Extensions of the Stone Duality to the category BooleSp
aa r X i v : . [ m a t h . GN ] S e p Extensions of the Stone Dualityto the category BooleSp
G. Dimov and E. Ivanova-Dimova ∗ Faculty of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria
Abstract
In [5], extending the Stone Duality Theorem, we proved two duality theo-rems for the category
ZDHaus of zero-dimensional Hausdorff spaces and con-tinuous maps. Now we derive from them the extension of the Stone DualityTheorem to the category
BooleSp of zero-dimensional locally compact Haus-dorff spaces and continuous maps obtained in [4], as well as two new dualitytheorems for the category
BooleSp . In 1937, M. Stone [12] proved that there exists a bijective correspondence T l betweenthe class of all (up to homeomorphism) zero-dimensional locally compact Hausdorffspaces (briefly, Boolean spaces ) and the class of all (up to isomorphism) generalizedBoolean algebras (briefly, GBAs) (or, equivalently, Boolean rings with or withoutunit). In the class of compact Boolean spaces (briefly,
Stone spaces ) this bijectioncan be extended to a duality T : Stone −→ Boole between the category
Stone ofStone spaces and continuous maps and the category
Boole of Boolean algebras andBoolean homomorphisms; this is the classical Stone Duality Theorem. In 1964, H.P. Doctor [6] showed that the Stone bijection T l can be extended even to a dualitybetween the category BooleSp perf of all Boolean spaces and all perfect maps betweenthem and the category
GBoole of all GBAs and suitable morphisms between them. ∗ The authors were supported by the Bulgarian National Fund of Science, contract no.DN02/15/19.12.2016. Keywords: (complete atomic) Boolean algebra, Boolean (l)(d)z-algebra, Boolean (l)(m)z-map,Stone space, Boolean space, duality. E-mail addresses: [email protected]fia.bg, [email protected]fia.bg
BooleSp ofBoolean spaces and continuous maps. Finally, in [5], we extended the Stone Dualityto the category
ZDHaus of zero-dimensional Hausdorff spaces and continuous maps.In this paper, which can be regarded as a continuation of [5], we show how theDimov Duality Theorem mentioned above can be derived from one of our generalduality theorems for the category
ZDHaus proved in [5]. Moreover, with the help ofour results from [5], we obtain two new extensions of the Stone Duality Theorem tothe category
BooleSp , one of which is in the spirit of the recent Duality Theorem ofBezhnanishvili, Morandi and Olberding [2] for the category
Tych of Tychonoff spacesand continuous maps.The paper is organized as follows. In Section 2 we collect all preliminary factswhich are needed for the exposition of our results. In particular, we briefly recall andfix the notation pertaining to all, the Stone Duality, the Tarski Duality and our twoduality theorems from [5] with which we extended the Stone Duality Theorem to thecategory
ZDHaus . In Section 3 we introduce the notion of
Boolean ldz-algebra andpresent our first new duality theorem for the category
BooleSp (see Theorem 3.3).It extends the Stone Duality Theorem and is obtained with the help of our dualitytheorem [5, Theorem 3.15]. After that, using it, we give a new proof of the DimovDuality Theorem for the category
BooleSp (see 3.5 and Theorem 3.4). Finally, inSection 4, we introduce the notion of lmz-map and with its help we obtain our secondnew duality theorem for the category
BooleSp (see Theorem 4.5). Its proof is basedon our duality theorem [5, Theorem 4.8] for the category
ZDHaus .We now fix the general notation.Throughout, ( B, ∧ , ∨ , ∗ , ,
1) will denote a Boolean algebra unless indicated oth-erwise; we do not assume that 0 = 1. With some abuse of language, we shall usuallyidentify algebras with their universe, if no confusion can arise.We denote by the simplest Boolean algebra containing only 0 and 1, where0 = 1, and by N + the set of positive integers.If A is a Boolean algebra, then A + df = A \ { } and At( A ) is the set of all atomsof A . If X is a set, we denote by P ( X ) the power set of X ; clearly, ( P ( X ) , ∪ , ∩ , \ , ∅ , X )(= ( P ( X ) , ⊆ )) is a complete atomic Boolean algebra.If X is a topological space, we denote by CO( X ) the set of all clopen (= closedand open) subsets of X , and by KO( X ) the set of all compact open subsets of X .Obviously, (CO( X ) , ∪ , ∩ , \ , ∅ , X ) (= (CO( X ) , ⊆ )) is a Boolean algebra. If M is asubset of X , we denote by cl X ( M ) (or simply by cl( M )) the closure of M in ( X, T ).If C is a category, we denote by | C | the class of the objects of C and by C ( X, Y )the set of all C -morphisms between two C -objects X and Y .We denote by: • Set the category of sets and functions, • Top the category of topological spaces and continuous maps, • ZDHaus the category of all zero-dimensional Hausdorff spaces and continuousmaps, 2
Stone the category of all compact Hausdorff zero-dimensional spaces (=
Stonespaces ) and their continuous maps, • BooleSp the category of all locally compact Hausdorff zero-dimensional spaces(=
Boolean spaces ) and their continuous maps, • Boole the category of Boolean algebras and Boolean homomorphisms, • Caba the category of all complete atomic Boolean algebras and all completeBoolean homomorphisms between them.The main reference books for all notions which are not defined here are [1, 11,10, 7].
We will first recall briefly the Stone Duality Theorem and the Tarski Duality Theorem,and will fix the notation.
We will denote by CO : Top −→ Boole the contravariant functor which assignsto every X ∈ | Top | the Boolean algebra (CO( X ) , ⊆ ) and to every f ∈ Top ( X, Y ), theBoolean homomorphism CO ( f ) : CO ( Y ) −→ CO ( X ) defined by CO ( f )( U ) df = f − ( U ),for every U ∈ CO( Y ).Now we will briefly describe the Stone duality [12] between the categories Boole and
Stone using its presentation given in [8]. We will define two contravariant func-tors S : Boole −→ Stone and T : Stone −→ Boole . For any Boolean algebra A , we let the space S ( A ) to be the set X A df = Boole ( A, )endowed with a topology T A having as a closed base the family { s A ( a ) | a ∈ A } ,where s A ( a ) df = { x ∈ X A | x ( a ) = 1 } , (1)for every a ∈ A ; then S ( A ) = ( X A , T A ) is a Stone space. Note that the family { s A ( a ) | a ∈ A } is also an open base of the space ( X A , T A ).If ϕ ∈ Boole ( A, B ), then we define S ( ϕ ) : S ( B ) −→ S ( A ) by the formula S ( ϕ )( y ) df = y ◦ ϕ for every y ∈ S ( B ). It is easy to see that S is a contravariant functor.The contravariant functor T is defined to be the restriction of the contravariantfunctor CO to the category Stone .Also, we recall that the Stone map s A : A −→ T ( S ( A )) , a s A ( a ) , (2)is a Boole -isomorphism. 3 .2.
We will need the Tarski Duality between the categories
Set and
Caba . Itconsists of two contravariant functors P : Set −→ Caba and At : Caba −→ Set which are defined as follows. For every set X , P ( X ) df = ( P ( X ) , ⊆ ) . If f ∈ Set ( X, Y ), then P ( f ) : P ( Y ) −→ P ( X ) is defined by the formula P ( f )( M ) df = f − ( M ) , for every M ∈ P ( Y ). Further, for every B ∈ | Caba | , At ( B ) df = At( B );if σ ∈ Caba ( B, B ′ ), then At ( σ ) : At ( B ′ ) −→ At ( B ) is defined by the formula At ( σ )( x ′ ) df = ^ { b ∈ B | x ′ ≤ σ ( b ) } , for every x ′ ∈ At( B ′ ).Now we will list some simple facts about some well known constructions ofBoolean homomorphism. The notation which we will introduce will be used through-out the paper. Let α ∈ Boole ( A, B ) and x ∈ At( B ). Then it is easy to see that the map α x : A −→ defined by α x ( a ) = 1 ⇔ x ≤ α ( a ), for a ∈ A , is a Boolean homomorphism. We put X α df = { α x | x ∈ At( B ) } . Note that if α is a complete Boolean homomorphism, then, for every x ∈ At( B ) , α x is a complete Boolean homomorphism as well. We put h α : At( B ) −→ X α , x α x . It is easy to see that if every atom of B is a meet of some elements of α ( A ) , then h α is a bijection. If X is a topological space, A = (CO( X ) , ⊆ ), B = P ( X ) and α is the inclusionmap, then, obviously, X = At( B ) and the map α x is defined by α x ( U ) = 1 ⇔ x ∈ U ,for every U ∈ A and every x ∈ X . In order to simplify the notation, for such A and B , we will write ˆ x instead of α x . ( Note that every ˆ x is a complete Booleanhomomorphism. ) Thus, in such a case, byˆ x : CO ( X ) −→
4e will understand the map defined by ˆ x ( U ) = 1 ⇔ x ∈ U , for every U ∈ CO ( X );also, we will write ˆ X instead of X α , and ˆ h X instead of h α , i.e.,ˆ X = { ˆ x : CO ( X ) −→ | x ∈ X } and ˆ h X : X −→ ˆ X, x ˆ x. Note that if the family CO( X ) T -separates the points of X (i.e., for every x, y ∈ X such that x = y , there exists U ∈ CO( X ) with | U ∩ { x, y }| = 1), then the map ˆ h X isa bijection. We have to recall some definitions and results from [5] which will be of greatimportance for our exposition.
Definition 2.4. [5] A pair (
A, X ), where A ∈ | Boole | and X ⊆ Boole ( A, ), iscalled a Boolean z-algebra (briefly, z-algebra ; abbreviated as ZA) if for each a ∈ A + there exists x ∈ X such that x ( a ) = 1. Fact 2.5. [5]
A pair ( A, X ) is a z-algebra if and only if A is a Boolean algebra and X is a dense subset of S ( A ) . Notation 2.6. If A is a Boolean algebra and X ⊆ Boole ( A, ), we set s XA ( a ) df = X ∩ s A ( a )for each a ∈ A , defining in such a way a map s XA : A −→ P ( X ) , a s XA ( a ) . Definition 2.7. [5] A z-algebra (
A, X ) is called a
Boolean dz-algebra (briefly, dz-algebra ; abbreviated as DZA) if s XA ( A ) = CO( X ) (where X is regarded as a subspaceof S ( A )). Proposition 2.8. [5]
There is a category dzBoole whose objects are all dz-algebrasand whose morphisms between any two dzBoole -objects ( A, X ) and ( A ′ , X ′ ) are allpairs ( ϕ, f ) such that ϕ ∈ Boole ( A, A ′ ) , f ∈ Set ( X ′ , X ) and x ′ ◦ ϕ = f ( x ′ ) forevery x ′ ∈ X ′ . The composition ( ϕ ′ , f ′ ) ◦ ( ϕ, f ) between two dzBoole -morphisms ( ϕ, f ) : ( A, X ) −→ ( A ′ , X ′ ) and ( ϕ ′ , f ′ ) : ( A ′ , X ′ ) −→ ( A ′′ , X ′′ ) is defined to be the dzBoole -morphism ( ϕ ′ ◦ ϕ, f ◦ f ′ ) : ( A, X ) −→ ( A ′′ , X ′′ ) ; the identity morphism ofa dzBoole ′ -object ( A, X ) is defined to be ( id A , id X ) . Theorem 2.9. [5]
The categories
ZDHaus and dzBoole are dually equivalent.Sketch of the proof.
For every X ∈ | ZDHaus | , we let F ( X ) df = ( CO ( X ) , ˆ X ) . Then F ( X ) ∈ | dzBoole | . For f ∈ ZDHaus ( X, Y ), set F ( f ) df = ( CO ( f ) , ˆ f ) , f : ˆ X −→ ˆ Y is defined by ˆ f (ˆ x ) df = d f ( x )for every x ∈ X . Then F ( f ) ∈ dzBoole ( F ( Y ) , F ( X )) and F : ZDHaus −→ dzBoole is a contravariant functor.For every ( A, X ) ∈ | dzBoole | , we set G ( A, X ) df = X, where X is regarded as a subspace of S ( A ). Then G ( A, X ) ∈ | ZDHaus | .If ( ϕ, f ) : ( A, X ) −→ ( A ′ , X ′ ) is a dzBoole -morphism, we put G ( ϕ, f ) df = f. Then G ( ϕ, f ) is a continuous function and G : dzBoole −→ ZDHaus is a contravariant functor. Moreover, the functors F ◦ G and G ◦ F are naturallyisomorphic to the corresponding identity functors. Definition 2.10. [5] Let A be a Boolean algebra and B ∈ | Caba | . A Booleanmonomorphism α : A −→ B is said to be a Boolean z-map (briefly, z-map ) if everyatom of B is a meet of some elements of α ( A ). A z-map α : A −→ B is called a maximal Boolean z-map (briefly, mz-map ) if CO( X α ) = s X α A ( A ), where X α is regardedas a subspace of S ( A ) (see 2.3 and 2.6 for the notation). Proposition 2.11. [5]
There is a category mzMaps whose objects are all mz-mapsand whose morphisms between any two mzMaps -objects α : A −→ B and α ′ : A ′ −→ B ′ are all pairs ( ϕ, σ ) such that ϕ ∈ Boole ( A, A ′ ) , σ ∈ Caba ( B, B ′ ) and α ′ ◦ ϕ = σ ◦ α , i.e., the diagram A ϕ / / α (cid:15) (cid:15) A ′ α ′ (cid:15) (cid:15) B σ / / B ′ is commutative. The composition ( ϕ ′ , σ ′ ) ◦ ( ϕ, σ ) between two mzMaps -morphisms ( ϕ, σ ) : α −→ α ′ and ( ϕ ′ , σ ′ ) : α ′ −→ α ′′ is defined to be the mzMaps -morphism ( ϕ ′ ◦ ϕ, σ ′ ◦ σ ) : α −→ α ′′ ; the identity map of an mzMaps -object α : A −→ B isdefined to be ( id A , id B ) . Theorem 2.12. [5]
The categories
ZDHaus and mzMaps are dually equivalent. ketch of the proof. We will define two contravariant functors F : ZDHaus −→ mzMaps and G : mzMaps −→ ZDHaus . For every X ∈ | ZDHaus | , we put F ( X ) df = i X , where i X : CO ( X ) −→ P ( X ) is the inclusion map. Then i X ∈ | mzMaps | . For f ∈ ZDHaus ( X, Y ), we set F ( f ) df = ( CO ( f ) , P ( f )) . Then F ( f ) is a mzMaps -morphism.For ( α : A −→ B ) ∈ | mzMaps | , we put G ( α ) df = X α . Clearly, the set X α endowed with the subspace topology from the space S ( A ) is a ZDHaus -object. For ( ϕ, σ ) ∈ mzMaps ( α, α ′ ), with α : A −→ B and α ′ : A ′ −→ B ′ , we set G ( ϕ, σ ) df = f σ , where f σ : X α ′ −→ X α is defined by the formula f σ ( α ′ x ′ ) = a At ( σ )( x ′ ) , for every x ′ ∈ At( B ′ ). Then G ( ϕ, σ ) is a ZDHaus -morphism.We obtain that F and G are contravariant functors. Their compositions arenaturally isomorphic to the corresponding identity functors.We have to recall as well some results from [4] concerning the Dimov extensionof the Stone Duality Theorem to the category BooleSp . Recall that if ( A, ≤ ) is a poset and B ⊆ A then B is said to be a dense subsetof A if for any a ∈ A + there exists b ∈ B + such that b ≤ a ; when ( B, ≤ ) is a posetand f : A −→ B is a map, then we will say that f is a dense map if f ( A ) is a densesubset of B .Recall that a frame is a complete lattice L satisfying the infinite distributivelaw a ∧ W S = W { a ∧ s | s ∈ S } , for every a ∈ L and every S ⊆ L .Let A be a distributive { } -pseudolattice and Idl ( A ) be the frame of all idealsof A . If J ∈ Idl ( A ) then we will write ¬ A J (or simply ¬ J ) for the pseudocomplementof J in Idl ( A ) (i.e., ¬ J = W { I ∈ Idl ( A ) | I ∧ J = { }} ). Note that ¬ J = { a ∈ A | ( ∀ b ∈ J )( a ∧ b = 0) } (see Stone [12]). Recall that an ideal J of A is called simple (Stone [12]) if J ∨ ¬ J = A (i.e., J is a complemented element of the frame Idl ( A )).As it is proved in [12], the set Si ( A ) of all simple ideals of A is a Boolean algebrawith respect to the lattice operations in Idl ( A ). Definition 2.14. [4] A pair (
A, I ), where A is a Boolean algebra and I is an idealof A (possibly non proper) which is dense in A (shortly, dense ideal), is called a localBoolean algebra (abbreviated as LBA). 7et LBA be the category whose objects are all LBAs and whose morphisms areall functions ϕ : ( A, I ) −→ ( B, J ) between the objects of
LBA such that ϕ : A −→ B is a Boolean homomorphism satisfying the following condition:(LBA) For every b ∈ J there exists a ∈ I such that b ≤ ϕ ( a );let the composition between the morphisms of LBA be the usual composition betweenfunctions, and the
LBA -identities be the identity functions.Note that two LBAs (
A, I ) and (
B, J ) are
LBA -isomorphic if there exists aBoolean isomorphism ϕ : A −→ B such that ϕ ( I ) = J . Definition 2.15. [4] An LBA (
A, I ) is called a
ZLB-algebra (briefly,
ZLBA ) if, forevery J ∈ Si ( I ), the join W A J (= W A { a | a ∈ J } ) exists. Notation 2.16.
For every LBA ( A, I ) , we put L AI = { x ∈ Boole ( A, ) | ∈ x ( I ) } . Proposition 2.17. [4]
Let ( A, I ) be an LBA. Then ( A, I ) is a ZLBA iff L AI ∩ s A ( A ) =CO( L AI ) (where L AI is regarded as a subspace of S ( A ) ). Let
ZLBA be the full subcategory of the category
LBA having as objects allZLBAs.
Theorem 2.18. [4]
The categories
BooleSp and
ZLBA are dually equivalent.Sketch of the proof.
We will define two contravariant functorsΘ ad : ZLBA −→ BooleSp and Θ td : BooleSp −→ ZLBA as follows. For X ∈ | BooleSp | , we setΘ td ( X ) df = ( CO ( X ) , KO( X )) . Then Θ td ( X ) is a ZLBA. For every f ∈ BooleSp ( X, Y ), we putΘ td ( f ) df = CO ( f ) . Then Θ t ( f ) is a ZLBA -morphism.For every ZLBA (
A, I ), we setΘ ad ( A, I ) df = L AI . Then Θ ad ( A, I ) ∈ | BooleSp | .If ϕ ∈ ZLBA (( A, I ) , ( B, J )), then we define the map Θ ad ( ϕ ) : Θ ad ( B, J ) −→ Θ ad ( A, I ) by the formulaΘ ad ( ϕ )( x ′ ) df = x ′ ◦ ϕ, ∀ x ′ ∈ Θ ad ( B, J ) . Then Θ ad ( ϕ ) is a BooleSp -morphism.Finally, we show that the compositions Θ ad ◦ Θ td and Θ td ◦ Θ ad are naturallyisomorphic to the corresponding identity functors.We will need as well the following theorem of M. Stone [13] (see also [10, The-orem 7.25]): 8 heorem 2.19. Let A ∈ | Boole | and ( X A , T A ) = S ( A ) . Then there exists an order-preserving bijection (and hence, frame isomorphism) ι : ( Idl ( A ) , ≤ ) −→ ( T A , ⊆ ) , J [ { s A ( a ) | a ∈ J } . If U ∈ T A then J = ι − ( U ) = { a ∈ A | s A ( a ) ⊆ U } . In this section we will introduce the notion of
Boolean ldz-algebra and with its help wewill obtain our first new duality theorem for the category
BooleSp , namely, Theorem3.3. Using it, we will present a new proof of the Dimov Duality Theorem [4] for thecategory
BooleSp (cited here as Theorem 2.18).
Definition 3.1.
A dz-algebra (
A, X ) is called a
Boolean ldz-algebra (briefly, ldz-algebra ; abbreviated as LDZA) if for every x ∈ X there exists a ∈ A such that x ∈ s A ( a ) ⊆ X .The following assertion is obvious. Fact 3.2.
A dz-algebra ( A, X ) is an ldz-algebra if and only if X is an open subsetof S ( A ) . Let ldzBoole be the full subcategory of the category dzBoole having as objectsall ldz-algebras.
Theorem 3.3.
The categories
BooleSp and ldzBoole are dually equivalent.Proof.
Let X ∈ | BooleSp | . Then, by Theorem 2.9, X is homeomorphic to the space G ( F ( X )) = ˆ X . Therefore, ˆ X is a locally compact dense subset of the space S ( CO ( X )).Then [7, Theorems 3.3.9 and 3.3.8] imply that ˆ X is an open subset of S ( CO ( X )).Hence, F ( X ) = ( CO ( X ) , ˆ X ) is an ldz-algebra. Thus, F ( BooleSp ) ⊆ ldzBoole .Let ( A, X ) be an ldz-algebra. Then X is an open subspace of the Stone space S ( A ). Thus, G ( A, X ) = X is a Boolean space, i.e., G ( ldzBoole ) ⊆ BooleSp . Now,using Theorem 2.9, we complete the proof of our assertion.Now it becomes clear that for proving Theorem 2.18 it is enough to show thatthe categories ldzBoole and
ZLBA are isomorphic.
Theorem 3.4.
The categories ldzBoole and
ZLBA are isomorphic. roof. We will define two functors E : ldzBoole −→ ZLBA and E ′ : ZLBA −→ ldzBoole , and will show that their compositions are equal to the corresponding identity functors.Let ( A, X ) ∈ | ldzBoole | . Then X is an open subset of the space S ( A ). Set I X df = { a ∈ A | s A ( a ) ⊆ X } . We will show that (
A, I X ) ∈ | ZLBA | . Indeed, by Theorem 2.19, I X is an ideal of A (proper or non proper). We have to prove that I X is a dense subset of A . Let b ∈ A + .Then s A ( b ) = ∅ . By Fact 2.5, X is a dense subset of S ( A ). Hence X ∩ s A ( b ) is an opennon-empty subset of S ( A ). Then there exists a ∈ A + such that s A ( a ) ⊆ X ∩ s A ( b ).Thus a ∈ I X and a ≤ b . So, I X is a dense ideal of A and, hence, ( A, I X ) is an LBA.For proving that it is a ZLBA, we will use Proposition 2.17 (and its notation). Wewill first show that L AI X = X . Indeed, let x ∈ X . Then there exists a ∈ A suchthat x ∈ s A ( a ) ⊆ X . Hence a ∈ I X and x ( a ) = 1. Thus, X ⊆ L AI X . Conversely,let x ∈ L AI X . Then there exists a ∈ I X such that x ( a ) = 1. Thus x ∈ s A ( a ) ⊆ X .Therefore, x ∈ X . So, L AI X = X. Since (
A, X ) is a DZA, we have that X ∩ s A ( A ) = CO( X ). Hence, L AI X ∩ s A ( A ) =CO( L AI X ). Then Proposition 2.17 shows that ( A, I X ) is a ZLBA. We set now: E ( A, X ) df = ( A, I X ) . Further, let ( ϕ, f ) ∈ ldzBoole (( A, X ) , ( B, X ′ )). Then ϕ ∈ Boole ( A, B ), f ∈ Set ( X ′ , X ) and f ( x ′ ) = x ′ ◦ ϕ , for every x ′ ∈ X ′ . We will show that ϕ : ( A, I X ) −→ ( B, I X ′ ) is a ZLBA -morphism. Set, for short, I df = I X and J df = I X ′ . We need only toprove that ϕ satisfies condition (LBA), i.e., we have to show that for every b ∈ J thereexists a ∈ I such that b ≤ ϕ ( a ). Let b ∈ J . Then s B ( b ) ⊆ X ′ and thus, f ( s B ( b )) ⊆ X .Set Φ df = ↑ ( b )(= { c ∈ B | b ≤ c } ). Then Φ is a filter in B . Suppose that Φ ∩ ϕ ( I ) = ∅ .Let I be the ideal in B generated by ϕ ( I ). We will prove that Φ ∩ I = ∅ . Let c ∈ I . Then c ≤ b ∨ . . . ∨ b n for some n ∈ N + and some b , . . . , b n ∈ ϕ ( I ). Forevery i = 1 , . . . , n , there exists a i ∈ I such that b i = ϕ ( a i ). Setting a = W ni =1 a i , weobtain that a ∈ I and b ∨ . . . ∨ b n = ϕ ( a ). Thus, c ≤ ϕ ( a ). Suppose that c ∈ Φ.Then ϕ ( a ) ∈ Φ, a contradiction. Therefore, Φ ∩ I = ∅ . Then, by the MaximalIdeal Theorem [12] (see also [9, Lemma 2.3]), there exists an ideal I ′ of B which ismaximal amongst those containing I and disjoint from Φ. Clearly, I ′ is an ideal of B which is maximal amongst those disjoint from Φ. Then, as it is well known (see,e.g., [9, Theorem 2.4]), I ′ is a prime ideal of B . Thus there exists a homomorphism x ′ ∈ Boole ( B, ) such that I ′ = ( x ′ ) − (0) (see, e.g., [9, Proposition 2.2]). Hence x ′ ( b ) = 1, i.e., x ′ ∈ s B ( b ) and therefore x ′ ∈ X ′ . Then f ( x ′ ) = x ′ ◦ ϕ ∈ X . Since X is an open subset of S ( A ), there exists a ∈ A such that x ′ ◦ ϕ ∈ s A ( a ) ⊆ X . Then a ∈ I and thus ϕ ( a ) ∈ ϕ ( I ) ⊆ I ⊆ I ′ . Hence, x ′ ( ϕ ( a )) = 0. Since x ′ ◦ ϕ ∈ s A ( a ), we10btain a contradiction. Therefore, ϕ ( I ) ∩ Φ = ∅ . This means that there exists a ∈ I such that ϕ ( a ) ≥ b . So, ϕ is a ZLBA -morphism. Now we set E ( ϕ, f ) df = ϕ. It is easy to see that E is a functor.Let now ( A, I ) ∈ | ZLBA | . We set X I df = S { s A ( a ) | a ∈ I } . Then, clearly, X I isan open subset of S ( A ). We will show that ( A, X I ) ∈ | ldzBoole | . Let b ∈ A + . Since I is dense in A , there exists a ∈ I + such that a ≤ b . Then ∅ 6 = s A ( a ) ⊆ X I ∩ s A ( b ).Thus X I is a dense subset of S ( A ). Now, Fact 2.5 implies that ( A, X I ) is a z-algebra.For proving that it is a dz-algebra, we will first show that X I = L AI (see Notation2.16). Let x ∈ X I . Then there exists a ∈ I such that x ∈ s A ( a ). Thus x ( a ) = 1 and,hence, x ∈ L AI . So, X I ⊆ L AI . Conversely, if x ∈ L AI , then there exists a ∈ I with x ( a ) = 1. Thus x ∈ s A ( a ) and, hence, x ∈ X I . Therefore, X I = L AI . Since, by Proposition 2.17, we have that L AI ∩ s A ( A ) = CO( L AI ), we obtain that X I ∩ s A ( A ) = CO( X I ). So, ( A, X I ) ∈ | ldzBoole | . We set E ′ ( A, I ) df = ( A, X I ) . Let ϕ ∈ ZLBA (( A, I ) , ( B, J )) and x ′ ∈ X J . Then there exists b ∈ J with x ′ ( b ) = 1. Since ϕ satisfies condition (LBA), there exists a ∈ I such that b ≤ ϕ ( a ).Then 1 = x ′ ( b ) ≤ x ′ ( ϕ ( a )). Hence, x ′ ( ϕ ( a )) = 1. This means that x ′ ◦ ϕ ∈ X I . So,the function f ϕ : X J −→ X I , x ′ x ′ ◦ ϕ, is well-defined. Thus ( ϕ, f ϕ ) ∈ ldzBoole ( E ′ ( A, I ) , E ′ ( B, J )) and we set E ′ ( ϕ ) df = ( ϕ, f ϕ ) . It is easy to see that E ′ is a functor.We will show that E ′ ◦ E = Id ldzBoole . Let ( A, X ) ∈ | ldzBoole | . Then E ( A, X ) = (
A, I X ), where I X = { a ∈ A | s A ( a ) ⊆ X } , and E ′ ( E ( A, X )) = (
A, X I X ),where X I X = S { s A ( a ) | a ∈ I X } . Clearly, X I X ⊆ X . Conversely, let x ∈ X . Thenthere exists a ∈ A such that x ∈ s A ( a ) ⊆ X . Thus a ∈ I X and, hence, s A ( a ) ⊆ X I X .This shows that x ∈ X I X . Therefore, X = X I X . So, E ′ ( E ( A, X )) = (
A, X ). Further, let ( ϕ, f ) ∈ ldzBoole (( A, X ) , ( B, X ′ )). Then E ′ ( E ( ϕ, f )) = E ′ ( ϕ ) = ( ϕ, f ϕ ), where f ∈ Set ( X ′ , X ), f ( x ′ ) = x ′ ◦ ϕ , for every x ′ ∈ X ′ , f ϕ ∈ Set ( X I X ′ , X I X ) and f ϕ ( x ′ ) = x ′ ◦ ϕ , for every x ′ ∈ X I X ′ . Since X I X ′ = X ′ and X I X = X (as we have shown above), we obtain that f = f ϕ . Therefore, E ′ ◦ E = Id ldzBoole . 11inally, we will show that E ◦ E ′ = Id ZLBA . Let (
A, I ) ∈ | ZLBA | . Then E ( E ′ ( A, I )) = E ( A, X I ) = ( A, I X I ), where X I = S { s A ( a ) | a ∈ I } and I X I = { a ∈ A | s A ( a ) ⊆ X I } . If a ∈ I , then s A ( a ) ⊆ X I and thus a ∈ I X I . Hence, I ⊆ I X I . Forproving that I X I ⊆ I , let a ∈ I X I . Then s A ( a ) ⊆ X I . We have already proved that X I = L AI = { x ∈ Boole ( A, ) | ∈ x ( I ) } . Thus, for every x ∈ s A ( a ) there exists b x ∈ I such that x ∈ s A ( b x ). Therefore, { s A ( b x ) ∩ s A ( a ) | x ∈ s A ( a ) } is an open coverof the compact space s A ( a ). Hence, there exist n ∈ N + and x , . . . , x n ∈ s A ( a ) suchthat s A ( a ) = s A ( a ) ∩ S ni =1 s A ( b x i ) = s A ( a ∧ W ni =1 b x i ). Setting b df = W ni =1 b x i , we obtainthat b ∈ I and s A ( a ) = s A ( a ∧ b ). Thus a = a ∧ b , i.e., a ≤ b . Since b ∈ I , we concludethat a ∈ I . So, I X I = I. Thus E ( E ′ ( A, I )) = (
A, I ). Let now ϕ ∈ ZLBA (( A, I ) , ( B, J )). Then E ( E ′ ( ϕ )) = E ( ϕ, f ϕ ) = ϕ . Hence, E ◦ E ′ = Id ZLBA .All this shows that the categories ldzBoole and
ZLBA are isomorphic.
Clearly, Theorems 3.4 and 3.3 imply Theorem 2.18. We will show that eventhe equalities E ◦ F l = Θ td and G l ◦ E ′ = Θ ad take place, where F l : BooleSp −→ ldzBoole and G l : ldzBoole −→ BooleSp are the restrictions of the contravariantfunctors F and G from the proof of Theorem 2.9, respectively. Indeed, for every( A, I ) ∈ | ZLBA | , we have G l ( E ′ ( A, I )) = G l ( A, X I ) = X I and Θ ad ( A, I ) = L AI . Since X I = L AI (see the proof of Theorem 3.4), we obtain that G l ( E ′ ( A, I )) = Θ ad ( A, I ).If ϕ ∈ ZLBA (( A, I ) , ( B, J )), then G l ( E ′ ( ϕ )) = G l ( ϕ, f ϕ ) = f ϕ , where f ϕ : X J −→ X I , x ′ x ′ ◦ ϕ . Also, Θ ad ( ϕ ) : Θ ad ( B, J ) −→ Θ ad ( A, I ) is defined by Θ ad ( ϕ )( x ′ ) = x ′ ◦ ϕ for every x ′ ∈ L BJ = X J . Therefore, G l ( E ′ ( ϕ )) = Θ ad ( ϕ ). So, G l ◦ E ′ =Θ ad . Further, for every X ∈ | BooleSp | , E ( F l ( X )) = E ( CO ( X ) , ˆ X ) = ( CO ( X ) , I ˆ X )and Θ td ( X ) = ( CO ( X ) , KO( X )). We have that ˆ X = { ˆ x : CO ( X ) −→ | x ∈ X } , where ˆ x ( U ) = 1 ⇔ x ∈ U , for every x ∈ X and every U ∈ CO ( X ), and I ˆ X = { U ∈ CO ( X ) | s CO ( X ) ( U ) ⊆ ˆ X } . Also, ˆ X ∩ s CO ( X ) ( U ) = { ˆ x ∈ ˆ X | ˆ x ( U ) =1 } = { ˆ x ∈ ˆ X | x ∈ U } = ˆ U . As it is shown in [5, Example 3.9], the map ˆ h X : X −→ ˆ X, x ˆ x , is a homeomorphism (regarding ˆ X as a subspace of S ( CO ( X ))).Therefore, ˆ h X (CO( X )) = CO( ˆ X ) and for every U ∈ CO( X ), U is homeomorphicto ˆ h ( U ) = ˆ U . Thus, if U ∈ I ˆ X then ˆ U = s CO ( X ) ( U ) and, hence, ˆ U is compact, sothat U ∈ KO( X ). Therefore, I ˆ X ⊆ KO( X ). Conversely, if U ∈ KO( X ) then ˆ U iscompact, i.e., ˆ X ∩ s CO ( X ) ( U ) is compact. Since ( CO ( X ) , ˆ X ) is a ZA (by [5, Example3.9]), ˆ X is a dense subset of S ( CO ( X )). Thus, setting Y df = S ( CO ( X )), we obtainthat ˆ U = cl Y ( ˆ U ) = cl Y ( ˆ X ∩ s CO ( X ) ( U )) = cl Y ( s CO ( X ) ( U )) = s CO ( X ) ( U ). Therefore, s CO ( X ) ( U ) ⊆ ˆ X , i.e., U ∈ I ˆ X . Hence, I ˆ X = KO( X ). Thus ( E ◦ F l )( X ) = Θ td ( X ).Finally, if f ∈ BooleSp ( X, Z ), then E ( F l ( f )) = E ( CO ( f ) , ˆ f ) = CO ( f ) = Θ td ( f ).This shows that E ◦ F l = Θ td . 12 One more extension of the Stone Duality Theo-rem to the category BooleSp
In this section we will introduce the notion of lmz-map and with its help we willobtain our second new duality theorem for the category
BooleSp , namely, Theorem4.5. For doing this, we have first to recall the following theorem from [5]:
Theorem 4.1. [5]
The categories mzMaps and dzBoole are equivalent.Sketch of the proof.
We will first define two functors F ′ : dzBoole −→ mzMaps and G ′ : mzMaps −→ dzBoole .For every ( A, X ) ∈ | dzBoole | , we set F ′ ( A, X ) df = s XA . Then s XA ∈ | mzMaps | . If ( ϕ, f ) ∈ dzBoole (( A, X ) , ( A ′ , X ′ )), we put F ′ ( ϕ, f ) df = ( ϕ, P ( f )) . Then F ′ ( ϕ, f ) ∈ mzMaps ( F ′ ( A, X ) , F ′ ( A ′ , X ′ )).For every ( α : A −→ B ) ∈ | mzMaps | , we set G ′ ( α ) df = ( CO ( X α ) , c X α ) , where X α is regarded as a subspace of S ( A ), c X α = { c α x : CO ( X α ) −→ | α x ∈ X α } and c α x ( U ) = 1 ⇔ α x ∈ U , for every U ∈ CO ( X α ). Then G ′ ( α ) ∈ | dzBoole | . Let( ϕ, σ ) ∈ mzMaps ( α, α ′ ), where α : A −→ B and α ′ : A ′ −→ B ′ . We set G ′ ( ϕ, σ ) df = ( CO ( f σ ) , b f σ ) , where f σ : X α ′ −→ X α is defined by α ′ x ′ α At ( σ )( x ′ ) and b f σ : d X α ′ −→ c X α is definedby c α ′ x ′ \ f σ ( α ′ x ′ ). Then G ′ ( ϕ, σ ) ∈ dzBoole ( G ′ ( α ) , G ′ ( α ′ )).It is easy to see that F ′ and G ′ are functors. Finally, the functors F ′ ◦ G ′ and G ′ ◦ F ′ are naturally isomorphic to the corresponding identity functors. Definition 4.2.
An mz-map α : A −→ B is called an lmz-map if for every x ∈ At( B )there exists a ∈ A such that α x ∈ s A ( a ) ⊆ X α . Fact 4.3.
An mz-map α : A −→ B is an lmz-map iff X α is an open subset of S ( A ) . Let lmzMaps be the full subcategory of the category mzMaps having as ob-jects all lmz-maps.
Theorem 4.4.
The categories ldzBoole and lmzMaps are equivalent. roof. We will show that the restriction F ′ l of the functor F ′ (defined in the proofof Theorem 4.1) to the category ldzBoole is in fact a functor between the categories ldzBoole and lmzMaps . Let ( A, X ) ∈ | ldzBoole | . Then F ′ ( A, X ) = s XA . Set α df = s XA . Then α : A −→ P ( X ) and, as it is easy to show (see also [5, page 17]), X α ≡ X . Since, by the definition of an ldz-algebra, X is an open subset of S ( A ), weobtain that X α is an open subset of S ( A ). Thus, F ′ ( A, X ) ∈ | lmzMaps | .Now we will show that the restriction G ′ l of the functor G ′ (defined in theproof of Theorem 4.1) to the category lmzMaps is in fact a functor between thecategories lmzMaps and ldzBoole . Let ( α : A −→ B ) ∈ | lmzMaps | . Then G ′ ( α ) = ( CO ( X α ) , c X α ). Set A ′ df = CO ( X α ). We need only to prove that c X α is anopen subset of S ( A ′ ). We have that X α is an open subset of S ( A ). As it is shown in [5,page 16], the restriction ¯ s X α A : A −→ CO ( X α ) = A ′ of the map s X α A : A −→ P ( X α ) isa Boolean isomorphism. Then, by the Stone Duality Theorem, the map f df = S (¯ s X α A ) : S ( A ′ ) −→ S ( A ) is a homeomorphism. We will prove that f ( c X α ) = X α . Clearly, thiswill imply that c X α is an open subset of S ( A ′ ). Let c α x ∈ c X α . Then x ∈ At( B ) and f ( c α x ) = c α x ◦ ¯ s X α A . Thus, for every a ∈ A , f ( c α x )( a ) = 1 ⇔ c α x (¯ s X α A ( a )) = 1 ⇔ α x ∈ ¯ s X α A ( a ) ⇔ α x ( a ) = 1. Hence, f ( c α x ) = α x . This shows that f ( c X α ) = X α . Therefore, G ′ ( α ) ∈ | ldzBoole | .So, F ′ l : ldzBoole −→ lmzMaps and G ′ l : lmzMaps −→ ldzBoole arerestrictions of the functors F ′ : dzBoole −→ mzMaps and G ′ : mzMaps −→ dzBoole , respectively. Now, our assertion follows from Theorem 4.1. Theorem 4.5.
The categories
BooleSp and lmzMaps are dually equivalent.Proof.
Obviously, our assertion follows immediately from Theorems 4.4 and 3.3. Inthe rest of this proof we will describe explicitly the contravariant functors between thecategories
BooleSp and lmzMaps which realize the requested duality and, moreover,we will simplify them.Set F l = F ′ l ◦ F l and G l = G l ◦ G ′ l . Then F l : BooleSp −→ lmzMaps and G l : lmzMaps −→ BooleSp . Clearly, these contravariant functors realize theduality between the categories
BooleSp and lmzMaps and they are restrictions ofthe functors F = F ′ ◦ F : ZDHaus −→ mzMaps and G = G ◦ G ′ : mzMaps −→ ZDHaus , respectively. As it is shown in [5, page 20], the contravariant functors F and F , as well as G and G , are naturally isomorphic (see the proof of Theorem 2.12for F and G ). Let F l : BooleSp −→ lmzMaps and G l : lmzMaps −→ BooleSp bethe restrictions of the contravariant functors F and G , respectively. Then, obviously,the conravariant functors F l and F l , as well as G l and G l , are naturally isomorphic.Thus, the contravariant functors F l and G l realize the duality between the categories BooleSp and lmzMaps . The formulas with which the conravariant functors F l and G l are described coincide with those describing the conravariant functors F and G ;they are given in the proof of Theorem 2.12.14 eferences [1] J. Ad´amek, H. Herrlich and G. E. Strecker,
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