separating subalgebra of C Ž M Ž B ... Then by Ž15., b Ž a, b . ) 0. On the other hand, by Ž13., b Ž1, b . s bŽ J1 . s 0, thus contradicting the monotony property b Ž a, b . F b Ž1, b . ŽProposition 2.2Ži... We conclude that Ja s J1 , and from Ž15. and Ž12., we get b Ž a, b . s b Ž a, 1. ? bŽ J1 . s aŽ I1 . ? bŽ J1 ., as required. Case 2. b Ž a, 1. s 0. Then by monotony, b Ž a, b . s 0 for all b g B. Recalling Ž12., it is now sufficient to note aŽ I1 . s b Ž a, 1. s 0. The claim is settled. The dependence of I1 and J1 on b is understood. Thus, upon defining Ib s I1 and Jb s J1 , the first statement is proved. To conclude the proof, it is sufficient to note that injectivity follows from A and B being separating subalgebras. To prove surjectivity, one notes that for each I g M Ž A. and J g M Ž B ., the map d : A = B ª w0, 1x given by d Ž a, b . s aŽ I . ? bŽ J . is a bimorphism of A = B into w0, 1x.
3. THE MV-ALGEBRAIC TENSOR PRODUCT DEFINITION 3.1. A bimorphism b g bimŽ A, B, C . is said to be uni¨ ersal iff for every MV-algebra C9 and b 9 g bimŽ A, B, C9. there is a unique homomorphism l: C ª C9 such that l ( b s b 9.
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Stated otherwise, we have a commutative diagram
6
6
b9
6
b
A=B
C
l
C9 We shall now routinely construct an MV-algebra A mMV B and a universal bimorphism mMV of A = B into A mMV B. Construction. Let F be the free MV-algebra w5x over the free generating set A = B. Let I be the ideal of F generated by elements of the following form, for all a, a1 , a2 g A and b, b1 , b 2 g B, where distŽ x, y . is Chang’s distance function Ž4.: distŽŽ0, 0., 0. distŽŽ1, 1., 1. distŽŽ a, b1 k b 2 ., Ž a, b1 . k Ž a, b 2 .. distŽŽ a, b1 n b 2 ., Ž a, b1 . n Ž a, b 2 .. and for all c1 , c 2 g B with c1(c 2 s 0, distŽŽ a, c1 . ( Ž a, c 2 ., 0. and distŽŽ a, c1 [ c 2 ., Ž a, c1 . [ Ž a, c 2 .., and their duals, by analogy with Definition 2.1. Let A mMV B s FrI, and let the map Ž a, b . ¬ a mMV b be the composite of the inclusion map i : A = B ª F and the quotient map p : F ª FrI. Specifically, a mMV b s p Ž a, b . g A mMV B is the equivalence class of Ž a, b . in FrI. A straightforward verification shows that the map mMV : A = B ª A mMV B is a bimorphism. As a matter of fact, the ideal I was just defined in such a way that the composite map A = B ª F ª FrI is a bimorphism. The MValgebra A mMV B is generated by elements of the form a mMV b. The following result establishes that the map A = B ª A mMV B is universal. THEOREM 3.2. For any bimorphism b : A = B ª C, there is precisely one homomorphism l: A mMV B ª C such that lŽ a mMV b . s b Ž a, b .. Proof. The freeness properties of F yield a homomorphism h : F ª C such that h ( i s b . This is the uniquely determined homomorphism that sends each generator Ž a, b . of F into the element b Ž a, b . of C. Since b is bilinear, h sends each generator of I into the zero element of C; in symbols, KerŽh . = I. Let p be the quotient map of F onto FrI. Then we
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235
get a homomorphism l: FrI ª C, also uniquely determined, such that l (p s h. We have a commutative diagram F h
p l
FrI s A mMV B
6
6
6
b
6
i
6
A=B
G In conclusion, l (p ( i s b and lŽ a mMV b . s b Ž a, b ., as required. Remark. A routine verification shows the uniqueness of A mMV B. As a matter of fact, if g : A = B ª D is another universal bimorphism, then there is an isomorphism u : A mMV B ª D such that u Ž a mMV b . s g Ž a, b .. We shall call A mMV B the MV-tensor product of A and B. EXAMPLE. Let Ł n s G ŽŽ1rn.Z, 1. s 0, 1rn, 2rn, . . . , Ž n y 1.rn, 14 be the subalgebra of w0, 1x with n q 1 elements. Then Ł n mMV Ł m s Ł n m . As a matter of fact, by the above construction there is a finitely generated ideal I of the free MV-algebra F over Ž n q 1.Ž m q 1. generators such that Ł n mMV Ł m s FrI. It follows w5, Theorem 3.3.9, Corollary 3.3.11x that Ł n mMV Ł m is semisimple. An application of Theorem 2.4 shows that Ł n mMV Ł m has precisely one maximal ideal. The identity Ł n mMV Ł m s Ł n m now follows from one more application of Theorem 2.4. Similarly, if A and B are finite MV-algebras, then so is A mMV B. Specifically, upon writing A s Ł n1 = ??? = Ł n p as an MV-algebraic product of finite chains Žsee w5, Corollary 4.2.20x. and B s Ł m = ??? = Ł m , using Theorem 2.4 1 q one sees that the semisimple MV-algebra A mMV B has precisely pq maximal ideals. By direct inspection, one can prove A mMV B s =i , j Ł n i mMV Ł m j s =i , j Ł n i m j . For more complex applications, the mMV -tensor product is not easy to visualize. One main source of difficulty is given by the following phenomenon. THEOREM 3.3. There is a semisimple MV-algebra A such that A mMV A is not semisimple. Proof. Let F1 denote the free MV-algebra over one free generator. Equivalently w5, Theorem 3.2.16; 4x, F1 is the MV-algebra of all McNaughton functions of one variable, those continuous functions f : w0, 1x ª w0, 1x consisting of finitely many linear pieces of the form ax q b, a, b g Z.
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For each j g w0, 1x, let Jj and Oj respectively denote the maximal ideal and the germinal ideal of F1 at j ; in symbols, Jj s f g F1 f Ž j . s 0 4 , Oj s f g F1 f s 0 on an open neighborhood of j in w 0, 1 x 4 . By w8, Proposition 3.6x, Oj s Jj iff j is irrational. Assume j to be irrational. Then the quotient MV-algebra A s F1rOj s F1rJj can be visualized in the following two equivalent ways: Ži. either as the subalgebra of w0, 1x given by all possible values of McNaughton functions of F1 at j . Thus, in particular, 04 is the only ideal of A, and A is semisimple; Žii. or, as the MV-algebra of germs at j of McNaughton functions. Recall that two functions p and q have the same germ at j Žin symbols, p ;j q . iff they coincide over some open neighborhood of j in w0, 1x. Let C Žw0, 1x 2 . denote the MV-algebra of all continuous w0, 1x-valued functions on the unit square w0, 1x 2 . Let the map b˜: F1 = F1 ª C Žw0, 1x 2 . be defined by b˜Ž f, g . s f Ž x . g Ž y . for all f, g g F1. Trivially, whenever f ;j f 9 and g ;j g 9, then the functions b˜Ž f, g . and b˜Ž f 9, g 9. coincide on an open neighborhood of Ž j , j . in w0, 1x 2 . Let the germinal ideal OŽ j , j . be defined by OŽ j , j . s h g C Ž w 0, 1 x
½
2
.
h s 0 on some open neighborhood of Ž j , j . .
5
Let the quotient MV-algebra H be defined by H s C Žw0, 1x 2 .rOŽ j , j . . Then H is the MV-algebra of germs at Ž j , j . of continuous w0, 1x-valued functions on w0, 1x 2 . From b˜ we obtain a map b : A = A ª H, sending each pair of germs Ž frOj , grOj . into the germ at Ž j , j . of the function f Ž x . g Ž y .. Let G be the subalgebra of H generated by the range of b . Then b is a bimorphism of A = A into G. Claim 1: G is not semisimple. As a matter of fact, let x and y be the two identity functions over w0, 1x 2 . The germ of x at Ž j , j . is a member of G, and so is the germ of y at Ž j , j .. Therefore, the free product MValgebra A@ A is a subalgebra of G. By w8, Corollary 4.4x together with Theorem 1.4Živ., A@ A is not semisimple. A fortiori, G is not semisimple. Claim 2: A mMV A is not semisimple. Assume A mMV A is semisimple Žabsurdum hypothesis.. A moment’s reflection shows that A mMV A has precisely one maximal ideal. For otherwise, by Theorem 1.1 there would exist at least two homomorphisms of A mMV A into w0, 1x, and the composition of mMV g bimŽ A, A, A mMV A. with these homomorphisms would
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give at least two distinct bimorphisms of A = A into w0, 1x, whence, by Theorem 2.4, at least two elements of M Ž A. = M Ž A., whence A s F1rJj would have at least two maximal ideals, which is impossible. Since, as we have just seen, A mMV A has precisely one maximal ideal and it is assumed to be semisimple, by Theorem 1.2, A mMV A coincides with a subalgebra of w0, 1x. By the universal property of A mMV A, there is a unique homomorphism l: A mMV A ª G such that l ( mMV s b . Since A mMV A is a subalgebra of w0, 1x, l is one-to-one. Since G is generated by the range of b , l is onto G. Thus, A mMV A is isomorphic to G, and by the first claim, A mMV A is not semisimple, a contradiction. 4. THE SEMISIMPLE TENSOR PRODUCT The following is a generalization of Definition 3.1. DEFINITION 4.1. Let K be a class of MV-algebras, and A, B g K. Then a bimorphism b of A = B into an MV-algera C is said to be K-uni¨ ersal iff C g K and for all C9 g K and b 9 g bimŽ A, B, C9., there is a unique homomorphism l: C ª C9 such that b 9 s l ( b . Throughout the rest of this paper we shall restrict attention to the case when K is the class of semisimple MV-algebras. We first construct the semisimple tensor product, as follows. Construction. Given semisimple MV-algebras A and B, let R be the radical of A mMV B, and A m B s Ž A mMV B .rR. Let r : A mMV B ª Ž A mMV B .rR be the quotient map. Then A m B is a semisimple MValgebra w5, Corollary 2.5.18x. Let the map m: Ž a, b . ¬ Ž a m b . send each pair Ž a, b . g A = B into the element Ž a m b .rR g A m B; in symbols, ms r ( mMV . Then m is well defined and is a bimorphism of A = B into A m B, whose range generates A m B. We shall call A m B the semisimple tensor product of A and B. The fundamental theorem of semisimple tensor products states that the bimorphism m: A = B ª A m B is universal for the class of semisimple MV-algebras. THEOREM 4.2. Let A and B be semisimple MV-algebras. Then for e¨ ery bimorphism b of A = B into a semisimple MV-algebra C, there is a unique homomorphism l: A m B ª C such that lŽ a m b . s b Ž a, b .. Proof. The universal property of A mMV B yields a unique homomorphism m : A mMV B ª C such that b s m ( mMV . Since C is semisimple, the kernel of m will include the radical of A mMV B. Thus the map
r Ž a mMV b . s Ž a mMV b . rR ¬ m Ž a mMV b .
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is a well-defined homomorphism l of A m B into C such that lŽ a m b . s b Ž a, b .. Uniqueness of l is routine, because elements of the form r Ž a mMV b . generate A m B. By contrast with the MV-tensor product, the semisimple tensor product allows a concrete visualization as follows. THEOREM 4.3. Let A and B be semisimple MV-algebras, respecti¨ ely identified with separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .. Ž whence the maps a ¬ a* and b ¬ b* are assumed to be identities.. Let p : A = B ª C Ž M Ž A. = M Ž B .. send each pair of elements a g A and b g B into the function f : M Ž A. = M Ž B . ª w0, 1x gi¨ en by f Ž I, J . s a Ž I . ? b Ž J . for all I g M Ž A. and J g M Ž B .. Let C be the MV-algebra generated by the range of p in C Ž M Ž A. = M Ž B ... Then p g bimŽ A, B, C . is uni¨ ersal for the class of semisimple MV-algebras, and hence, C is isomorphic to A m B. Further, C is a separating subalgebra of C Ž M Ž A. = M Ž B ... Proof. Let m: A = B ª A m B be the universal bimorphism of Theorem 4.2. Identify A m B with a separating subalgebra of C Ž M Ž A m B ... Then there is a unique homomorphism l: A m B ª C such that p s l ( m . Since the range of p generates C, then l is surjective. Let Y s F fy1 Ž0. < f g KerŽ l.4 . Then Y is a closed subspace of M Ž A m B ., and the map l amounts to restricting to Y each function in A m B : C Ž M Ž A m B ..; in symbols, lŽ f . s f ° Y. Claim: l is injective. Otherwise Žabsurdum hypothesis., Y is strictly contained in M Ž A m B .. Let H g M Ž A m B . be a maximal ideal which is not a member of Y. Let r H : A m B ª w0, 1x be the quotient map. From our identification of A m B with an MV-algebra of functions, it follows that r H amounts to evaluating at H each function f g A m B; in symbols,
rH Ž f . s f Ž H .
for all f g A m B.
Ž 16 .
The composite map r H ( m is a bimorphism of A = B into w0, 1x. By Theorem 2.4, there is a unique pair Ž I, J . g M Ž A. = M Ž B . such that for all Ž a, b . g A = B, Ž r H ( m .Ž a, b . s aŽ I . ? bŽ J . s Ž c (p .Ž a, b ., where the evaluation map c : C Ž M Ž A. = M Ž B .. ª w0, 1x is defined by c Ž a m b . s aŽ I . ? bŽ J .. It follows that c ( l ( m s c (p s r H ( m , whence Žfrom the fact that the range of m generates A m B . we can write
rH s c ( l .
Ž 17 .
Stated otherwise, for all Ž a, b . g A = B, Ž a m b .Ž H . s aŽ I . ? bŽ J .. By Theorem 1.1, the kernel of c is a maximal ideal P g M Ž C .. The inverse
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image ly1 Ž P . is a maximal ideal Q g M Ž A m B ., and, necessarily, Q g Y. The following diagram illustrates the situation: AmB l
w0, 1x
c
6
p
rH
66
m
6 6
A=B
C
Recalling Ž17., for each f g A m B, we have f Ž H . s rH Ž f . s Ž c ( l. Ž f . s c Ž f ° Y . s Ž f ° Y . Ž Q . s f Ž Q . . Since f Ž H . s f Ž Q . for all f g A m B, and A m B is a separating subalgebra of C Ž M Ž A m B .., it follows that Q s H, thus contradicting the fact that Q g Y and H f Y. The claim is settled and l is an isomorphism of C onto A m B. It is now easy to see that p g bimŽ A, B, C . is a universal bimorphism for the class of semisimple MV-algebras. A moment’s reflection also shows that C is a separating subalgebra of the MV-algebra C Ž M Ž A. = M Ž B ... Dropping the assumption that the maps a ¬ a* and b ¬ b* are identities, their ranges A* and B* will be separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .., respectively. The uniqueness of the semisimple tensor product can then be expressed as follows. COROLLARY 4.4. Let A and B be semisimple MV-algebras. Let the map Ž), ).: A = B ª A* = B* send each pair Ž a, b . into Ž a*, b*.. Let m*: A* = B* ª C Ž M Ž A. = M Ž B .. transform each Ž a*, b*. into the function g: M Ž A. = M Ž B . ª w0, 1x gi¨ en by g Ž I, J . s a*Ž I . ? b*Ž J .. Let us agree to denote by A* m*B* the subalgebra of C Ž M Ž A. = M Ž B .. generated by the range of m*. It follows that A* m*B* is a separating subalgebra of C Ž M Ž A. = M Ž B .., and m* is a bimorphism of A* = B* into A* m*B*, which is uni¨ ersal for semisimple MV-algebras. Further, Ži. There is a unique isomorphism l: A m B ª A* m*B* such that lŽ a m b . s a* m*b*; graphically, m
6
m*
A* m*B*
6
6
A* = B*
AmB l
Ž), ) .
6
A=B
Ž18.
Žii. The map I ¬ lŽ I . is an isomorphism of the lattice of ideals of A m B onto the lattice of ideals of A* m B*. For any ideal I of A m B, let r I : A m B ª Ž A m B .rI be the quotient map; further, let VlŽ I . s F fy1 Ž0. N f g lŽ I .4 : M Ž A. = M Ž B . be the intersection of the zerosets of functions in
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lŽ I .. Then, letting ° denote the restriction operation, there is a unique isomorphism m I : Ž A m B .rI ª Ž A* m*B*. ° VlŽ I . such that rlŽ I . ( l s m I ( r I ; graphically, rI
6
6
Žiii.
rlŽ I .
Ž A* m*B*. ° VlŽ I .
6
A* m*B*
Ž A m B .rI mI
l
6
AmB
Ž19.
For all Ž a, b . g A = B, we ha¨ e the identity
m I Ž Ž a m b . rI . s Ž a* m*b* . ° VlŽ I . .
Ž 20 .
5. MULTIPLICATIVE MV-ALGEBRAS Construction. Let A be a semisimple MV-algebra. The flip automorphism w : A m A ª A m A is generated by the map Ž a1 , a2 . ¬ Ž a2 , a1 ., for all a1 , a2 g A. An ideal J of A m A is said to be in¨ ariant iff w J s J. The diagonal ideal D is the intersection of all maximal invariant ideals of A m A. Let r D : A m A ª Ž A m A.rD be the quotient map. Note that Ž A m A.rD is semisimple, because the ideal D is an intersection of maximal ideals w5, Theorem 2.5.17x. We then define the diagonal map d : A ª Ž A m A.rD by stipulating that for all a g A,
d Ž a. s rD Ž a m 1. .
Ž 21 .
The flip automorphism of A* m*A* canonically induces a flip homeomorphism of M Ž A. = M Ž A. onto itself. Let the diagonal D of M Ž A. = M Ž A. be defined by D s Ž I, I . I g M Ž A . 4 . Recalling from Corollary 4.4Ži. that A* m*A* is a separating subalgebra of C Ž M Ž A. = M Ž A.., let Ž I, J . ¬ f g A* m*A* < f Ž I, J . s 04 be the canonical correspondence between points Ž I, J . g M Ž A. = M Ž A. and maximal ideals K of A* m*A*. ŽThe inverse of this correspondence maps each maximal ideal K of A* m*A* into the intersection Z of the zerosets of all functions in K. Note that Z s Ž I, J .4 is a singleton, because A* m*A* separates points.. Then a maximal ideal K of A* m*A* is invariant iff its corresponding point Ž I, J . is invariant under the flip homeomorphism, iff I s J, iff Ž I, J . g D. By Corollary 4.4, for all a, b g A, we have a m b g D iff a m b g KerŽ r D . iff a* m*b* g KerŽ rlŽ D . . if a* m*b* g lŽ D . iff a* m*b* identically vanishes over D iff a* m*b* belongs to all invariant
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maximal ideals of A* m*A*. Since the zerosets of functions in A* m*A* form a basis of closed sets for the spectral topology of A* m*A*, we get VlŽ D . s F fy1 Ž 0 . f g lŽ D . 4 s F fy1 Ž 0 . f s 0 over D 4 s D . Ž 22 . Let the map e : A* ª Ž A* m*A*. ° D be defined by stipulating that, for all a* g A*, e Ž a*. s Ž a* m*1. ° D. In other words, for all Ž I, I . g D,
Ž e Ž a*. . Ž I, I . s a* Ž I . .
Ž 23 .
Then e is a homomorphism, and we have a commutative diagram 6
d
mD
6
6
A* m*A* ° D
A )
e
6
Ž A m A.rD
A*
Ž24.
As a matter of fact, from Ž20. ] Ž23. we get Ž m D ( d .Ž a. s Ž m D ( r D .Ž a m 1. s Ž a* m*1. ° VlŽ D . s Ž a* m*1. ° D s e Ž a*.. PROPOSITION 5.1. e are one-to-one.
For e¨ ery semisimple MV-algebra A, both maps d and
Proof. In the light of diagram Ž24., since both maps ) and m D are isomorphisms, it suffices to prove that e is one-to-one. Let 0 / a* g A*, with the intent of proving 0 / e Ž a*.. By Theorems 1.2 and 4.3, A* and A* m*A* are separating subalgebras of C Ž M Ž A.. and C Ž M Ž A. = M Ž A.., respectively. Let J g M Ž A. be such that a*Ž J . / 0. Then by Ž23., since Ž J, J . g D we can write 0 / a*Ž J . s Ž e Ž a*..Ž J, J ., whence e Ž a*. / 0, as required. PROPOSITION 4.2. For any semisimple MV-algebra A, the following conditions are equi¨ alent: Ži. The diagonal map d : A ª Ž A m A.rD is onto; Žii. The map e is an isomorphism of A* onto Ž A* m*A*. ° D; Žiii. d isomorphically maps A onto Ž A m A.rD; Živ. A* is closed under pointwise multiplication. Proof. The equivalences of Ži. ] Žiii. follow from Proposition 5.1, recalling that both maps ) and m D are isomorphisms. For the implication Živ. ª Žii., by Corollary 4.4 every element of Ž A* m*A*. ° D is in the MV-algebra generated by elements of the form Ž a* m*b*. ° D, i.e., elements of the form a* ? b*, where a*, b* g A* and ? is pointwise multiplication. Since by hypothesis, a* ? b* g A*, we conclude by Ž23. that e is surjective. Conversely, in order to prove Žii. ª Živ., adopting the notation
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>
of the commutative diagrams Ž18., Ž19., Ž24., let us define the operation : A* = A* ª A* by a* b* s Ž ey1 ( rlŽ D . ( m*.Ž a*, b*.. By Corollary 4.4, Ž rlŽ D . ( m*.Ž a*, b*. s Ž a* m*b*. ° D. By definition of e , together with Corollary 4.4, the function Ž a* m*b*. ° D: D ª w0, 1x is the e-image of the function a* ? b*: M Ž A. ª w0, 1x given by Ž a* ? b*.Ž I . s a*Ž I . ? b*Ž I . for all I g M Ž A.. This shows that the binary operation coincides with pointwise multiplication.
>
>
DEFINITION 5.3. A semisimple MV-algebra A is called multiplicati¨ e iff it satisfies any of the equivalent conditions of the above proposition. PROPOSITION 5.4. Let A be a semisimple multiplicati¨ e MV-algebra. Let the map w: A = A ª A be defined by a1w a2 s Ž dy1 ( r D . Ž a1 m a2 . . Then w has the following properties: Ži. Ž a1w a2 .* s aU1 ? aU2 , where ? is pointwise multiplication; Žii. w is commutati¨ e, associati¨ e, has neutral element 1, and is a bimorphism of A = A into A. Thus, for all a, b, c g A, aw0 s 0, and w distributes o¨ er the lattice operations of A in the following sense: Žiii. cwŽ a k b . s Ž cw a. k Ž cwb ., and Živ. cwŽ a n b . s Ž cw a. n Ž cwb .. For all a, b g A with a(b s 0, we ha¨ e Žv. Ž cw a. ( Ž cwb . s 0 and Žvi. cwŽ a [ b . s Ž cw a. [ Ž cwb .. Proof. Ži. Consider the following commutative diagram, as given by Corollary 4.4 and the assumption that A is multiplicative: dy1
mD
)
ey1
6
6
Ž A* m*A*. ° D
A
A*
6
rlŽ D .
6
6
A* m*A*
6
m*
Ž A m A.rD
6
rD
l
Ž), ) .
6
A* = A*
AmA
6
m
6
A=A
We then have
Ž a1w a2 . * s Ž *( dy1 ( r D . Ž a1 m a2 . s Ž ey1 ( rlŽ D . ( m*( Ž ), ) . . Ž a1 , a2 . s Ž ey1 ( rlŽ D . . Ž aU1 m*aU2 . s Ž ey1 . Ž Ž aU1 m*aU2 . ° D . s aU1 ? aU2 ,
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
243
as required. The proof of Žii. follows from the definition of semisimple tensor product, together with the commutativity of the above diagram. The fact that w is a bimorphism of A = A into A is proved by direct inspection of the above diagram. The proof of conditions Žiii. ] Žvi. now follows by Definition 2.1 using Propositions 2.2 and 2.3. DEFINITION 5.5. The above operation w: A = A ª A is called the natural product of the semisimple multiplicative MV-algebra A. The following is an immediate consequence of the definitions and of Proposition 5.2. PROPOSITION 5.6. Ži. E¨ ery boolean algebra Ž i.e., e¨ ery MV-algebra satisfying the idempotence equation x [ x s x . is multiplicati¨ e, and the natural product coincides with the lattice operation n. Žii. E¨ ery subalgebra of w0, 1x which is closed under multiplication of real numbers, is multiplicati¨ e. More generally, if A is an MV-algebra of w0, 1x-¨ alued continuous functions o¨ er some compact Hausdorff space X and A is closed under pointwise multiplication of functions, then A is multiplicati¨ e and its natural product coincides with pointwise multiplication. By Corollary 1.3 together with Propositions 5.2 and 5.4, the class of MV-algebras given in Žii. above is the most general possible example of a Žsemisimple. multiplicative MV-algebra. Remark. Our analysis of tensor products in semisimple MV-algebras allows an intrinsic, representation-free formulation of the intuitive notion of A being ‘‘closed under multiplication.’’ This fact may be of help in further investigations of intrinsic multiplication operations in classes of nonsemisimple MV-algebras}e.g., the MV-algebras arising from a nonstandard model of the multiplicative algebra w0, 1x.
6. THE MV-ALGEBRAIC LOOMIS]SIKORSKI THEOREM In this section, we assume familiarity with the spectral topology w1x, and states w6x of l-groups. Let A be a s-complete MV-algebra, and let G be its corresponding l-group with strong unit u, as given by Theorem 1.4. Then by Proposition 1.5, A is semisimple, and by Proposition 1.5, every bounded sequence of elements of G has a supremum in G: for short, following w6x, G is Dedekind s-complete. For every maximal ideal J of A, the quotient MV-algebra ArJ is canonically identified with a subalgebra of w0, 1x, in the light of Theorem 1.1. We say that J is discrete iff ArJ is finite. Similarly, a maximal l-ideal I of G is said to be discrete iff the quotient l-group GrI is cyclic. In the light of Theorem 1.4Živ., it is easy to
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DANIELE MUNDICI
see that I is discrete iff so is the ideal J s I l w0, u x g M Ž A.. For any maximal l-ideal I of G, there is precisely one embedding of GrI into the totally ordered group R in such a way that the strong unit urI is mapped into the element 1. This is a well-known consequence of Holder’s theorem ¨ w1, 2.6, 12.2.1x. Let SŽ G, u. be the state space of G, i.e., the convex set of group homomorphisms j : G ª R such that j Ž u. s 1 and 0 F j Ž x . for all 0 F x g G. Let e SŽ G, u. be the space of extremal points of S Ž G, u. with the natural topology w6x. By w6, Theorem 15.32, Corollary 12.19x, e S Ž G, u. is canonically homeomorphic to the space MaxspecŽ G . Ži.e., m G, in the notation of w1x. of maximal l-ideals of G with the spectral topology w1x. By Theorem 1.4Žiii. ] Živ., MaxspecŽ G . is in turn canonically homeomorphic to M Ž A.. For each extremal state s g e SŽ G, u., and corresponding maximal l-ideal I of G, the range sŽ G. of s coincides with the quotient l-group GrI, once Ž GrI, urI . is canonically embedded into ŽR, 1.. LEMMA 6.1. Let A be a s-complete MV-algebra. Let X s M Ž A. be the space of maximal ideals of A with the spectral topology. For each J g X, let ArJ be identified with the set of possible ¨ alues at J of functions a*, for all a g A. Then the map a ¬ a* of Ž2. is an isomorphism of A onto the MV-algebra E consisting of all continuous functions g: X ª w0, 1x such that g Ž J . g ArJ for each discrete J g X. This map preserves all countable suprema in the sense that, for e¨ ery sequence a i g A, the element ŽE a i .* is the supremum in C Ž X . of the functions aUi , and coincides with the supremum in E of the aUi . Ž Note that ŽE a i .* need not coincide with the pointwise supremum of the functions aUi .. Proof. Let G be the corresponding l-group of A with strong unit u; in symbols, A s G Ž G, u.. Then G is Dedekind s-complete. In the light of Theorem 1.4Živ., let us canonically identify the homeomorphic spaces X and MaxspecŽ G .. From w6, Corollary 9.10x, we get that X is basically disconnected; in other words, the closure of every open Fs subset of X is open. Let G h be the l-group of all real-valued continuous functions over X. Note that the constant function 1 is a strong unit for G h. By w6, Corollary 9.3x, G h is Dedekind s-complete. By w6, Corollary 9.14x, together with our introductory discussion in this section, there is an l-isomorphism c of G onto the l-subgroup B of G h consisting of all continuous functions f : X ª R such that f Ž J . g GrJ : R for each discrete maximal l-ideal J of G. Again, we are canonically identifying Ž GrJ, urJ . with a subgroup of ŽR, 1.. A moment’s reflection shows that B is large in G h Žequivalently, G h is an essential extension of B .: to this purpose, it is sufficient to note that for each 0 - f g G h, there is an element 0 - g g B and a positive integer n such that g F nf. By w1, Corollary 12.1.12x, all existing suprema in B are preserved in G h. By w6, Lemma 9.12x, countable suprema of G are mapped by c into countable suprema of B or, what is
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
245
the same, countable suprema in G h. By definition of the canonical homeomorphism e SŽ G, u. ( MaxspecŽ G ., we obtain that the l-isomorphism G ( B is generated by the map a ¬ a*. Thus, by Theorem 1.4, the map a ¬ a* coincides with the isomorphism c ° A of A onto the subalgebra E of C Ž X . given by the w0, 1x-valued functions of B. By Proposition 1.5, both C Ž X . s G Ž G h, 1. and E s G Ž B, 1. are s-complete. Since countable suprema are preserved by the map c , and B is large in G h, then, going back to MV-algebras, preservation of countable suprema is a consequence of Theorem 1.4Žiii. ] Živ.. By a s-field of sets over a nonempty set X we mean a s-complete boolean algebra of 0, 14 -valued functions over X, where countable suprema are given by pointwise sups. This is equivalent to the usual definition Žsee, e.g., w10x.. Similarly, following w9, 8.1.1x, by a tribe over X we mean an MV-algebra F of w0, 1x-valued functions on X such that, for each sequence f 1 , f 2 , . . . g F , the pointwise supremum f s sup i f i also belongs to F. Thus, in particular, in every tribe F we have Ži. 0 g F ; Žii. whenever f g F , then ! f s 1 y f g F ; Žiii. if f 1 , f 2 , . . . g F , then 1 n Ý`is1 f i g F , where it is understood that 1 n ` s 1. Note that F is a s-complete MV-algebra. On the other hand, not every s-complete MV-algebra is isomorphic to a tribe Žalready a s-complete boolean algebra need not be isomorphic to a s-field of sets.. Let A be a s-complete MV-algebra. Then a homomorphism h : F ª A is said to be a s-homomorphism iff for each sequence f 1 , f 2 , f 3 . . . g F , letting f s sup i f i be their pointwise supremum, h Ž f . coincides with the supremum Ei h Ž f i . in A. The following result generalizes the well-known Loomis]Sikorski theorem w10, 29.1, p. 93x. THEOREM 6.2. Let A be a s-complete MV-algebra. Let X s M Ž A.. Then there is a tribe F o¨ er X and a s-homomorphism h of F onto A. Proof. As in the proof of the previous lemma, the preservation properties of the G functor Ž1.4Živ.., together with w6, Corollary 9.10x, ensure that X is basically disconnected, and by w6, Corollary 9.3x, C Ž X . is s-complete. By Lemma 6.1, we can safely identify A with a s-complete and separating subalgebra of C Ž X .. While countable suprema in A need not coincide with pointwise suprema of functions, the proof of Lemma 6.1 has shown that all existing suprema in A : C Ž X . are preserved in C Ž X .. For any function f : X ª w0, 1x, let us agree to say that f coincides almost e¨ erywhere with a continuous function g g A, in symbols, f f g, iff the set
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DANIELE MUNDICI
J g X < f Ž J . / g Ž J .4 is meager Žalso called a set of first category.. Stated otherwise, the set is the countable union of subsets of X whose closure has empty interior. Note that if we also have f f h and h g A, then necessarily g s h. Thus, letting F : w0, 1x X be the set of functions f that coincide almost everywhere with some function g g A, we have a map h of F onto A. It is easy to see that F is an MV-algebra and that h is a homomorphism. There remains to be shown that F is a tribe and that h is a s-homomorphism. It is sufficient to prove that, for each sequence f 1, f 2 , . . . g F, sup f i f sup h Ž f i . f i
i
E h Ž fi . . i
Since by construction, f i f h Ž f i ., the fact that sup i f i f sup i h Ž f i . immediately follows by definition of meager set. To conclude the proof, let f s sup i h Ž f i . and fˆs Ei h Ž f i .. Recall that fˆ is also the supremum Ei h Ž f i . ˆ Further, since each h Ž f i . is continuous, the set in C Ž X .. Trivially, f F f. < Y s J g X f is not continuous at J 4 is meager in X, by the Baire category theorem w11, Baire’s Theorem 2, pp. 12]13x.
ˆ For otherwise Žabsurdum hypothesis., assume D s J g Claim: f f f. X < f Ž J . / fˆŽ J .4 is not meager. Then D l Ž X R Y . contains at least one element J, because D cannot be contained in the meager set Y. Let 0 - e s fˆŽ J . y f Ž J .. Since both f and fˆ are continuous at J, there is an open neighborhood U of J such that fˆŽ I . y f Ž I . ) 2 er3 for all I g U. Since X is basically disconnected, U may be assumed clopen. Let the function d : X ª w0, 1x be defined by d Ž J . s 0 outside U, and d Ž J . s er2 over U. Then by definition of U, d g C Ž X .. Now the function fˇs fˆy d is ˆ and for all i s 1, 2, . . . , an element of C Ž X ., is strictly smaller than f, ˇ This contradicts our assumption that fˆ is the supremum Ei h Ž f i . f i F f F f. in C Ž X .. ˆ We have proved that f and fˆ only differ on a meager set, i.e., f f f. This completes the proof. Closing a circle of ideas, and putting the two conditions of s-completeness and multiplicativeness together, we have the following. COROLLARY 6.3. Let A be a s-complete multiplicati¨ e MV-algebra. Let X s M Ž A.. Then there is a tribe F o¨ er X which is closed under pointwise multiplication, and a s-homomorphism h of F onto A such that, for all f, g g F , h Ž fg . s h Ž f .wh Ž g .. Proof. We adopt the same notation and terminology of the previous proof. Since A is multiplicative, by Proposition 5.6Žii., whenever J is a discrete maximal ideal of A, then the quotient algebra ArJ : w0, 1x must
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coincide with the two-element boolean algebra 0, 14 . Again letting F : w0, 1x X be the set of functions almost everywhere coinciding with some function in A, we have a map h of F onto A. F is an MV-algebra of functions closed under pointwise multiplication, and h is a homomorphism. By Proposition 5.6, h Ž fg . s h Ž f .wh Ž g .. For each sequence f 1 , f 2 , . . . g F , we must prove sup i f i f sup i h Ž f i . f Ei h Ž f i .. Letting f s ˆ Again, sup i h Ž f i . and fˆs Ei h Ž f i ., we immediately obtain sup i f i f f F f. the set Y of discontinuity points of f is meager in X. In order to prove ˆ by way of contradiction, assume the set D of points where f / fˆ that f f f, is not meager. Let J g D l Ž X R Y .. Let 0 - e s fˆŽ J . y f Ž J .. There is a clopen neighborhood U of J such that fˆŽ I . y f Ž I . ) 2 er3 for all I g U. Case 1. The set of maximal ideals I such that ArI s 0, 14 is dense over U. Then, by continuity, over U, each f i , as well as f, identically vanishes, and fˆs 1. Let the function fˇ be defined by fˇs fˆ outside U, and fˇs 0 over U. Then fˇg A, and for all i, f i F f F fˇ; on the other hand, fˇ is ˆ thus contradicting the definition of f.ˆ strictly smaller than f, Case 2. For some nonempty open set W : U, there are no ideals I such that ArI s 0, 14 . Then, again, we can safely assume W clopen. Let the function d : X ª w0, 1x be defined by d Ž J . s 0 outside W, and d Ž J . s er2 over W. Then ˇ d g A. Similarly, the function fˇs fˆy d is in A, and for all i, f i F f F f, ˇ ˆ ˆ whence f is strictly smaller than f, thus contradicting the definition of f.
ACKNOWLEDGMENTS The author is indebted to Alberto Marcone for his advice concerning Baire category arguments. He is also greatly indebted to Anatolij Dvurecenskij and Beloslav Riecan ˇ ˇ for many inspiring discussions concerning the relationships between tribes and MV-algebraic measure theory.
REFERENCES 1. A. Bigard, K. Keimel, and S. Wolfenstein, ‘‘Groupes et Anneaux Reticules,’’ ´ ´ Lecture Notes in Mathematics, Volume 608, Springer-Verlag, Berlin, 1971. 2. C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 Ž1958., 467]490. 3. C. C. Chang, A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93 Ž1959., 74]80.
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4. R. Cignoli and D. Mundici, An invitation to Chang’s MV-algebras, in ‘‘Advances in Algebra and Model Theory’’ ŽM. Droste and R. Gobel, Eds.., pp. 171]197, Gordon and ¨ Breach Publishing Group, Reading, UK Ž1997.. 5. R. Cignoli, I. M. L. D’Ottaviano, and D. Mundici, ‘‘Algebras of Łukasiewicz Logics,’’ second edition. Editions CLE, State University of Campinas, Campinas, S.P., Brazil, 1995. ŽIn Portuguese.. 6. K. R. Goodearl, ‘‘Partially Ordered Abelian Groups with Interpolation,’’ American Mathematical Society, Providence, RI, 1986. 7. D. Mundici, Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 Ž1986., 15]63. 8. D. Mundici, Free products in the category of abelian l-groups with strong unit, J. Algebra 113 Ž1988., 89]109. 9. B. Riecan ˇ and T. Neubrunn, ‘‘Integral, Measure, and Ordering,’’ Kluwer Academic Publishers, Dordrecht, 1997. 10. R. Sikorski, ‘‘Boolean Algebras,’’ Springer-Verlag, Berlin, 1960. 11. K. Yosida, ‘‘Functional Analysis,’’ sixth ed., Springer-Verlag, Berlin, 1980.
Tensor Products and the Loomis]Sikorski Theorem for MV-Algebras Daniele Mundici* Department of Computer Science, Uni¨ ersity of Milan, Via Comelico 39-41, 20135 Milan, Italy E-mail: [email protected] Received May 15, 1998; accepted September 19, 1998
MV-algebras are the models of the time-honored equational theory of magnitudes with unit. Introduced by Chang as a counterpart of the infinite-valued sentential calculus of Łukasiewicz, they are currently investigated for their relations with AF C*-algebras, toric desingularizations, and lattice-ordered abelian groups. Using tensor products, in this paper we shall characterize multiplicatively closed MV-algebras. Generalizing work of Loomis and Sikorski, we shall investigate the relationships between s-complete multiplicatively closed MV-algebras, Q 1999 Academic and pointwise s-complete MV-algebras of w0, 1x-valued functions. Press
1. INTRODUCTION Introduced by Chang in w2, 3x, MV-algebras are the models of the equational theory of magnitudes with a distinguished archimedean unit w1, pp. 106]107x, in a sense that will be made precise by Theorem 1.4 below. The monograph w5x, as well as the survey paper w4x, provide all the necessary background information. The prototypical MV-algebra is given by the real unit interval w0, 1x equipped with the operations ! x s 1 y x,
x [ y s min Ž 1, x q y . ,
x( y s max Ž 0, x q y y 1 . . Ž 1.
MV-algebras are the algebras satisfying precisely the same equations that are satisfied by w0, 1x. Equivalently, by Chang’s completeness theorem w2, *Partially supported by CNR-GNSAGA Project on Symbolic Computation, and by the European Cost Action 15 on Many-valued logics for computer science applications. 227 0196-8858r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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3x, an MV-algebra A s Ž A, 0, 1, !, [ , ( . is an abelian monoid Ž A, 0, [ . with an involution ! such that x [ 1 s 1 s !0, x( y s !Ž! x [ ! y ., and y [ !Ž y [ ! x . s x [ !Ž x [ ! y .. An MV-algebra is called tri¨ ial iff it only consists of the zero element. Unless otherwise specified, all MV-algebras considered in this paper shall be nontrivial. Every MV-algebra A s Ž A, 0, 1, !, [ , ( . is further equipped with the operations k and n given by a k b s !Ž! a [ b . [ b and a n b s !Ž! a k ! b .. The binary relation F on A given by a F b iff a( ! b s 0, is a partial order; as a matter of fact, Chang w2x proved that Ž A, k, n, 0, 1. is a distributive lattice with least element 0 and greatest element 1; this is called the underlying lattice of A. We say that MV-algebra A is s-complete iff its underlying lattice is s-complete, i.e., every nonempty countable subset of A has a supremum in A. Ideals are kernels of homomorphisms. Unless otherwise specified, every ideal I of an MV-algebra A considered in this paper shall be assumed to be proper, i.e., I / A. An ideal I is said to be maximal iff there is no ideal of A strictly containing I Žother than the improper ideal A.. Thus, maximal ideals are automatically proper. We let M Ž A. denote the set of all maximal ideals of A. Since I is maximal iff it is maximal for the property of not containing the unit element 1, we easily conclude that M Ž A. / B. The following result is an MV-algebraic variant of Holder’s theorem ¨ w1, 2.6, 12.2.1x. THEOREM 1.1. For any MV-algebra A and maximal ideal I of A, there is a unique isomorphism i I of the quotient MV-algebra ArI onto a subalgebra of w0, 1x. Let r I : A ª ArI be the quotient map. Then the map I ¬ i I ( r I is a one-to-one correspondence of the set of maximal ideals of A onto the set of homomorphisms of A into w0, 1x. The in¨ erse of this map sends any such homomorphism into its kernel. Proof. See, e.g., w5, Theorems 2.4.14, 2.5.7x. The set RadŽ A. s F K < K g M Ž A.4 is called the radical of A. A is said to be semisimple iff RadŽ A. s 04 . Equivalently, for each nonzero element x g A, there is a homomorphism h : A ª w0, 1x with h Ž x . / 0. In order to define the spectral Žalso called hull-kernel . topology on the set M Ž A. of all maximal ideals of A, for every ideal I of A, let OI s K g M Ž A. < K W I 4 . Then the collection of all sets of the form OI is a compact Hausdorff topology on M Ž A.. Any closed subset is of the form CI s K g M Ž A. < K = I 4 , for some ideal I of A. For any compact Hausdorff space X, let us agree to denote by C Ž X . the MV-algebra of all continuous w0, 1x-valued functions on X with pointwise
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229
operations as in Ž1.; let, as usual, w0, 1x X denote the MV-algebra of all w0, 1x-valued functions on X. In the light of Theorem 1.1, let the map a g A ¬ a* g w0, 1x X be defined by a* Ž I . s Ž i I ( x I . Ž a . g w 0, 1 x
for all I g M Ž A . .
Ž 2.
The following theorem gives a concrete representation of semisimple MV-algebras; its proof is a fundamental consequence of Chang’s completeness theorem. THEOREM 1.2. Let A be a semisimple MV-algebra. Then the map a ¬ a* is an isomorphism of A onto a separating subalgebra A* of C Ž M Ž A.., in the sense that whene¨ er I, J are distinct maximal ideals of A, there are elements a*, b* g A* with a*Ž I . / b*Ž J .. COROLLARY 1.3. An MV-algebra A is semisimple iff A is isomorphic to a separating MV-algebra of w0, 1x-¨ alued continuous functions defined o¨ er some compact Hausdorff space, iff A is isomorphic to an MV-algebra of w0, 1x¨ alued functions defined o¨ er some set. We assume familiarity with lattice-ordered abelian groups, for short
l-groups, for which we refer to w1x. For any l-group G, an element u g G
is said to be a strong unit of G iff for all g g G there is an integer n G 1 such that nu G g. By a morphism f : Ž G, u. ª Ž G9, u9. we mean a group homomorphism f : G ª G9 that also preserves the lattice structure and satisfies the condition f Ž u. s u9. THEOREM 1.4 w7x. For any l-group G with a strong unit u, let G Ž G, u. be the unit inter¨ al w0, u x s h g G < 0 F h F u4 , equipped with the operations ! g s u y g, g [ h s u n Ž g q h., and g(h s 0 k Ž g q h y u.. Then Ži. A s Žw0, u x, !, [ , ( . s G Ž G, u. is an MV-algebra. Žii. Letting, for any morphism f : Ž G, u. ª Ž G9, u9., G Ž f . be the restriction of f to w0, u x, then G is a categorical equi¨ alence Ž i.e., a full, faithful, dense functor . from l-groups with strong unit to MV-algebras. Žiii. The lattice operations on A agree with those of G. Živ. The map J ¬ J l w0, u x is an isomorphism between the lattice of l-ideals of G Ž equipped with set-theoretic inclusion. and the lattice of ideals of A Ž also equipped with inclusion.. All infinite suprema and infima are preser¨ ed by this correspondence. There is a natural isomorphism between G Ž GrJ, urJ . and G Ž G, u.rŽ J l w0, u x.. Thus, in particular, up to isomorphism, every MV-algebra A can be identified with the unit interval of a unique l-group G with strong unit u;
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in symbols, A s G Ž G, u . .
Ž 3.
We say that G is the l-group with strong unit u corresponding to A. As an application of the G functor we have the following. PROPOSITION 1.5. An MV-algebra is s-complete iff e¨ ery bounded sequence of elements in its corresponding l-group G, has a supremum in G. Any s-complete MV-algebra A is semisimple. Proof. The first statement follows from the preservation properties of the G functor w9, Proposition 9.4.3x. The second statement now immediately follows from its counterpart for l-groups Žsee, e.g., w1, Proposition 11.2.2x., in the light of Theorem 1.4Živ..
2. BIMORPHISMS OF MV-ALGEBRAS Let A and B be MV-algebras, and A = B be their cartesian product Žas a set.. DEFINITION 2.1. A bimorphism b of A = B into an MV-algebra C is a function b : A = B ª C such that b Ž1, 1. s 1 and for all a, a1 , a2 g A and b, b1 , b 2 g B, we have ŽI. b Ž a, 0. s 0 s b Ž0, b .; ŽII. b Ž a, b1 k b 2 . s b Ž a, b1 . k b Ž a, b 2 .; b Ž a1 k a2 , b . s b Ž a1 , b . k b Ž a2 , b .; ŽIII. b Ž a, b1 n b 2 . s b Ž a, b1 . n b Ž a, b 2 .; b Ž a1 n a2 , b . s b Ž a1 , b . n b Ž a2 , b .; ŽIV. if b1(b 2 s 0, then b Ž a, b1 . ( b Ž a, b 2 . s 0 and b Ž a, b1 [ b 2 . s Ž b a, b1 . [ b Ž a, b 2 .; symmetrically, if a1(a2 s 0, then b Ž a1 , b . ( b Ž a2 , b . s 0 and b Ž a1 [ a2 , b . s b Ž a1 , b . [ b Ž a2 , b .. We denote by bimŽ A, B, C . the set of all biomorphisms b : A = B ª C. If h : C ª C9 is a homomorphism of MV-algebras and b g bimŽ A, B, C ., then h ( b g bimŽ A, B, C9.. To increase readability, we shall adopt the usual convention that ! is more binding than ( , the latter being more binding than [. We also assume that the lattice operations k and n are less binding than any other operation. Following Chang w2x, in every MV-algebra A we define the distance function distŽ x, y . by dist Ž x, y . s x( ! y [ ! x( y.
Ž 4.
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
PROPOSITION 2.2. Ži. Žii. Žiii.
231
Let b g bimŽ A, B, C .. Then
if b1 F b 2 , then b Ž a, b1 . F b Ž a, b 2 .; b Ž a, ! b . s b Ž a, 1. ( ! b Ž a, b .; b Ž a, b1 [ b 2 . s b Ž a, 1. n Ž b Ž a, b1 . [ b Ž a, b 2 ...
Proof. Ži. This is an immediate consequence of ŽIII. in the above definition. To prove Žii., from b( ! b s 0 and b [ ! b s 1 we get by ŽIV. 0 s b Ž a, b . ( b Ž a, ! b .
whence b Ž a, ! b . F ! b Ž a, b .
Ž 5.
and
b Ž a, ! b . [ b Ž a, b . s b Ž a, 1 . .
Ž 6.
By Ži., b Ž a, b . F b Ž a, 1., i.e., ! b Ž a, 1 . ( b Ž a, b . s 0.
Ž 7.
Recalling w2, 3.15x, we get b Ž a, b . [ distŽ b Ž a, 1., b Ž a, b .. s b Ž a, 1., and hence, b Ž a, b . [ b Ž a, 1. ( ! b Ž a, b . [ ! b Ž a, 1. ( b Ž a, b . s b Ž a, 1., whence by Ž7.
b Ž a, 1 . ( ! b Ž a, b . [ b Ž a, b . s b Ž a, 1 . .
Ž 8.
From Ž6. and Ž8. together with the inequalities b Ž a, ! b . F ! b Ž a, b . and b Ž a, 1. ( ! b Ž a, b . F ! b Ž a, b ., it follows that cancellation can be applied Žsee w2, 1.13x., whence b Ž a, ! b . s b Ž a, 1. ( ! b Ž a, b ., as required. Finally, to prove Žiii. let us write
b Ž a, b1 [ b 2 . s b Ž a, b1 k Ž b1 [ b 2 . . s b Ž a, b1 [ Ž ! b1( Ž b1 [ b 2 . . . s b Ž a, b1 . [ b Ž a, ! b1( Ž b1 [ b 2 . . s b Ž a, b1 . [ b Ž a, ! b1 n b 2 . s b Ž a, b1 . [ Ž b Ž a, ! b1 . n b Ž a, b 2 . . s Ž b Ž a, b1 . [ b Ž a, ! b1 . . n Ž b Ž a, b1 . [ b Ž a, b 2 . . s b Ž a, 1 . n b Ž a, b1 . [ b Ž a, b 2 . , as required. For every element w in an MV-algebra A, the inter¨ al MV-algebra A w s Žw0, w x, 0, w, !w , [w , (w . is obtained by equipping the set w0, w x s x g A < 0 F x F w4 with the operations !w x s w( ! x, x [w y s w n Ž x [ y . , x(w y s !w Ž !w x [w !w y . .
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Note that A w is a trivial MV-algebra iff w s 0. For any b g bimŽ A, B, C ., a g A and b g B, we define the maps b a : B ª C and b b : A ª C by stipulating that for all x g A and y g B,
b a Ž y . s b Ž a, y .
and
b b Ž x . s b Ž x, b . .
Ž 9.
PROPOSITION 2.3. Let b g bimŽ A, B, C . and a g A. Then b a is a homomorphism of B into the inter¨ al MV-algebra w0, b Ž a, 1.x. Similarly, for each b g B, b b is a homomorphism of A into w0, b Ž1, b .x. Proof. This is essentially the content of Proposition 2.2Žii. ] Žiii., upon noting that, by Ži., rangeŽ b a . Žresp., rangeŽ b b .. is contained in w0, b Ž a, 1.x Žresp., in w0, b Ž1, b .x.. THEOREM 2.4. Let A and B be semisimple MV-algebras, respecti¨ ely identified with separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .. using the maps a ¬ a* and b ¬ b* of Theorem 1.2 Ž whence in particular a* s a and b* s b .. Then for each b g bimŽ A, B, w0, 1x., there is a unique pair of maximal ideals Ib g M Ž A. and Jb g M Ž B . such that, for all a g A and b g B,
b Ž a, b . s a Ž Ib . ? b Ž Jb . , where ? denotes natural pointwise multiplication. The function b ¬ Ž Ib , Jb . is a one-to-one correspondence of bimŽ A, B, w0, 1x. onto M Ž A. = M Ž B .. Proof. For any element a g A, let w s b aŽ1. g R, where b a : B ª w0, w x is the homomorphism of Proposition 2.3 given by Ž9.. In the light of Theorem 1.4, write B s G Ž G, u. as in Ž3.. Similarly, write w0, w x s G Ž L, w ., for a suitable l-subgroup L of the additive group R with natural order. Also write b a s G Žh . for a uniquely determined l-homomorphism h : G ª L such that h Ž u. s w. We shall now prove the existence of a maximal ideal Ja g M Ž A. and a real number m a G 0 such that, for all b g B, b aŽ b . s m a bŽ Ja .. In case w s 0 upon choosing m a s 0, any arbitrary maximal ideal in the nonempty set M Ž A. will do. In case w / 0, by Holder’s theorem there is a real number m a such that the map c : ¨ x ¬ xrm a is the only l-homomorphism of L into R sending w into 1. It follows that the composite map c (h : G ª L ª R is an l-homomorphism of G into R sending u to 1, and G Ž c (h . is a homomorphism of B into w0, 1x. By Theorem 1.1, there is a unique maximal ideal Ja g M Ž A. such that, for all b g B,
b a Ž b . s m a ? b Ž Ja . .
Ž 10 .
Dually, for each b g B, there is a maximal ideal Ib g M Ž B . and a real number n b G 0 such that, for all a g A,
b b Ž a . s n b ? a Ž Ib . .
Ž 11 .
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233
In particular, when a s 1 and b s 1, we get 1 s b 1Ž1. s m1 ? 1 s m1 and symmetrically, 1 s b 1 Ž1. s n1 , whence m1 s n1 s 1. By Ž10. and Ž11., we can write a Ž I1 . s n1 ? a Ž I1 . s b 1 Ž a . s b Ž a, 1 . s b a Ž 1 . s m a ,
Ž 12 .
and symmetrically, b Ž J1 . s m1 ? b Ž J1 . s b 1 Ž b . s b Ž 1, b . s b b Ž 1 . s n b ,
Ž 13 .
whence a Ž I1 . ? b Ž Ja . s m a ? b Ž Ja . s b Ž a, b . s n b ? a Ž Ib . s b Ž J1 . ? a Ž Ib . . Ž 14 . Claim: bŽ J1 ..
For all a g A and b g B, we have the identity b Ž a, b . s aŽ I1 . ?
Case 1. b Ž a, 1. / 0. Then by Ž12. and Ž14., there is an ideal Ja g M Ž B . such that, for all b g B,
b Ž a, b . s b Ž a, 1 . ? b Ž Ja . .
Ž 15 .
Assume Ja / J1 Žabsurdum hypothesis., and pick b g B such that bŽ Ja . ) 0 and bŽ J1 . s 0. The existence of such b is ensured by the fact that B is a separating subalgebra of C Ž M Ž B ... Then by Ž15., b Ž a, b . ) 0. On the other hand, by Ž13., b Ž1, b . s bŽ J1 . s 0, thus contradicting the monotony property b Ž a, b . F b Ž1, b . ŽProposition 2.2Ži... We conclude that Ja s J1 , and from Ž15. and Ž12., we get b Ž a, b . s b Ž a, 1. ? bŽ J1 . s aŽ I1 . ? bŽ J1 ., as required. Case 2. b Ž a, 1. s 0. Then by monotony, b Ž a, b . s 0 for all b g B. Recalling Ž12., it is now sufficient to note aŽ I1 . s b Ž a, 1. s 0. The claim is settled. The dependence of I1 and J1 on b is understood. Thus, upon defining Ib s I1 and Jb s J1 , the first statement is proved. To conclude the proof, it is sufficient to note that injectivity follows from A and B being separating subalgebras. To prove surjectivity, one notes that for each I g M Ž A. and J g M Ž B ., the map d : A = B ª w0, 1x given by d Ž a, b . s aŽ I . ? bŽ J . is a bimorphism of A = B into w0, 1x.
3. THE MV-ALGEBRAIC TENSOR PRODUCT DEFINITION 3.1. A bimorphism b g bimŽ A, B, C . is said to be uni¨ ersal iff for every MV-algebra C9 and b 9 g bimŽ A, B, C9. there is a unique homomorphism l: C ª C9 such that l ( b s b 9.
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Stated otherwise, we have a commutative diagram
6
6
b9
6
b
A=B
C
l
C9 We shall now routinely construct an MV-algebra A mMV B and a universal bimorphism mMV of A = B into A mMV B. Construction. Let F be the free MV-algebra w5x over the free generating set A = B. Let I be the ideal of F generated by elements of the following form, for all a, a1 , a2 g A and b, b1 , b 2 g B, where distŽ x, y . is Chang’s distance function Ž4.: distŽŽ0, 0., 0. distŽŽ1, 1., 1. distŽŽ a, b1 k b 2 ., Ž a, b1 . k Ž a, b 2 .. distŽŽ a, b1 n b 2 ., Ž a, b1 . n Ž a, b 2 .. and for all c1 , c 2 g B with c1(c 2 s 0, distŽŽ a, c1 . ( Ž a, c 2 ., 0. and distŽŽ a, c1 [ c 2 ., Ž a, c1 . [ Ž a, c 2 .., and their duals, by analogy with Definition 2.1. Let A mMV B s FrI, and let the map Ž a, b . ¬ a mMV b be the composite of the inclusion map i : A = B ª F and the quotient map p : F ª FrI. Specifically, a mMV b s p Ž a, b . g A mMV B is the equivalence class of Ž a, b . in FrI. A straightforward verification shows that the map mMV : A = B ª A mMV B is a bimorphism. As a matter of fact, the ideal I was just defined in such a way that the composite map A = B ª F ª FrI is a bimorphism. The MValgebra A mMV B is generated by elements of the form a mMV b. The following result establishes that the map A = B ª A mMV B is universal. THEOREM 3.2. For any bimorphism b : A = B ª C, there is precisely one homomorphism l: A mMV B ª C such that lŽ a mMV b . s b Ž a, b .. Proof. The freeness properties of F yield a homomorphism h : F ª C such that h ( i s b . This is the uniquely determined homomorphism that sends each generator Ž a, b . of F into the element b Ž a, b . of C. Since b is bilinear, h sends each generator of I into the zero element of C; in symbols, KerŽh . = I. Let p be the quotient map of F onto FrI. Then we
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235
get a homomorphism l: FrI ª C, also uniquely determined, such that l (p s h. We have a commutative diagram F h
p l
FrI s A mMV B
6
6
6
b
6
i
6
A=B
G In conclusion, l (p ( i s b and lŽ a mMV b . s b Ž a, b ., as required. Remark. A routine verification shows the uniqueness of A mMV B. As a matter of fact, if g : A = B ª D is another universal bimorphism, then there is an isomorphism u : A mMV B ª D such that u Ž a mMV b . s g Ž a, b .. We shall call A mMV B the MV-tensor product of A and B. EXAMPLE. Let Ł n s G ŽŽ1rn.Z, 1. s 0, 1rn, 2rn, . . . , Ž n y 1.rn, 14 be the subalgebra of w0, 1x with n q 1 elements. Then Ł n mMV Ł m s Ł n m . As a matter of fact, by the above construction there is a finitely generated ideal I of the free MV-algebra F over Ž n q 1.Ž m q 1. generators such that Ł n mMV Ł m s FrI. It follows w5, Theorem 3.3.9, Corollary 3.3.11x that Ł n mMV Ł m is semisimple. An application of Theorem 2.4 shows that Ł n mMV Ł m has precisely one maximal ideal. The identity Ł n mMV Ł m s Ł n m now follows from one more application of Theorem 2.4. Similarly, if A and B are finite MV-algebras, then so is A mMV B. Specifically, upon writing A s Ł n1 = ??? = Ł n p as an MV-algebraic product of finite chains Žsee w5, Corollary 4.2.20x. and B s Ł m = ??? = Ł m , using Theorem 2.4 1 q one sees that the semisimple MV-algebra A mMV B has precisely pq maximal ideals. By direct inspection, one can prove A mMV B s =i , j Ł n i mMV Ł m j s =i , j Ł n i m j . For more complex applications, the mMV -tensor product is not easy to visualize. One main source of difficulty is given by the following phenomenon. THEOREM 3.3. There is a semisimple MV-algebra A such that A mMV A is not semisimple. Proof. Let F1 denote the free MV-algebra over one free generator. Equivalently w5, Theorem 3.2.16; 4x, F1 is the MV-algebra of all McNaughton functions of one variable, those continuous functions f : w0, 1x ª w0, 1x consisting of finitely many linear pieces of the form ax q b, a, b g Z.
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For each j g w0, 1x, let Jj and Oj respectively denote the maximal ideal and the germinal ideal of F1 at j ; in symbols, Jj s f g F1 f Ž j . s 0 4 , Oj s f g F1 f s 0 on an open neighborhood of j in w 0, 1 x 4 . By w8, Proposition 3.6x, Oj s Jj iff j is irrational. Assume j to be irrational. Then the quotient MV-algebra A s F1rOj s F1rJj can be visualized in the following two equivalent ways: Ži. either as the subalgebra of w0, 1x given by all possible values of McNaughton functions of F1 at j . Thus, in particular, 04 is the only ideal of A, and A is semisimple; Žii. or, as the MV-algebra of germs at j of McNaughton functions. Recall that two functions p and q have the same germ at j Žin symbols, p ;j q . iff they coincide over some open neighborhood of j in w0, 1x. Let C Žw0, 1x 2 . denote the MV-algebra of all continuous w0, 1x-valued functions on the unit square w0, 1x 2 . Let the map b˜: F1 = F1 ª C Žw0, 1x 2 . be defined by b˜Ž f, g . s f Ž x . g Ž y . for all f, g g F1. Trivially, whenever f ;j f 9 and g ;j g 9, then the functions b˜Ž f, g . and b˜Ž f 9, g 9. coincide on an open neighborhood of Ž j , j . in w0, 1x 2 . Let the germinal ideal OŽ j , j . be defined by OŽ j , j . s h g C Ž w 0, 1 x
½
2
.
h s 0 on some open neighborhood of Ž j , j . .
5
Let the quotient MV-algebra H be defined by H s C Žw0, 1x 2 .rOŽ j , j . . Then H is the MV-algebra of germs at Ž j , j . of continuous w0, 1x-valued functions on w0, 1x 2 . From b˜ we obtain a map b : A = A ª H, sending each pair of germs Ž frOj , grOj . into the germ at Ž j , j . of the function f Ž x . g Ž y .. Let G be the subalgebra of H generated by the range of b . Then b is a bimorphism of A = A into G. Claim 1: G is not semisimple. As a matter of fact, let x and y be the two identity functions over w0, 1x 2 . The germ of x at Ž j , j . is a member of G, and so is the germ of y at Ž j , j .. Therefore, the free product MValgebra A@ A is a subalgebra of G. By w8, Corollary 4.4x together with Theorem 1.4Živ., A@ A is not semisimple. A fortiori, G is not semisimple. Claim 2: A mMV A is not semisimple. Assume A mMV A is semisimple Žabsurdum hypothesis.. A moment’s reflection shows that A mMV A has precisely one maximal ideal. For otherwise, by Theorem 1.1 there would exist at least two homomorphisms of A mMV A into w0, 1x, and the composition of mMV g bimŽ A, A, A mMV A. with these homomorphisms would
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237
give at least two distinct bimorphisms of A = A into w0, 1x, whence, by Theorem 2.4, at least two elements of M Ž A. = M Ž A., whence A s F1rJj would have at least two maximal ideals, which is impossible. Since, as we have just seen, A mMV A has precisely one maximal ideal and it is assumed to be semisimple, by Theorem 1.2, A mMV A coincides with a subalgebra of w0, 1x. By the universal property of A mMV A, there is a unique homomorphism l: A mMV A ª G such that l ( mMV s b . Since A mMV A is a subalgebra of w0, 1x, l is one-to-one. Since G is generated by the range of b , l is onto G. Thus, A mMV A is isomorphic to G, and by the first claim, A mMV A is not semisimple, a contradiction. 4. THE SEMISIMPLE TENSOR PRODUCT The following is a generalization of Definition 3.1. DEFINITION 4.1. Let K be a class of MV-algebras, and A, B g K. Then a bimorphism b of A = B into an MV-algera C is said to be K-uni¨ ersal iff C g K and for all C9 g K and b 9 g bimŽ A, B, C9., there is a unique homomorphism l: C ª C9 such that b 9 s l ( b . Throughout the rest of this paper we shall restrict attention to the case when K is the class of semisimple MV-algebras. We first construct the semisimple tensor product, as follows. Construction. Given semisimple MV-algebras A and B, let R be the radical of A mMV B, and A m B s Ž A mMV B .rR. Let r : A mMV B ª Ž A mMV B .rR be the quotient map. Then A m B is a semisimple MValgebra w5, Corollary 2.5.18x. Let the map m: Ž a, b . ¬ Ž a m b . send each pair Ž a, b . g A = B into the element Ž a m b .rR g A m B; in symbols, ms r ( mMV . Then m is well defined and is a bimorphism of A = B into A m B, whose range generates A m B. We shall call A m B the semisimple tensor product of A and B. The fundamental theorem of semisimple tensor products states that the bimorphism m: A = B ª A m B is universal for the class of semisimple MV-algebras. THEOREM 4.2. Let A and B be semisimple MV-algebras. Then for e¨ ery bimorphism b of A = B into a semisimple MV-algebra C, there is a unique homomorphism l: A m B ª C such that lŽ a m b . s b Ž a, b .. Proof. The universal property of A mMV B yields a unique homomorphism m : A mMV B ª C such that b s m ( mMV . Since C is semisimple, the kernel of m will include the radical of A mMV B. Thus the map
r Ž a mMV b . s Ž a mMV b . rR ¬ m Ž a mMV b .
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is a well-defined homomorphism l of A m B into C such that lŽ a m b . s b Ž a, b .. Uniqueness of l is routine, because elements of the form r Ž a mMV b . generate A m B. By contrast with the MV-tensor product, the semisimple tensor product allows a concrete visualization as follows. THEOREM 4.3. Let A and B be semisimple MV-algebras, respecti¨ ely identified with separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .. Ž whence the maps a ¬ a* and b ¬ b* are assumed to be identities.. Let p : A = B ª C Ž M Ž A. = M Ž B .. send each pair of elements a g A and b g B into the function f : M Ž A. = M Ž B . ª w0, 1x gi¨ en by f Ž I, J . s a Ž I . ? b Ž J . for all I g M Ž A. and J g M Ž B .. Let C be the MV-algebra generated by the range of p in C Ž M Ž A. = M Ž B ... Then p g bimŽ A, B, C . is uni¨ ersal for the class of semisimple MV-algebras, and hence, C is isomorphic to A m B. Further, C is a separating subalgebra of C Ž M Ž A. = M Ž B ... Proof. Let m: A = B ª A m B be the universal bimorphism of Theorem 4.2. Identify A m B with a separating subalgebra of C Ž M Ž A m B ... Then there is a unique homomorphism l: A m B ª C such that p s l ( m . Since the range of p generates C, then l is surjective. Let Y s F fy1 Ž0. < f g KerŽ l.4 . Then Y is a closed subspace of M Ž A m B ., and the map l amounts to restricting to Y each function in A m B : C Ž M Ž A m B ..; in symbols, lŽ f . s f ° Y. Claim: l is injective. Otherwise Žabsurdum hypothesis., Y is strictly contained in M Ž A m B .. Let H g M Ž A m B . be a maximal ideal which is not a member of Y. Let r H : A m B ª w0, 1x be the quotient map. From our identification of A m B with an MV-algebra of functions, it follows that r H amounts to evaluating at H each function f g A m B; in symbols,
rH Ž f . s f Ž H .
for all f g A m B.
Ž 16 .
The composite map r H ( m is a bimorphism of A = B into w0, 1x. By Theorem 2.4, there is a unique pair Ž I, J . g M Ž A. = M Ž B . such that for all Ž a, b . g A = B, Ž r H ( m .Ž a, b . s aŽ I . ? bŽ J . s Ž c (p .Ž a, b ., where the evaluation map c : C Ž M Ž A. = M Ž B .. ª w0, 1x is defined by c Ž a m b . s aŽ I . ? bŽ J .. It follows that c ( l ( m s c (p s r H ( m , whence Žfrom the fact that the range of m generates A m B . we can write
rH s c ( l .
Ž 17 .
Stated otherwise, for all Ž a, b . g A = B, Ž a m b .Ž H . s aŽ I . ? bŽ J .. By Theorem 1.1, the kernel of c is a maximal ideal P g M Ž C .. The inverse
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
239
image ly1 Ž P . is a maximal ideal Q g M Ž A m B ., and, necessarily, Q g Y. The following diagram illustrates the situation: AmB l
w0, 1x
c
6
p
rH
66
m
6 6
A=B
C
Recalling Ž17., for each f g A m B, we have f Ž H . s rH Ž f . s Ž c ( l. Ž f . s c Ž f ° Y . s Ž f ° Y . Ž Q . s f Ž Q . . Since f Ž H . s f Ž Q . for all f g A m B, and A m B is a separating subalgebra of C Ž M Ž A m B .., it follows that Q s H, thus contradicting the fact that Q g Y and H f Y. The claim is settled and l is an isomorphism of C onto A m B. It is now easy to see that p g bimŽ A, B, C . is a universal bimorphism for the class of semisimple MV-algebras. A moment’s reflection also shows that C is a separating subalgebra of the MV-algebra C Ž M Ž A. = M Ž B ... Dropping the assumption that the maps a ¬ a* and b ¬ b* are identities, their ranges A* and B* will be separating subalgebras of C Ž M Ž A.. and C Ž M Ž B .., respectively. The uniqueness of the semisimple tensor product can then be expressed as follows. COROLLARY 4.4. Let A and B be semisimple MV-algebras. Let the map Ž), ).: A = B ª A* = B* send each pair Ž a, b . into Ž a*, b*.. Let m*: A* = B* ª C Ž M Ž A. = M Ž B .. transform each Ž a*, b*. into the function g: M Ž A. = M Ž B . ª w0, 1x gi¨ en by g Ž I, J . s a*Ž I . ? b*Ž J .. Let us agree to denote by A* m*B* the subalgebra of C Ž M Ž A. = M Ž B .. generated by the range of m*. It follows that A* m*B* is a separating subalgebra of C Ž M Ž A. = M Ž B .., and m* is a bimorphism of A* = B* into A* m*B*, which is uni¨ ersal for semisimple MV-algebras. Further, Ži. There is a unique isomorphism l: A m B ª A* m*B* such that lŽ a m b . s a* m*b*; graphically, m
6
m*
A* m*B*
6
6
A* = B*
AmB l
Ž), ) .
6
A=B
Ž18.
Žii. The map I ¬ lŽ I . is an isomorphism of the lattice of ideals of A m B onto the lattice of ideals of A* m B*. For any ideal I of A m B, let r I : A m B ª Ž A m B .rI be the quotient map; further, let VlŽ I . s F fy1 Ž0. N f g lŽ I .4 : M Ž A. = M Ž B . be the intersection of the zerosets of functions in
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DANIELE MUNDICI
lŽ I .. Then, letting ° denote the restriction operation, there is a unique isomorphism m I : Ž A m B .rI ª Ž A* m*B*. ° VlŽ I . such that rlŽ I . ( l s m I ( r I ; graphically, rI
6
6
Žiii.
rlŽ I .
Ž A* m*B*. ° VlŽ I .
6
A* m*B*
Ž A m B .rI mI
l
6
AmB
Ž19.
For all Ž a, b . g A = B, we ha¨ e the identity
m I Ž Ž a m b . rI . s Ž a* m*b* . ° VlŽ I . .
Ž 20 .
5. MULTIPLICATIVE MV-ALGEBRAS Construction. Let A be a semisimple MV-algebra. The flip automorphism w : A m A ª A m A is generated by the map Ž a1 , a2 . ¬ Ž a2 , a1 ., for all a1 , a2 g A. An ideal J of A m A is said to be in¨ ariant iff w J s J. The diagonal ideal D is the intersection of all maximal invariant ideals of A m A. Let r D : A m A ª Ž A m A.rD be the quotient map. Note that Ž A m A.rD is semisimple, because the ideal D is an intersection of maximal ideals w5, Theorem 2.5.17x. We then define the diagonal map d : A ª Ž A m A.rD by stipulating that for all a g A,
d Ž a. s rD Ž a m 1. .
Ž 21 .
The flip automorphism of A* m*A* canonically induces a flip homeomorphism of M Ž A. = M Ž A. onto itself. Let the diagonal D of M Ž A. = M Ž A. be defined by D s Ž I, I . I g M Ž A . 4 . Recalling from Corollary 4.4Ži. that A* m*A* is a separating subalgebra of C Ž M Ž A. = M Ž A.., let Ž I, J . ¬ f g A* m*A* < f Ž I, J . s 04 be the canonical correspondence between points Ž I, J . g M Ž A. = M Ž A. and maximal ideals K of A* m*A*. ŽThe inverse of this correspondence maps each maximal ideal K of A* m*A* into the intersection Z of the zerosets of all functions in K. Note that Z s Ž I, J .4 is a singleton, because A* m*A* separates points.. Then a maximal ideal K of A* m*A* is invariant iff its corresponding point Ž I, J . is invariant under the flip homeomorphism, iff I s J, iff Ž I, J . g D. By Corollary 4.4, for all a, b g A, we have a m b g D iff a m b g KerŽ r D . iff a* m*b* g KerŽ rlŽ D . . if a* m*b* g lŽ D . iff a* m*b* identically vanishes over D iff a* m*b* belongs to all invariant
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
241
maximal ideals of A* m*A*. Since the zerosets of functions in A* m*A* form a basis of closed sets for the spectral topology of A* m*A*, we get VlŽ D . s F fy1 Ž 0 . f g lŽ D . 4 s F fy1 Ž 0 . f s 0 over D 4 s D . Ž 22 . Let the map e : A* ª Ž A* m*A*. ° D be defined by stipulating that, for all a* g A*, e Ž a*. s Ž a* m*1. ° D. In other words, for all Ž I, I . g D,
Ž e Ž a*. . Ž I, I . s a* Ž I . .
Ž 23 .
Then e is a homomorphism, and we have a commutative diagram 6
d
mD
6
6
A* m*A* ° D
A )
e
6
Ž A m A.rD
A*
Ž24.
As a matter of fact, from Ž20. ] Ž23. we get Ž m D ( d .Ž a. s Ž m D ( r D .Ž a m 1. s Ž a* m*1. ° VlŽ D . s Ž a* m*1. ° D s e Ž a*.. PROPOSITION 5.1. e are one-to-one.
For e¨ ery semisimple MV-algebra A, both maps d and
Proof. In the light of diagram Ž24., since both maps ) and m D are isomorphisms, it suffices to prove that e is one-to-one. Let 0 / a* g A*, with the intent of proving 0 / e Ž a*.. By Theorems 1.2 and 4.3, A* and A* m*A* are separating subalgebras of C Ž M Ž A.. and C Ž M Ž A. = M Ž A.., respectively. Let J g M Ž A. be such that a*Ž J . / 0. Then by Ž23., since Ž J, J . g D we can write 0 / a*Ž J . s Ž e Ž a*..Ž J, J ., whence e Ž a*. / 0, as required. PROPOSITION 4.2. For any semisimple MV-algebra A, the following conditions are equi¨ alent: Ži. The diagonal map d : A ª Ž A m A.rD is onto; Žii. The map e is an isomorphism of A* onto Ž A* m*A*. ° D; Žiii. d isomorphically maps A onto Ž A m A.rD; Živ. A* is closed under pointwise multiplication. Proof. The equivalences of Ži. ] Žiii. follow from Proposition 5.1, recalling that both maps ) and m D are isomorphisms. For the implication Živ. ª Žii., by Corollary 4.4 every element of Ž A* m*A*. ° D is in the MV-algebra generated by elements of the form Ž a* m*b*. ° D, i.e., elements of the form a* ? b*, where a*, b* g A* and ? is pointwise multiplication. Since by hypothesis, a* ? b* g A*, we conclude by Ž23. that e is surjective. Conversely, in order to prove Žii. ª Živ., adopting the notation
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DANIELE MUNDICI
>
of the commutative diagrams Ž18., Ž19., Ž24., let us define the operation : A* = A* ª A* by a* b* s Ž ey1 ( rlŽ D . ( m*.Ž a*, b*.. By Corollary 4.4, Ž rlŽ D . ( m*.Ž a*, b*. s Ž a* m*b*. ° D. By definition of e , together with Corollary 4.4, the function Ž a* m*b*. ° D: D ª w0, 1x is the e-image of the function a* ? b*: M Ž A. ª w0, 1x given by Ž a* ? b*.Ž I . s a*Ž I . ? b*Ž I . for all I g M Ž A.. This shows that the binary operation coincides with pointwise multiplication.
>
>
DEFINITION 5.3. A semisimple MV-algebra A is called multiplicati¨ e iff it satisfies any of the equivalent conditions of the above proposition. PROPOSITION 5.4. Let A be a semisimple multiplicati¨ e MV-algebra. Let the map w: A = A ª A be defined by a1w a2 s Ž dy1 ( r D . Ž a1 m a2 . . Then w has the following properties: Ži. Ž a1w a2 .* s aU1 ? aU2 , where ? is pointwise multiplication; Žii. w is commutati¨ e, associati¨ e, has neutral element 1, and is a bimorphism of A = A into A. Thus, for all a, b, c g A, aw0 s 0, and w distributes o¨ er the lattice operations of A in the following sense: Žiii. cwŽ a k b . s Ž cw a. k Ž cwb ., and Živ. cwŽ a n b . s Ž cw a. n Ž cwb .. For all a, b g A with a(b s 0, we ha¨ e Žv. Ž cw a. ( Ž cwb . s 0 and Žvi. cwŽ a [ b . s Ž cw a. [ Ž cwb .. Proof. Ži. Consider the following commutative diagram, as given by Corollary 4.4 and the assumption that A is multiplicative: dy1
mD
)
ey1
6
6
Ž A* m*A*. ° D
A
A*
6
rlŽ D .
6
6
A* m*A*
6
m*
Ž A m A.rD
6
rD
l
Ž), ) .
6
A* = A*
AmA
6
m
6
A=A
We then have
Ž a1w a2 . * s Ž *( dy1 ( r D . Ž a1 m a2 . s Ž ey1 ( rlŽ D . ( m*( Ž ), ) . . Ž a1 , a2 . s Ž ey1 ( rlŽ D . . Ž aU1 m*aU2 . s Ž ey1 . Ž Ž aU1 m*aU2 . ° D . s aU1 ? aU2 ,
LOOMIS ] SIKORSKI THEOREM FOR MV-ALGEBRAS
243
as required. The proof of Žii. follows from the definition of semisimple tensor product, together with the commutativity of the above diagram. The fact that w is a bimorphism of A = A into A is proved by direct inspection of the above diagram. The proof of conditions Žiii. ] Žvi. now follows by Definition 2.1 using Propositions 2.2 and 2.3. DEFINITION 5.5. The above operation w: A = A ª A is called the natural product of the semisimple multiplicative MV-algebra A. The following is an immediate consequence of the definitions and of Proposition 5.2. PROPOSITION 5.6. Ži. E¨ ery boolean algebra Ž i.e., e¨ ery MV-algebra satisfying the idempotence equation x [ x s x . is multiplicati¨ e, and the natural product coincides with the lattice operation n. Žii. E¨ ery subalgebra of w0, 1x which is closed under multiplication of real numbers, is multiplicati¨ e. More generally, if A is an MV-algebra of w0, 1x-¨ alued continuous functions o¨ er some compact Hausdorff space X and A is closed under pointwise multiplication of functions, then A is multiplicati¨ e and its natural product coincides with pointwise multiplication. By Corollary 1.3 together with Propositions 5.2 and 5.4, the class of MV-algebras given in Žii. above is the most general possible example of a Žsemisimple. multiplicative MV-algebra. Remark. Our analysis of tensor products in semisimple MV-algebras allows an intrinsic, representation-free formulation of the intuitive notion of A being ‘‘closed under multiplication.’’ This fact may be of help in further investigations of intrinsic multiplication operations in classes of nonsemisimple MV-algebras}e.g., the MV-algebras arising from a nonstandard model of the multiplicative algebra w0, 1x.
6. THE MV-ALGEBRAIC LOOMIS]SIKORSKI THEOREM In this section, we assume familiarity with the spectral topology w1x, and states w6x of l-groups. Let A be a s-complete MV-algebra, and let G be its corresponding l-group with strong unit u, as given by Theorem 1.4. Then by Proposition 1.5, A is semisimple, and by Proposition 1.5, every bounded sequence of elements of G has a supremum in G: for short, following w6x, G is Dedekind s-complete. For every maximal ideal J of A, the quotient MV-algebra ArJ is canonically identified with a subalgebra of w0, 1x, in the light of Theorem 1.1. We say that J is discrete iff ArJ is finite. Similarly, a maximal l-ideal I of G is said to be discrete iff the quotient l-group GrI is cyclic. In the light of Theorem 1.4Živ., it is easy to
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DANIELE MUNDICI
see that I is discrete iff so is the ideal J s I l w0, u x g M Ž A.. For any maximal l-ideal I of G, there is precisely one embedding of GrI into the totally ordered group R in such a way that the strong unit urI is mapped into the element 1. This is a well-known consequence of Holder’s theorem ¨ w1, 2.6, 12.2.1x. Let SŽ G, u. be the state space of G, i.e., the convex set of group homomorphisms j : G ª R such that j Ž u. s 1 and 0 F j Ž x . for all 0 F x g G. Let e SŽ G, u. be the space of extremal points of S Ž G, u. with the natural topology w6x. By w6, Theorem 15.32, Corollary 12.19x, e S Ž G, u. is canonically homeomorphic to the space MaxspecŽ G . Ži.e., m G, in the notation of w1x. of maximal l-ideals of G with the spectral topology w1x. By Theorem 1.4Žiii. ] Živ., MaxspecŽ G . is in turn canonically homeomorphic to M Ž A.. For each extremal state s g e SŽ G, u., and corresponding maximal l-ideal I of G, the range sŽ G. of s coincides with the quotient l-group GrI, once Ž GrI, urI . is canonically embedded into ŽR, 1.. LEMMA 6.1. Let A be a s-complete MV-algebra. Let X s M Ž A. be the space of maximal ideals of A with the spectral topology. For each J g X, let ArJ be identified with the set of possible ¨ alues at J of functions a*, for all a g A. Then the map a ¬ a* of Ž2. is an isomorphism of A onto the MV-algebra E consisting of all continuous functions g: X ª w0, 1x such that g Ž J . g ArJ for each discrete J g X. This map preserves all countable suprema in the sense that, for e¨ ery sequence a i g A, the element ŽE a i .* is the supremum in C Ž X . of the functions aUi , and coincides with the supremum in E of the aUi . Ž Note that ŽE a i .* need not coincide with the pointwise supremum of the functions aUi .. Proof. Let G be the corresponding l-group of A with strong unit u; in symbols, A s G Ž G, u.. Then G is Dedekind s-complete. In the light of Theorem 1.4Živ., let us canonically identify the homeomorphic spaces X and MaxspecŽ G .. From w6, Corollary 9.10x, we get that X is basically disconnected; in other words, the closure of every open Fs subset of X is open. Let G h be the l-group of all real-valued continuous functions over X. Note that the constant function 1 is a strong unit for G h. By w6, Corollary 9.3x, G h is Dedekind s-complete. By w6, Corollary 9.14x, together with our introductory discussion in this section, there is an l-isomorphism c of G onto the l-subgroup B of G h consisting of all continuous functions f : X ª R such that f Ž J . g GrJ : R for each discrete maximal l-ideal J of G. Again, we are canonically identifying Ž GrJ, urJ . with a subgroup of ŽR, 1.. A moment’s reflection shows that B is large in G h Žequivalently, G h is an essential extension of B .: to this purpose, it is sufficient to note that for each 0 - f g G h, there is an element 0 - g g B and a positive integer n such that g F nf. By w1, Corollary 12.1.12x, all existing suprema in B are preserved in G h. By w6, Lemma 9.12x, countable suprema of G are mapped by c into countable suprema of B or, what is
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the same, countable suprema in G h. By definition of the canonical homeomorphism e SŽ G, u. ( MaxspecŽ G ., we obtain that the l-isomorphism G ( B is generated by the map a ¬ a*. Thus, by Theorem 1.4, the map a ¬ a* coincides with the isomorphism c ° A of A onto the subalgebra E of C Ž X . given by the w0, 1x-valued functions of B. By Proposition 1.5, both C Ž X . s G Ž G h, 1. and E s G Ž B, 1. are s-complete. Since countable suprema are preserved by the map c , and B is large in G h, then, going back to MV-algebras, preservation of countable suprema is a consequence of Theorem 1.4Žiii. ] Živ.. By a s-field of sets over a nonempty set X we mean a s-complete boolean algebra of 0, 14 -valued functions over X, where countable suprema are given by pointwise sups. This is equivalent to the usual definition Žsee, e.g., w10x.. Similarly, following w9, 8.1.1x, by a tribe over X we mean an MV-algebra F of w0, 1x-valued functions on X such that, for each sequence f 1 , f 2 , . . . g F , the pointwise supremum f s sup i f i also belongs to F. Thus, in particular, in every tribe F we have Ži. 0 g F ; Žii. whenever f g F , then ! f s 1 y f g F ; Žiii. if f 1 , f 2 , . . . g F , then 1 n Ý`is1 f i g F , where it is understood that 1 n ` s 1. Note that F is a s-complete MV-algebra. On the other hand, not every s-complete MV-algebra is isomorphic to a tribe Žalready a s-complete boolean algebra need not be isomorphic to a s-field of sets.. Let A be a s-complete MV-algebra. Then a homomorphism h : F ª A is said to be a s-homomorphism iff for each sequence f 1 , f 2 , f 3 . . . g F , letting f s sup i f i be their pointwise supremum, h Ž f . coincides with the supremum Ei h Ž f i . in A. The following result generalizes the well-known Loomis]Sikorski theorem w10, 29.1, p. 93x. THEOREM 6.2. Let A be a s-complete MV-algebra. Let X s M Ž A.. Then there is a tribe F o¨ er X and a s-homomorphism h of F onto A. Proof. As in the proof of the previous lemma, the preservation properties of the G functor Ž1.4Živ.., together with w6, Corollary 9.10x, ensure that X is basically disconnected, and by w6, Corollary 9.3x, C Ž X . is s-complete. By Lemma 6.1, we can safely identify A with a s-complete and separating subalgebra of C Ž X .. While countable suprema in A need not coincide with pointwise suprema of functions, the proof of Lemma 6.1 has shown that all existing suprema in A : C Ž X . are preserved in C Ž X .. For any function f : X ª w0, 1x, let us agree to say that f coincides almost e¨ erywhere with a continuous function g g A, in symbols, f f g, iff the set
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J g X < f Ž J . / g Ž J .4 is meager Žalso called a set of first category.. Stated otherwise, the set is the countable union of subsets of X whose closure has empty interior. Note that if we also have f f h and h g A, then necessarily g s h. Thus, letting F : w0, 1x X be the set of functions f that coincide almost everywhere with some function g g A, we have a map h of F onto A. It is easy to see that F is an MV-algebra and that h is a homomorphism. There remains to be shown that F is a tribe and that h is a s-homomorphism. It is sufficient to prove that, for each sequence f 1, f 2 , . . . g F, sup f i f sup h Ž f i . f i
i
E h Ž fi . . i
Since by construction, f i f h Ž f i ., the fact that sup i f i f sup i h Ž f i . immediately follows by definition of meager set. To conclude the proof, let f s sup i h Ž f i . and fˆs Ei h Ž f i .. Recall that fˆ is also the supremum Ei h Ž f i . ˆ Further, since each h Ž f i . is continuous, the set in C Ž X .. Trivially, f F f. < Y s J g X f is not continuous at J 4 is meager in X, by the Baire category theorem w11, Baire’s Theorem 2, pp. 12]13x.
ˆ For otherwise Žabsurdum hypothesis., assume D s J g Claim: f f f. X < f Ž J . / fˆŽ J .4 is not meager. Then D l Ž X R Y . contains at least one element J, because D cannot be contained in the meager set Y. Let 0 - e s fˆŽ J . y f Ž J .. Since both f and fˆ are continuous at J, there is an open neighborhood U of J such that fˆŽ I . y f Ž I . ) 2 er3 for all I g U. Since X is basically disconnected, U may be assumed clopen. Let the function d : X ª w0, 1x be defined by d Ž J . s 0 outside U, and d Ž J . s er2 over U. Then by definition of U, d g C Ž X .. Now the function fˇs fˆy d is ˆ and for all i s 1, 2, . . . , an element of C Ž X ., is strictly smaller than f, ˇ This contradicts our assumption that fˆ is the supremum Ei h Ž f i . f i F f F f. in C Ž X .. ˆ We have proved that f and fˆ only differ on a meager set, i.e., f f f. This completes the proof. Closing a circle of ideas, and putting the two conditions of s-completeness and multiplicativeness together, we have the following. COROLLARY 6.3. Let A be a s-complete multiplicati¨ e MV-algebra. Let X s M Ž A.. Then there is a tribe F o¨ er X which is closed under pointwise multiplication, and a s-homomorphism h of F onto A such that, for all f, g g F , h Ž fg . s h Ž f .wh Ž g .. Proof. We adopt the same notation and terminology of the previous proof. Since A is multiplicative, by Proposition 5.6Žii., whenever J is a discrete maximal ideal of A, then the quotient algebra ArJ : w0, 1x must
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coincide with the two-element boolean algebra 0, 14 . Again letting F : w0, 1x X be the set of functions almost everywhere coinciding with some function in A, we have a map h of F onto A. F is an MV-algebra of functions closed under pointwise multiplication, and h is a homomorphism. By Proposition 5.6, h Ž fg . s h Ž f .wh Ž g .. For each sequence f 1 , f 2 , . . . g F , we must prove sup i f i f sup i h Ž f i . f Ei h Ž f i .. Letting f s ˆ Again, sup i h Ž f i . and fˆs Ei h Ž f i ., we immediately obtain sup i f i f f F f. the set Y of discontinuity points of f is meager in X. In order to prove ˆ by way of contradiction, assume the set D of points where f / fˆ that f f f, is not meager. Let J g D l Ž X R Y .. Let 0 - e s fˆŽ J . y f Ž J .. There is a clopen neighborhood U of J such that fˆŽ I . y f Ž I . ) 2 er3 for all I g U. Case 1. The set of maximal ideals I such that ArI s 0, 14 is dense over U. Then, by continuity, over U, each f i , as well as f, identically vanishes, and fˆs 1. Let the function fˇ be defined by fˇs fˆ outside U, and fˇs 0 over U. Then fˇg A, and for all i, f i F f F fˇ; on the other hand, fˇ is ˆ thus contradicting the definition of f.ˆ strictly smaller than f, Case 2. For some nonempty open set W : U, there are no ideals I such that ArI s 0, 14 . Then, again, we can safely assume W clopen. Let the function d : X ª w0, 1x be defined by d Ž J . s 0 outside W, and d Ž J . s er2 over W. Then ˇ d g A. Similarly, the function fˇs fˆy d is in A, and for all i, f i F f F f, ˇ ˆ ˆ whence f is strictly smaller than f, thus contradicting the definition of f.
ACKNOWLEDGMENTS The author is indebted to Alberto Marcone for his advice concerning Baire category arguments. He is also greatly indebted to Anatolij Dvurecenskij and Beloslav Riecan ˇ ˇ for many inspiring discussions concerning the relationships between tribes and MV-algebraic measure theory.
REFERENCES 1. A. Bigard, K. Keimel, and S. Wolfenstein, ‘‘Groupes et Anneaux Reticules,’’ ´ ´ Lecture Notes in Mathematics, Volume 608, Springer-Verlag, Berlin, 1971. 2. C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 Ž1958., 467]490. 3. C. C. Chang, A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93 Ž1959., 74]80.
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4. R. Cignoli and D. Mundici, An invitation to Chang’s MV-algebras, in ‘‘Advances in Algebra and Model Theory’’ ŽM. Droste and R. Gobel, Eds.., pp. 171]197, Gordon and ¨ Breach Publishing Group, Reading, UK Ž1997.. 5. R. Cignoli, I. M. L. D’Ottaviano, and D. Mundici, ‘‘Algebras of Łukasiewicz Logics,’’ second edition. Editions CLE, State University of Campinas, Campinas, S.P., Brazil, 1995. ŽIn Portuguese.. 6. K. R. Goodearl, ‘‘Partially Ordered Abelian Groups with Interpolation,’’ American Mathematical Society, Providence, RI, 1986. 7. D. Mundici, Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 Ž1986., 15]63. 8. D. Mundici, Free products in the category of abelian l-groups with strong unit, J. Algebra 113 Ž1988., 89]109. 9. B. Riecan ˇ and T. Neubrunn, ‘‘Integral, Measure, and Ordering,’’ Kluwer Academic Publishers, Dordrecht, 1997. 10. R. Sikorski, ‘‘Boolean Algebras,’’ Springer-Verlag, Berlin, 1960. 11. K. Yosida, ‘‘Functional Analysis,’’ sixth ed., Springer-Verlag, Berlin, 1980.