GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS

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1 IJMMS 2003:11, PII. S X Hindawi Publishing Corp. GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS SERGIO CELANI Received 14 November 2001 We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boolean equivalence, and open filters Mathematics Subject Classification: 06E25, 03G Introduction. In [4], Halmos generalizes the notion of Boolean homomorphism introducing the notion of join-hemimorphism between two Boolean algebras. A join-hemimorphism is a mapping between two Boolean algebras preserving 0 and. As it is shown by Halmos in [4] and by Wright in [10], there exists a duality between join-hemimorphism and Boolean relations. On the other hand, Jónsson and Tarski in [5, 6] introduce the class of Boolean algebras with operators BAO. They showed that a Boolean algebra endowed with a family of operators can be represented as a subalgebra of a power algebra X, where the operators of X are defined by means of certain finitary relations on X. This class of algebras plays a key role in modal logic, and has very important applications in theoretical computer science see, e.g., [1, 2]. The Halmos-Wright duality can be extended to a duality between BAO and Boolean spaces endowed with a set of finitary relations, which are a generalization of the Boolean relations. The aim of this paper is the study of an extension of these dualities. In Section 2, we recall some notions on Boolean duality. In Section 3, we define the notion of generalized join-hemimorphism as a mapping between a finite product of a family of Boolean algebras {B 1,...,B n } into a Boolean algebras B 0 such that it preserves 0 and in each coordinate. We will prove that there exists a duality between generalized join-hemimorphism and certain n + 1-ary relations called generalized Boolean relations. This duality extends the duality given by Halmos and Wright. In Section 4, we define the generalized modal algebra as a pair {B 0,B 1,...,B n },, where B 0,B 1,...,B n are Boolean algebras and : n i=1 B i B 0 is a generalized join-hemimorphism.

2 682 SERGIO CELANI In Section 5, we define the notions of generalized subalgebra and generalized Boolean equivalence and we prove that these notions are duals. Similarly, in Section 6, we introduce the generalized congruences and generalized open filters and we will prove that there exists a bijective correspondence between them. 2. Preliminaries. A topological space is a pair X, X, orx, for short, where X is a subset of X that is closed under finite intersections and arbitrary unions. The set X is called the set of open sets of the topological space. The collection of all closed sets of a topological space X, X is denoted by X. The set ClopX is the set of closed and open sets of X, X. A Boolean space X, X is a topological space that is compact and totally disconnected, that is, given distinct points x,y X, there is a clopen subset U of X such that x U and y U. If X, X is a Boolean space, then ClopX is a basis for X and is a Boolean algebra under set-theoretical complement and intersection. Also, the application H X : X Ul ClopX, 2.1 given by H X x ={U ClopX : x U}, is a bijective and continuous function. To each Boolean algebra A, we can associate a Boolean space SpecA whose points are the elements of UlA with the topology determined by the clopen basis β A A ={β A a : a A}, where β A : A UlA is the Stone mapping defined by β A a = { P UlA : a P }. 2.2 By the above considerations, we have that, if X is a Boolean space, then X SpecClopX, and if A is a Boolean algebra, then A ClopSpecA. Let B be a Boolean algebra. The filter ideal generated by a set H A will be denoted by FH IH. The lattice of all filters ideals of B is denoted by FiA IdA. Let Y be a subset of a set X. The theoretical complement of Y is denoted by Y c = X Y. 3. Generalized join-hemimorphisms Definition 3.1. Let B 0,B 1,...,B n be Boolean algebras. A generalized joinhemimorphism is a function : n i=1 B i B 0 such that 1 a 1,...,a i 1,0,a i+1,...,a n = 0, 2 a 1,...,a i 1,x y,a i+1,...,a n = a 1,...,a i 1,x,a i+1,...,a n a 1,...,a i 1,y,a i+1,...,a n.

3 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 683 It is easy to see that each generalized join-hemimorphism h : n i=1 B i B 0 is monotonic in each variable, that is, x,y B i, x y, then a 1,...,x,...,a n a1,...,y,...,a n. 3.1 A generalized join-hemimorphism h : n i=1 B i B 0 defines a generalized meethemimorphism : n i=1 B i B 0 as follows: a 1,a 2,...,a n = a1, a 2,..., a n. 3.2 It is clear that preserves 1 and, and is monotonic in each variable. Let B 0,B 1,...,B n be Boolean algebras and let : n i=1 B i B 0 be a generalized join-hemimorphism. Let F = F 1 F 2 F n, where F i is a filter of B i,for 1 i n. We consider the set F = { y B 0 : x i F i x1,x 2,...,x n y }. 3.3 Theorem 3.2. Let B 0,B 1,...,B n be Boolean algebras and let : n i=1 B i B 0 be a generalized join-hemimorphism. Let F i be a proper filter of B i,for1 i n. Then 1 F is a proper filter of B 0, 2 if P UlB 0 and F P, then there exist Q i UlB i,for1 i n such that F i Q i, Q 1 Q 2 Q n 1 P. 3.4 Proof. 1 It is easy to take into account that the function is monotonic in each variable. 2 Consider the family 1 = { Q 1 F i B1 : F1 Q 1, Q 1 F 2 F n P }. 3.5 We note that 1 because F 1 1. By Zorn s lemma, there exists a maximal element Q 1 in 1. We prove that Q 1 UlB 1.Leta B 1 and suppose that a, a Q 1. Consider the filter F a = FQ 1 {a} and F a = FQ 1 { a}.sinceq 1 is maximal in 1,thenF a, F a 1. So, there exist x 1,x 2,...,x n F a F 2 F n and y 1,y 2,...,y n F a F 2 F n such that x 1,x 2,...,x n P and y 1,y 2,...,y n P. Sincex 1 F a and y 1 F a,thenq 1 a x 1 and q 2 a y 1, for some q 1,q 2 Q 1.AsQ 1 is a filter of B 1, q = q 1 q 2 Q 1, q a x 1, and q a y 1, and as each F i is a filter, we get z i = x i y i F i, for i = 2,...,n.Then q,z 2,...,z n = q a a,z2,...,z n = q a q a,z 2,...,z n = 3.6 q a,z 2,...,z n q a,z2,...,z n P.

4 684 SERGIO CELANI If q a,z 2,...,z n P, then q a,z 2,...,z n q1 a,x 2,...,x n x1,x 2,...,x n P, 3.7 which is a contradiction. If q a,z 2,...,z n P, then we deduce that y 1,y 2,...,y n P, which also is a contradiction. Thus, a Q 1 or a Q 1. Suppose that we have determinate ultrafilters Q 1,...,Q k in B 1,...,B k, respectively, such that F i Q i,for1 i k, and Q 1 Q k F n P. Consider the set k+1 = { Q k+1 B k+1 : F i Q i, Q 1 Q k+1 F n P }. 3.8 We note that k+1 because F k+1 k+1. By the Zorn s lemma, there exists a maximal element Q k+1 in k+1. As in the above case, we can prove that Q k+1 UlB k+1. Therefore, we have ultrafilters Q 1,...,Q n in B 1,...,B n, respectively, such that F i Q i and Q 1 Q 2 Q n P. It is easy to check that the last inclusion implies that Q 1 Q 2 Q n 1 P. Example 3.3. Let X 0,...,X n be sets. Let R n i=0 X i. Then, the function R : X 1 X n X 0, defined by R U1,U 2,...,U n = { x0 X 0 : R x 0 U1 U 2 U n }, 3.9 where Rx 0 ={x 1,...,x n n i=1 X i : x 0,x 1,...,x n R}, is a generalized join-homomorphism. Definition 3.4. Let X 0,...,X n be Boolean spaces. Consider a relation R n i=0 X i.thenr is a generalized Boolean relation if 1 Rx is a closed subset in the product topology of X 1 X n, for each x X 0 ; 2 for all U i ClopX i,with1 i n, R U 1,...,U n ClopX 0. We note that if 1 i 2, then we have the notion of Boolean relation as defined in [4]. Theorem 3.5. Let B 0,B 1,...,B n be Boolean algebras and let : n i=1 B i B 0 be a generalized join-hemimorphism. Then the relation R n i=0 UlB i, defined by P,P1,...,P n R P 1 P n 1 P, 3.10

5 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 685 is a generalized Boolean relation such that β B0 a1,...,a n = R βb1 a1,...,βbn an, 3.11 for all a 1,...,a n n i=1 B i. Proof. We prove that for P UlB 0, R P = { βb1 a1,...,βbn an c : a1,...,a n P } If P 1,...,P n R P and P1,...,P n { βb1 a1,...,βbn an c : a1,...,a n P }, 3.13 then, for some a 1,...,a n P, wegetp 1,...,P n β B1 a 1,...,β Bn a n, that is, a i P i for 1 i n. It follows that a 1,...,a n P, which is a contradiction. The other direction is similar. Thus, R P is a closed subset of n i=1 UlB i. Equality 3.11 follows by Theorem 3.2. We note that the relation R n i=0 UlB i defined in Theorem 3.5 also can be defined using the notion of generalized meet-hemimorphism in the following way: P,P1,...,P n R 1 P P 1 + +P n, 3.14 where P 1 + +P n ={a 1,...,a n n i=1 B i : a i P i, for some 1 i n}. Lemma 3.6. Let X 0,...,X n be Boolean spaces. Consider a relation R n i=0 X i. Suppose that for all U i ClopX i,with1 i n, R U 1,...,U n ClopX 0. Then the following conditions are equivalent: 1 R is a generalized Boolean relation; 2 if H x0 x 0,...,H Xn x n R R, then x 0,...,x n R. Proof Suppose that H X0 x 0,...,H Xn x n R R and x 0,...,x n R. SinceRx 0 is a closed subset of n i=1 X i,thereexistu i ClopX i such that Rx 0 U 1,...,U n = and x i U i. Then, x 0 R U 1,...,U n, thatis, U 1,...,U n H X1 x 1 H Xn x n, which is a contradiction We have to prove that Rx is a closed subset of X 0 X 1. Suppose that x 1,...,x n Rx. Then, H X0 x,...,h Xn x n R R, that is, for each 1 i n, thereexistu i D i such that Rx U 1,...,U n = and x i U i. Thus, Rx is a closed subset.

6 686 SERGIO CELANI By the above results, we deduce that there exists a duality between generalized Boolean relations and generalized join-hemimorphisms. Theorem 3.7. Let X 0,...,X n be Boolean spaces. Let R n i=0 X i be a generalized Boolean relation. Then the mapping R :ClopX 1 ClopX n ClopX 0, defined as in Example 3.3, is a generalized join-hemimorphism such that H x0 x 0,...,H Xn x n R R if and only if x 0,...,x n R, for all x 0,...,x n n i=0 X i. Proof. It is clear that if x 0,...,x n R, thenh x0 x 0,...,H Xn x n R R. The other direction follows by Lemma 3.6. As an application of the above duality, we prove a generalization of the result that asserts that the Boolean homomorphisms are the minimal elements in the set of all join-hemimorphisms between two Boolean algebras see [3]. Let {B i }={B 0,...,B n } be a family of Boolean algebras. Let GJH n i=1 B i,b 0 be the set of all generalized join-hemimorphisms between n i=1 B i and B 0 endowed with the pointwise order. Similarly, let GBR n i=0 X i be the set of all generalized Boolean relations defined in n i=0 X i endowed with the pointwise order. Let 1 and 2 GJH n i=1 B i,b 0 and let R 1 and R 2 GBR n i=0 UlB i be the associate generalized Boolean relations. It is clear that 1 2 if and only if R 1 R 2. Theorem 3.8. An element of GBR n i=0 X i is minimal if and only if it is a continuous function. Proof. Let R n i=0 X i be a minimal element in GBR n i=0 X i. We prove that it is a function. Let x X 0 and let x, y n i=1 X i such that x, y Rx. Suppose that x y. Then, x i y i for some 1 i n. Then, there exist U i ClopX i such that x i U i and y i U i. Consider the sequence U = X 1,...,U i,...,x n. Then, x U and y U. We define an auxiliary relation RU n i=0 X i by Rx, if x R U, RU x = Rx U, if x R U We prove that RU is a generalized Boolean relation. It is clear that R U is closed. Let V = V 1,...,V n ClopX 1 ClopX n.then V = { x X 0 : R U x V } R U = { x X 0 : x R U, Rx U V } { x X 0 : x R U, Rx V } 3.16 = R U V R V R U c Clop X0.

7 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 687 Thus, RU is a generalized Boolean relation. It is clear that R U R, that is, R is not a minimal element in GBR n i=0 X i, which is a contradiction. Thus, R is a continuous function. If R is continuous function, then it is easy to see that R is a minimal element in GBR n i=0 X i. 4. Generalized modal algebras. Now we consider a finite family of Boolean algebras endowed with a generalized join-hemimorphism. This class of structures is a generalization of the notion of Boolean algebra with operators. In the sequel, we will write {X i } to denote the family of sets {X i :1 i n}. Definition 4.1. Let {B i } be a family of Boolean algebras. A generalized modal algebra is a pair B = {B i },, where : n i=1 B i B 0 is a generalized join-hemimorphism. Definition 4.2. A generalized modal space is a structure = {X i },R, where X 0,X 1,...,X n are Boolean spaces and R is generalized Boolean relation. Definition 4.3. Let B = {B i }, B and A = {A i }, A be two generalized modal algebras. A generalized homomorphism between B and A is a finite sequence h = h 0,h 1,...,h n such that 1 h i : B i A i is a Boolean homomorphism for each 1 i n, 2 h 0 B a 1,...,a n = A h 1 a 1,...,h n a n. We write h : B A to denote that there exists a generalized homomorphism h between the generalized modal algebras B and A. We say that a generalized homomorphismh between two generalized modal algebras A and B isinjective if each Boolean homomorphism h i,1 i n, is injective, and h is surjective if each h i surjective. Finally, h is a generalized isomorphism if h is bijective generalized homomorphism. Theorem 4.4. Let B = {B i }, be a generalized modal algebra. Then the structure B = {UlB i },R is a generalized modal space such that B is isomorphic to the generalized modal algebra A B = {ClopUlB i },R R. Proof. It is clear that R is a generalized join-hemimorphism. By Theorem 3.5 we have that β = β B0,β B1,...,β Bn is a generalized homomorphism, and since each β Bi : B i ClopUlB i, for1 i n, is a Boolean isomorphism, then β is an generalized isomorphism. Definition 4.5. Let = {X i },R and g = {Y i },S be two generalized modal spaces. A generalized morphism between and is a sequence f = f 0,f 1,...,f n such that 1 f i : X i Y i are continuous functions, 2 if x 0,x 1,...,x n R, thenf 0 x 0,f i x 1,...,f n x n S,

8 688 SERGIO CELANI 3 if f 0 x 0,y 1,...,y n S, then there exist x i X i,with1 i n, such that x 0,x 1,...,x n R and f i x i = y i with 1 i n. We write f : to denote that there exists a generalized morphism f between the generalized modal spaces and. Proposition 4.6. Let = {X i },R and g = {Y i },S be two generalized modal spaces. If f : is a generalized morphism, then the map Af : A A, defined by Af = f0 1,...,f 1 n with f 1 i :ClopY i ClopX i, is a generalized homomorphism. Proof. It is clear that each map f 1 i :ClopY i ClopX i is a Boolean homomorphism. Let U i ClopX i,1 i n, and x 0 X 0 such that x 0 f0 1 SU 1,...,U n.thensf 0 x 0 U 1,...,U n. It follows that there exist y i Y i,withi = 1,...,n, such that f 0 x 0,y 1,...,y n S.Sincef is a generalized morphism, there exist x i X i,withi = 1,...,n, such that x 0,x 1,...,x n R and f i x i = y i.so,x i f 1 i U i,with1 i n and this implies that R x 0 f 1 1 U1,...,f 1 n Un, 4.1 that is, x 0 R f 1 1 U 1,...,f 1 n U n. The other direction is easy and left to the reader. Theorem 4.7. Let B = {B i }, B and A = {A i }, A be two generalized modal algebras and let h = h 0,...,h n be a generalized homomorphism. Then the sequence f = h 1 0,...,h 1 n is a generalized morphism between the dual spaces A and B. Proof. It is clear that each h 1 i : UlB i UlA i is a continuous function. We prove conditions 2 and 3 of Definition Let P 0,...,P n R B and let a 1,...,a n h 1 1 P 1 h 1 n P n. Since P P n 1 B P 0, A h1 a1,...,hn an = h0 B a1,...,a n P So, a 1,...,a n A h 1 0 P 0. 3 Let h 1 1 P 0,Q 1,...,Q n R B. We prove that A h1 Q1 hn Qn P Let q i Q i,with1 i n, such that h 1 q 1,...,h n q n 1 A P 0. Then A h1 q1,...,hn qn = h0 B q1,...,q n P0, 4.4

9 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 689 that is, q 1,...,q n 1 B h 1 0 P 0, which is a contradiction. Thus, A h1 Q1 hn Qn P We consider the filter F i = Fh i Q i, for1 i n. Then it is clear that F 1 F 2 F n 1 B P By Theorem 3.2, there exist ultrafilters P i UlA i,for1 i n, such that F i P i and P 1 P 2 P n 1 A P 0. Sinceh i Q i F i P i,wegetq i = h 1 i P i,1 i n. We denote by the class of generalized modal algebras with generalized homomorphisms and denote by the class of generalized modal spaces with generalized morphism. By the above results and by the Boolean duality, we can say that the classes and are dually equivalents. 5. Generalized subalgebras. It is known that there exists a duality between Boolean subalgebras of a Boolean algebra A and equivalence relations defined on the dual space UlA [7]. The duality is given as follows. Let X be a Boolean space and let E be an equivalence relation on X. A subset U X is said to be E-closed if for any x,y X, such that x,y E and x U,theny U, that is, U E = { y X : x,y E, x U } U. 5.1 A Boolean equivalence is an equivalence E defined on X such that, for any x,y X, ifx,y E, there exists an E-closed U ClopX such that x U and y U. The Boolean subalgebra of ClopX associated with a Boolean equivalence E is defined by BE = { U ClopX : U E = U }. 5.2 If A is a Boolean algebra and B is a Boolean subalgebra of A, then the relation EB UlA 2,givenby P,Q EB P B = Q B, 5.3 is a Boolean equivalence. Theorem 5.1 [7]. Let A be a Boolean algebra. Then there exists a dual orderisomorphism between Boolean subalgebras of A and Boolean equivalences defined on UlA. Definition 5.2. Let B = {B i }, be a generalized modal algebra. A subalgebra of B is a sequence A = A 0,...,A n such that, for each 0 i n, A i is a subalgebra of B i, and for each a n i=1 B i,weget a B 0.

10 690 SERGIO CELANI Definition 5.3. Let = {X i },R be a generalized modal spaces. A generalized Boolean equivalence is a sequence E = E 0,E 1,...,E n such that, for each 0 i n, E i is a Boolean equivalence of X i,andifa,b E 0 and a,x 1,...,x n R, thenthereexistsy 1,...,y n n i=1 X i, such that b,y 1,..., y n R and x i,y i E i, for each 1 i n. Theorem 5.4. Let B = {B i }, be a generalized modal algebra. Then the following conditions are equivalent: 1 A = {A i }, is a subalgebra of B; 2 the sequence E A = E B0,E B1,...,E Bn is a generalized Boolean equivalence. Proof Let A = {A i }, be a subalgebra of B. LetP 0,Q 0 UlB 0 such that P 0 A 0 = Q 0 A 0 and let P 0,P 1,...,P n R B. We consider the set FP 1 A 1,...,FP n A n, where FP i A i is the filter generated by the set P i A i. We prove that F P 1 A 1,...,F Pn A n Q Let a = a 1,...,a n with a i P i A i,for1 i n. Since a P 1 P n 1 P 0, a P 0.As a n i=1 A i and A is a subalgebra of B, then a P 0 A 0 = Q 0 A 0. Thus, a Q 0 and 5.4 is valid. By Theorem 3.2, thereexist ultrafilters Q i UlB i,for1 i n, such that P i A i = Q i A i. Therefore, Q 0,Q 1,...,Q n R B and P i A i = Q i A i for each 1 i n Suppose that E A = E B0,E B1,...,E Bn is a generalized Boolean equivalence. Let a n i=1 A i and suppose that a A 0. Consider the set in B 0, F a A0 { a }. 5.5 This set has the finite intersection property. Suppose the contrary. Then, there exists x F a A 0 such that x a = 0. It follows that a x a, that is, x = a A 0, which is a contradiction. Thus, the set 5.5 has the finite intersection property. So, there exists an ultrafilter P 0 UlB 0 such that F a A 0 P 0, a P Consider the set { a } P0 A This set has the finite intersection property, because in contrary case there exists p P 0 A 0 such that a p. This implies that p F a A 0 P 0, which is a contradiction. Thus, there exists Q 0 UlB 0 such that a Q 0, P 0 A 0 = Q 0 A

11 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 691 Since a Q 0, there exists Q 1,...,Q n n i=1 UlB i such that Q0,Q 1,...,Q n RB, a Q 1,...,Q n. 5.9 By hypothesis, there exists P 1,...,P n R B P 0 such that P i A 0 = Q i A 0,for 1 i n. Hence,a i Q i A 0, we have a P 0, which is a contradiction by 5.6. Thus, a A Generalized congruences. Recall that, given a modal algebra B, there exists a bijective correspondence between congruences of B and filters F of B closed under, that is, a F when a F see, e.g., [8, 9]. This class of filters are called open filters. In this section, we introduce a generalization of the notion of congruences and open filter. Let B be a Boolean algebra. Recall that if F is a filter of B, the relation θf = { x,y : f : x f = y f } 6.1 is a Boolean congruence. On the other hand, if θ is a Boolean congruence, then Fθ= { x B : x,1 θ } 6.2 is a filter of B such that θfθ = θ and FθF= F. Let B = {B i }, be a generalized modal algebra, let F i be a filter of B i,1 i n, and let F 1 + +F n ={a 1,...,a n n i=1 B i : a i F i, for some 1 i n}. Definition 6.1. Let B = {B i }, be a generalized modal algebra. A generalized modal filter of B is a sequence F = F 0,F 1,...,F n such that 1 F i is a filter of B i,0 i n; 2 for any a F 1 + +F n, a F 0. Definition 6.2. Let B = {B i }, be a generalized modal algebra. A generalized modal congruence of B is a finite sequence θ = θ 0,...,θ n 6.3 such that 1 θ i is a Boolean congruence of B i, for each 0 i n; 2 if a i,b i θ i with 1 i n, then a 1,...,a n, b 1,...,b n θ 0. Theorem 6.3. Let B = {B i }, be a generalized modal algebra. There exists a bijective correspondence between congruences of B and the generalized modal filter of B.

12 692 SERGIO CELANI Proof. Let θ = θ 0,...,θ n be a generalized congruence of B. Define the sequence Fθ = Fθ 0,...,Fθ n. Leta 1,a 2,...,a n Fθ 1 + +Fθ n. Then there exist some 1 j n such that a j,1 θ j.sincea i,a i θ i for every 1 i n, then a1,...,a j,...,a n, a1,...,1,...,a n = a1,...,a j,...,a n,1 θ Thus, a 1,...,a j,...,a n Fθ 0. Let F = F 0,F 1,...,F n be a generalized modal filter. Consider the sequence θ F = θf 0,...,θF n. Let a i,b i θf i,for1 i n. Then, for each 1 i n, thereexistf i F i such that a i f i = b i f i. We prove that there exists f 0 F 0 such that a 1,...,a n f 0 = b 1,...,b n f 0. Suppose the contrary, that is, for every f 0 F 0, Then there exists P 0 UlA 0 such that a 1,...,a n f0 b 1,...,b n f F 0 P 0, a 1,...,a n P0, b 1,...,b n P So, there exists P 1,...,P n R B P 0 such that a i P i, 1 i n. 6.7 Since a 1,...,a n P 0,thena j P j, for some 1 j n, and as 0,...,f j,...,0 F 1 + +F n,weget 0,...,f j,...,0 F 0 P 0. It follows that f j P j and since a j f j = b j f j, b j P j, which is a contradiction by 6.7. Thus, there exists f 0 F 0 such that a 1,...,a n f 0 = b 1,...,b n f 0. References [1] M. M. Bonsangue and M. Z. Kwiatkowska, Re-interpreting the modal µ-calculus, Modal Logic and Process Algebra. A Bisimulation Perspective Amsterdam, 1994 A. Ponse, M. de Rijke, and Y. Venema, eds., CSLI Lecture Notes, vol. 53, CSLI Publications, California, 1995, pp [2] C. Brink and I. M. Rewitzky, Finite-cofinite program relations, Log. J. IGPL , no. 2, [3] S. Graf, A selection theorem for Boolean correspondences, J. reine angew. Math , [4] P. R. Halmos, Algebraic Logic, Chelsea Publishing, New York, [5] B. Jónsson and A. Tarski, Boolean algebras with operators. I, Amer. J. Math , [6], Boolean algebras with operators. II, Amer. J. Math , [7] S. Koppelberg, Topological duality, Handbook of Boolean Algebras. Vol. 1 J. D. Monk and R. Bonnet, eds., North-Holland Publishing, Amsterdam, 1989, pp [8] M. Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, vol. 142, North-Holland Publishing, Amsterdam, 1999.

13 GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS 693 [9] G. Sambin and V. Vaccaro, Topology and duality in modal logic, Ann. Pure Appl. Logic , no. 3, [10] F. B. Wright, Some remarks on Boolean duality, Portugal. Math , Sergio Celani: Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro, 7000-Tandil, Provincia of Buenos Aires, Argentina address:

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