De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces
Guram Bezhanishvili, Vincenzo Marra, Patrick J. Morandi, Bruce Olberding
aa r X i v : . [ m a t h . R A ] N ov DE VRIES POWERS: A GENERALIZATION OF BOOLEAN POWERSFOR COMPACT HAUSDORFF SPACES
G. BEZHANISHVILI, V. MARRA, P. J. MORANDI, B. OLBERDING
Abstract.
We generalize the Boolean power construction to the setting of compact Haus-dorff spaces. This is done by replacing Boolean algebras with de Vries algebras (completeBoolean algebras enriched with proximity) and Stone duality with de Vries duality. For acompact Hausdorff space X and a totally ordered algebra A , we introduce the concept of afinitely valued normal function f ∶ X → A . We show that the operations of A lift to the set F N ( X, A ) of all finitely valued normal functions, and that there is a canonical proximityrelation ≺ on F N ( X, A ) . This gives rise to the de Vries power construction, which whenrestricted to Stone spaces, yields the Boolean power construction.We prove that de Vries powers of a totally ordered integral domain A are axiomatized asproximity Baer Specker A -algebras, those pairs ( S, ≺) , where S is a torsion-free A -algebragenerated by its idempotents that is a Baer ring, and ≺ is a proximity relation on S . Weintroduce the category of proximity Baer Specker A -algebras and proximity morphismsbetween them, and prove that this category is dually equivalent to the category of compactHausdorff spaces and continuous maps. This provides an analogue of de Vries duality forproximity Baer Specker A -algebras. Introduction
For an algebra A of a given type and a Boolean algebra B , the Boolean power of A by B isthe algebra C ( X, A disc ) of all continuous functions from the Stone space X of B to A , where A is given the discrete topology and the operations of A are lifted to C ( X, A disc ) pointwise(see, e.g., [1, 9]). For convenience, we also refer to C ( X, A disc ) as the Boolean power of A by X . Boolean powers turned out to be a very useful tool in universal algebra, where theyhave been used to transfer results about Boolean algebras to other varieties [9].There is no obvious way to generalize the Boolean power construction to compact Haus-dorff spaces. Since X is compact and A is discrete, each f ∈ C ( X, A disc ) is finitely valued,and gives a partition of X into finitely many clopen (closed and open) sets. So if there arenot enough clopens in X , then C ( X, A disc ) is not representative enough. For example, if X = [ , ] , then C ( X, A disc ) degenerates to simply A . The goal of this article is to general-ize the Boolean power construction in such a way that it encompasses compact Hausdorffspaces. For this, instead of working with clopen sets, which form a basis only in the zero-dimensional case, we will work with regular open sets, which form a basis for any compactHausdorff space.One of the most natural generalizations of Stone duality to compact Hausdorff spaces is deVries duality [11]. We recall that a binary relation ≺ on a Boolean algebra B is a proximity if it satisfies the following axioms:(DV1) 1 ≺ Mathematics Subject Classification.
Key words and phrases.
Specker algebra, f -ring, Baer ring, Boolean algebra, Boolean power, proximity,de Vries algebra, Stone space, compact Hausdorff space. (DV2) a ≺ b implies a ≤ b .(DV3) a ≤ b ≺ c ≤ d implies a ≺ d .(DV4) a ≺ b, c implies a ≺ b ∧ c .(DV5) a ≺ b implies ¬ b ≺ ¬ a .(DV6) a ≺ b implies there is c ∈ B such that a ≺ c ≺ b .(DV7) a ≠ ≠ b ∈ B such that b ≺ a .A proximity Boolean algebra is a pair ( B, ≺ ) , where B is a Boolean algebra and ≺ is aproximity on B , and a de Vries algebra is a proximity Boolean algebra such that B iscomplete as a Boolean algebra.By de Vries duality, each compact Hausdorff space X gives rise to the de Vries algebra (RO( X ) , ≺ ) , where RO( X ) is the complete Boolean algebra of regular open subsets of X ,the Boolean operations on RO( X ) are given by ⋁ U i = Int ( Cl (⋃ i U i )) , ⋀ U i = Int (⋂ i U i ) , and ¬ U = Int ( X / U ) , and the proximity is given by U ≺ V iff Cl ( U ) ⊆ V , where Int and Cl arethe interior and closure operators. Moreover, each de Vries algebra ( B, ≺ ) is isomorphic tothe de Vries algebra (RO( X ) , ≺ ) for a unique (up to homeomorphism) compact Hausdorffspace X . This 1-1 correspondence extends to a dual equivalence between the categories ofde Vries algebras and compact Hausdorff spaces. To define the category of de Vries algebras,we recall that a map σ ∶ B → C between proximity Boolean algebras is a de Vries morphism provided(M1) σ ( ) = σ ( a ∧ b ) = σ ( a ) ∧ σ ( b ) .(M3) a ≺ b implies ¬ σ (¬ a ) ≺ σ ( b ) .(M4) σ ( a ) is the least upper bound of { σ ( b ) ∶ b ≺ a } .Note that function composition of two de Vries morphisms need not be a de Vries morphismbecause it need not satisfy (M4). Nevertheless, the de Vries algebras and de Vries morphismsbetween them form a category DeV , where the composition ρ ⋆ σ of two de Vries morphisms σ ∶ B → B and ρ ∶ B → B is given by ( ρ ⋆ σ )( a ) = ⋁{ ρσ ( b ) ∶ b ≺ a } . Each continuous function ϕ ∶ X → Y between compact Hausdorff spaces X, Y gives riseto the de Vries morphism ̂ ϕ ∶ RO( Y ) → RO( X ) , where ̂ ϕ ( U ) = Int ( Cl ( ϕ − ( U ))) for each U ∈ RO( Y ) . Moreover, each de Vries morphism between de Vries algebras comes about thisway. The upshot of all this is that DeV is dually equivalent to the category
KHaus ofcompact Hausdorff spaces and continuous maps, which is one of the key results of [11].We wish to use de Vries duality to define the de Vries power of an algebra by a compactHausdorff space the same way Stone duality is used to define the Boolean power of analgebra by a Stone space. As a motivating example, let X be a compact Hausdorff spaceand let f ∶ X → R be a finitely valued function. If f is continuous, then f − ( a, ∞) is clopenin X for each a ∈ R . On the other hand, we show that f − ( a, ∞) is regular open for all a ∈ R iff f is a normal function, where we recall that a lower semicontinuous function f is normal provided f − (−∞ , a ) is a union of regular closed sets for each a ∈ R [12, Sec. 3].Since for a finitely valued function f , we have f − ( a, ∞) = f − [ b, ∞) for some b > a (and f − (−∞ , a ) = f − (−∞ , b ] for some b < a ), this observation allows us to generalize the conceptof a finitely valued normal function as follows.Let A be a totally ordered algebra of a given type, let X be a compact Hausdorff space,and let f ∶ X → A be a finitely valued function. We call f normal if f − ( ↑ a ) is regular e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 3 open in X for each a ∈ A , where ↑ a = { b ∈ A ∶ a ≤ b } . Let F N ( X, A ) be the set of finitelyvalued normal functions from X to A . For a finitely valued function f ∶ X → A , we introducethe concept of normalization of f , and show that normalization lifts the operations of A to F N ( X, A ) . Thus, F N ( X, A ) has the algebra structure of A . In addition, F N ( X, A ) has acanonical proximity given by f ≺ g iff f − ( ↑ a ) ≺ g − ( ↑ a ) in RO( X ) for each a ∈ A . We callthe pair ( F N ( X, A ) , ≺ ) the de Vries power of A by X . Equivalently, if ( B, ≺ ) is a de Vriesalgebra and X is its dual compact Hausdorff space, then we call ( F N ( X, A ) , ≺ ) the de Vriespower of A by ( B, ≺ ) . We show that when X is a Stone space, this construction yields theBoolean power construction.The main goal of this article is to axiomatize de Vries powers of a totally ordered integraldomain, thus including such classic cases as Z , Q , and R . Our results generalize severalknown results in the literature. Boolean powers of Z were studied by Ribenboim [16]. Theyturn out to be exactly the Specker ℓ -groups introduced and studied by Conrad [10]. On theother hand, Boolean powers of R are the Specker R -algebras introduced and studied in [6].The category of Specker R -algebras is dually equivalent to the category of Stone spaces, andthis duality can be thought of as an economic version of Gelfand-Neumark-Stone duality inthe particular case of Stone spaces [6, Rem. 6.9]. In [5], these results were generalized toaxiomatize Boolean powers of a commutative ring.Let A be a commutative ring with 1, let S be a commutative A -algebra with 1, and letId ( S ) be the Boolean algebra of idempotents of S . A nonzero e ∈ Id ( S ) is faithful provided ae = a = a ∈ A . We call S a Specker A -algebra if S is generated as an A -algebra by a Boolean subalgebra B of Id ( S ) whose nonzero elements are faithful. In case A is an integral domain, S is a Specker A -algebra iff S is generated as an A -algebra by Id ( S ) and S is torsion-free as an A -module [5, Prop. 4.1]. By [5, Thm. 2.7], Boolean powers of A are precisely Specker A -algebras. Moreover, if A is a domain (or more generally if A is anindecomposable ring; that is, if Id ( A ) = { , } ), then the category of Specker A -algebras isequivalent to the category of Boolean algebras, and is dually equivalent to the category ofStone spaces [5, Thm. 3.8 and Cor. 3.9].In this article, for a totally ordered domain A , we enrich the concept of a Specker A -algebra to that of a proximity Specker A -algebra, and show that a de Vries power of atotally ordered domain is precisely a proximity Specker A -algebra that is also a Baer ring. Weprove that each proximity Specker A -algebra ( S, ≺ ) can be represented as a dense subalgebraof ( F N ( X, A ) , ≺ ) for a unique (up to homeomorphism) compact Hausdorff space X . Wealso prove that ( S, ≺ ) is isomorphic to ( F N ( X, A ) , ≺ ) iff S is a Baer ring. We introduceproximity morphisms between proximity Specker A -algebras, and show that the proximityBaer Specker A -algebras with proximity morphisms between them form a category PBSp A that is equivalent to DeV and is dually equivalent to
KHaus . In fact, the functor
KHaus → PBSp A is the de Vries power functor, while the functor PBSp A → KHaus associates witheach proximity Baer Specker A -algebra ( S, ≺ ) , the compact Hausdorff space of ends of ( S, ≺ ) .The obtained duality provides an analogue of de Vries duality for proximity Baer Specker A -algebras.The article is organized as follows. In Section 2 we introduce finitely valued normalfunctions and establish their basic properties. In Section 3, for a totally ordered algebra A , we generalize the notion of a Boolean power of A to that of a de Vries power of A . InSection 4 we specialize to the case of a totally ordered integral domain A , introduce thenotion of a proximity Specker A -algebra, and show that a de Vries power of A is a proximity G. Bezhanishvili, V. Marra, P. J. Morandi, B. Olberding
Baer Specker A -algebra. In Section 5 we prove our main representation theorem that everyproximity Specker A -algebra ( S, ≺ ) embeds in a de Vries power of A , and that the embeddingis an isomorphism iff S is Baer. In Section 6 we introduce proximity morphisms. For proxmitySpecker A -algebras ( S, ≺ ) and ( T, ≺ ) , we prove that there is a 1-1 correspondence betweenproximity morphisms S → T , de Vries morphisms Id ( S ) → Id ( T ) , and continuous maps Y → X , where X and Y are the de Vries duals of Id ( S ) and Id ( T ) , respectively. In Section7 we introduce ends of a proximity Specker A -algebra ( S, ≺ ) , give several characterizationsof ends, and show that the space of ends of ( S, ≺ ) is homeomorphic to the de Vries dual ofId ( S ) . Finally, in Section 8 we prove that the proximity Baer Specker A -algebras form acategory that is equivalent to the category of de Vries algebras and is dually equivalent tothe category of compact Hausdorff spaces.2. Finitely valued normal functions
Throughout this section we assume that X is a compact Hausdorff space and A is a totallyordered set. In Section 3 we specialize to the case in which A is a totally ordered algebraof a given type, and in Section 4 to the case when it is an integral domain. In this sectionthough the algebraic structure of A plays no role.For a ∈ A , let ↑ a = { b ∈ A ∶ a ≤ b } , ↓ a = { b ∈ A ∶ b ≤ a } , and [ a, b ] = ↑ a ∩ ↓ b = { x ∈ A ∶ a ≤ x ≤ b } .We write a < b provided a ≤ b and a ≠ b . We topologize A with the interval topology. In thistopology closed intervals [ a, b ] form a basis of closed sets. Notation 2.1. (1) We denote by F ( X ) = F ( X, A ) the set of all finitely valued functions from X to A ;that is, F ( X ) is the set of all functions f ∶ X → A whose image is finite.(2) We denote by F C ( X ) = F C ( X, A ) the set of all finitely valued continuous functionsfrom X to A , where A has the interval topology. As follows from [5, Prop. 5.4], F C ( X, A ) = C ( X, A disc ) .(3) For nonempty X , each a ∈ A gives rise to the constant function on X whose value is a . Clearly this function is in F C ( X ) , and we will view A as a subset of F C ( X ) .Under the pointwise order, F ( X ) is a lattice, where the join and meet operations arealso pointwise: sup ( f, g )( x ) = max { f ( x ) , g ( x )} and inf ( f, g )( x ) = min { f ( x ) , g ( x )} . Clearly F C ( X ) is a sublattice of F ( X ) .We make frequent use of the simple observation that a finitely valued function on X canalternatively be viewed as a function from A to the powerset of X . We formalize this in thefollowing lemma. If U is a subset of X , we denote by χ U the characteristic function of U . Lemma 2.2. (1) If f ∈ F ( X ) and a < ⋯ < a n are the values of f , set U i = f − ( ↑ a i ) for ≤ i ≤ n and U n + = ∅ . Then X = U ⊃ U ⊃ ⋯ ⊃ U n ⊃ U n + = ∅ . Moreover, f ( x ) = a i iff x ∈ U i − U i + , and f = a + ∑ ni = ( a i − a i − ) χ U i . (2) Conversely, if X = U ⊃ U ⊃ ⋯ ⊃ U n ⊃ U n + = ∅ and a < ⋯ < a n are elements of A ,then the function f ∶ X → A defined by f ( x ) = a i if x ∈ U i − U i + is finitely valued and f − ( ↑ a i ) = U i .Proof. Straightforward. (cid:3) e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 5
Therefore, to define a finitely valued function on X , it suffices to produce a finite sequence a < ⋯ < a n in A and a finite sequence X = U ⊃ U ⊃ ⋯ ⊃ U n ⊃ ∅ of subsets of X . The nextlemma shows that two elements f, g ∈ F ( X ) can be described in a compatible way. Lemma 2.3.
Let f, g ∈ F ( X ) . If the values of f and g are among a < ⋯ < a n and a n + ∈ A satisfies a n < a n + , then f ( x ) = a i if x ∈ f − ( ↑ a i ) − f − ( ↑ a i + ) and g ( x ) = a i if x ∈ g − ( ↑ a i ) − g − ( ↑ a i + ) . Furthermore, f ≤ g iff f − ( ↑ a i ) ⊆ g − ( ↑ a i ) for each i . Consequently, f = g iff f − ( ↑ a i ) = g − ( ↑ a i ) for each i .Proof. Straightforward. (cid:3)
In [12] Dilworth described the Dedekind-MacNeille completion of the lattice C ( X, R ) ofcontinuous real-valued functions by means of normal functions ; that is, lower semicontinuousfunctions f ∶ X → R for which f − ( −∞ , a ) is a union of regular closed sets for each a ∈ R (see[12, Thm. 3.2]; note that Dilworth worked with upper semicontinuous functions). We adaptDilworth’s notion of normal function to the setting of functions with finitely many valuesin A . To motivate our definition, we first describe finitely valued normal functions in thespecial case in which A = R ; this description is not needed later in the paper, but see [7] fora development of proximity in the setting of real-valued normal functions. Proposition 2.4.
Let f ∶ X → R be finitely valued. The following conditions are equivalent. (1) f is normal. (2) f − ( a, ∞ ) is regular open in X for each a ∈ R . (3) f − [ a, ∞ ) is regular open in X for each a ∈ R .Proof. (2) ⇔ (3): Since f is finitely valued, for each a ∈ R there is b > a with f − ( a, ∞ ) = f − [ b, ∞ ) ; we may choose b to be the smallest value of f greater than a if such a valueexists, or else b may be chosen to be any real number larger than a . (Similarly, f − ( −∞ , a ) = f − ( −∞ , b ] for some b < a .) From this it is evident that conditions (2) and (3) are equivalent.(1) ⇒ (2): It is known that a bounded real-valued function f is lower semicontinuous iff f − ( a, ∞ ) is open for each a ∈ R , and that a lower semicontinuous function f is normaliff f − ( −∞ , a ) is a union of regular closed sets for each a ∈ R [12, Thm. 3.2]. Suppose f is normal. Let a ∈ R . Since f is lower semicontinuous, f − ( a, ∞ ) is open in X . Thus, f − ( −∞ , a ] = X − f − ( a, ∞ ) is closed. Because f is finitely valued, there is c ∈ R with f − ( −∞ , a ] = f − ( −∞ , c ) . Since f is normal and regular closed sets are closures of opensets, there is a family { U i } of open sets such that f − ( −∞ , c ) = ⋃ i Cl ( U i ) . Let U = ⋃ i U i , anopen set. Clearly U ⊆ f − ( −∞ , c ) and, as f − ( −∞ , c ) is closed, Cl ( U ) ⊆ f − ( −∞ , c ) . On theother hand, U i ⊆ U implies Cl ( U i ) ⊆ Cl ( U ) , so f − ( −∞ , c ) = ⋃ i Cl ( U i ) ⊆ Cl ( U ) . Therefore, f − ( −∞ , c ) = Cl ( U ) , and as U is open, f − ( −∞ , c ) = f − ( −∞ , a ] is regular closed. Thus, itscomplement f − ( a, ∞ ) is regular open.(2) ⇒ (1): Suppose that f − ( a, ∞ ) is regular open in X for each a ∈ R . Then it is clear that f is lower semicontinuous. In addition, since f − ( −∞ , a ) = f − ( −∞ , b ] for some b < a , and f − ( −∞ , b ] = X − f − ( b, ∞ ) , which is regular closed as f − ( b, ∞ ) is regular open, we see that f is normal. (cid:3) We use this characterization of finitely valued normal functions f ∶ X → R to define finitelyvalued normal functions f ∶ X → A , where A is an arbitrary totally ordered set. Definition 2.5.
We define a finitely valued function f ∶ X → A to be normal provided f − ( ↑ a ) is regular open for each a ∈ A . We denote by F N ( X ) = F N ( X, A ) the set of allfinitely valued normal functions from X to A . G. Bezhanishvili, V. Marra, P. J. Morandi, B. Olberding
Remark 2.6. If f ∈ F ( X ) , with a < ⋯ < a n the values of f and U i = f − ( ↑ a i ) , then f ∈ F N ( X ) iff each U i is regular open. Thus, if each U i is regular open, then Lemma 2.2(1)implies that a + ∑ ni = ( a i − a i − ) χ U i is normal. We will use this fact throughout.While a function in F N ( X ) need not be continuous, the next proposition shows it is con-tinuous on an open dense subset of X . This relationship between finitely valued normal func-tions and continuous functions on open dense subsets is also considered in Proposition 2.10,and is made more explicit in Theorem 3.2. We remind the reader that we are using theinterval topology on A , and that F C ( X ) = C ( X, A disc ) as pointed out in Notation 2.1. Proposition 2.7. If f ∈ F N ( X ) , then f is continuous on an open dense subset of X .Proof. Let a < ⋯ < a n be the values of f , and let U i = f − ( ↑ a i ) . Then each U i is regularopen. We show that f is continuous on U ∶ = (⋃ n − i = ( U i − Cl ( U i + ))) ∪ U n , and that this unionis open dense in X . For continuity, since f ( U i − Cl ( U i + )) = { a i } and f ( U n ) = { a n } , we seethat f is constant, hence continuous on the open set U i − Cl ( U i + ) for each i , as well as onthe open set U n . Therefore, f is continuous on the open set U . To prove density, let V be anonempty open subset of X . There is a smallest m > V ∩ U m = ∅ . Then V ∩ U m − ≠ ∅ and V ∩ Cl ( U m ) = ∅ . Therefore, V ∩ U ≠ ∅ . Thus, U is open dense in X . (cid:3) Definition 2.8.
Let f ∈ F ( X ) and let a < ⋯ < a n be the values of f . For each i = , . . . , n ,set U i = Int ( Cl ( f − ( ↑ a i ))) , and let U n + = ∅ . Define f ∶ X → A by f ( x ) = a i provided x ∈ U i − U i + . By Lemma 2.2(2), f ∈ F N ( X ) , and we call f the normalization of f . Remark 2.9.
For f ∈ F ( X ) , the following facts are immediate:(1) f ∈ F N ( X ) iff f = f .(2) If f = χ U for U ⊆ X , then f = χ Int ( Cl ( U )) . More generally, write f = a + ∑ ni = ( a i − a i − ) χ U i as in Lemma 2.2(1). Then f = a + ∑ ni = ( a i − a i − ) χ Int ( Cl ( U i )) .(3) The image of f is contained in the image of f .(4) If f ∈ F C ( X ) , then f is normal.Let U be a nonempty subset of X and let f ∈ F ( U ) . Replacing X by U and using thesame idea as in Definition 2.8 allows us to define f ∈ F N ( X ) . Then f is characterized by ( f ) − ( ↑ a ) = Int ( Cl ( f − ( ↑ a ))) for each a ∈ A . Proposition 2.10.
Let U be an open dense subset of X and let f ∈ F C ( U ) . Then f is theunique function in F N ( X ) that restricts to f on U .Proof. Let a < ⋯ < a n be the values of f . If 1 ≤ i ≤ n , let V i = f − ( ↑ a i ) and U i = Int ( Cl ( V i )) ,and set U n + = V n + = ∅ . As f is continuous and ↑ a i is closed, V i is closed in U . This yields V i = Cl ( V i ) ∩ U . Therefore, V i ⊆ U i ∩ U = Int ( Cl ( V i )) ∩ U ⊆ Cl ( V i ) ∩ U = V i , so V i = U i ∩ U . Let x ∈ U , and suppose that f ( x ) = a i . Then x ∈ V i − V i + , and as x ∈ U , we have x ∈ U i − U i + .Thus, f ( x ) = a i = f ( x ) , and so f ∣ U = f .For uniqueness, let g ∈ F N ( X ) with g ∣ U = f and let a ∈ A . Then g − ( ↑ a ) ∩ U = f − ( ↑ a ) .Since g − ( ↑ a ) is regular open and U is open dense, Cl ( g − ( ↑ a )) = Cl ( g − ( ↑ a ) ∩ U ) , so g − ( ↑ a ) = Int ( Cl ( g − ( ↑ a ))) = Int ( Cl ( g − ( ↑ a ) ∩ U )) = Int ( Cl ( f − ( ↑ a ))) . This yields g − ( ↑ a ) = ( f ) − ( ↑ a ) for each a ∈ A , so g = f by Lemma 2.3. (cid:3) The partial order on F ( X ) restricts to F N ( X ) . By normalizing the join and meet oper-ations on F ( X ) , we obtain operations on F N ( X ) which we show are the join and meet in F N ( X ) with respect to the induced partial order on F N ( X ) . e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 7 Proposition 2.11.
F N ( X ) is a lattice, where the meet is the pointwise meet and the joinis the normalization of the pointwise join. In other words, if ∧ , ∨ denote the meet and joinoperations on F N ( X ) , then for f, g ∈ F N ( X ) , we have f ∧ g = inf ( f, g ) and f ∨ g = sup ( f, g ) . Proof.
First we claim that the normalization operation is order preserving; that is, if f, g ∈ F ( X ) with f ≤ g , then f ≤ g . Let a < ⋯ < a n be elements of A containing all valuesof f and g . By Lemma 2.3, f − ( ↑ a i ) ⊆ g − ( ↑ a i ) for each i . Therefore, Int ( Cl ( f − ( ↑ a i ))) ⊆ Int ( Cl ( g − ( ↑ a i ))) for each i . Thus, applying Lemma 2.3 again yields f ≤ g .Now we prove the proposition. Let f, g ∈ F N ( X ) . Then inf ( f, g ) − ( ↑ a ) = f − ( ↑ a ) ∩ g − ( ↑ a ) is regular open for each a ∈ A . Thus, inf ( f, g ) ∈ F N ( X ) , and so f ∧ g = inf ( f, g ) . Since f ≤ sup ( f, g ) , we see that f = f ≤ sup ( f, g ) . Similarly, g ≤ sup ( f, g ) . If k ∈ F N ( X ) with f, g ≤ k , then sup ( f, g ) ≤ k in F ( X ) , and so sup ( f, g ) ≤ k = k . Consequently, f ∨ g = sup ( f, g ) . (cid:3) de Vries powers of totally ordered algebras In this section we continue to assume X is a compact Hausdorff space, but we assume nowthat A is a totally ordered algebra of a given type. We introduce the notion of a de Vriespower of A by X (as a set it will be F N ( X ) ) in such a way that the power is an algebraof the same type as A and comes equipped with a canonical proximity relation. We firstindicate how to lift operations from A to F ( X ) ; once this is accomplished, we normalizethese operations to obtain an algebraic structure on F N ( X ) having the same type as thatof A .We extend the order and operations on A to F ( X ) pointwise. That is, for f, g ∈ F ( X ) ,we set f ≤ g iff f ( x ) ≤ g ( x ) for each x ∈ X , and if λ is an m -ary operation on A and f , . . . , f m ∈ F ( X ) , then we set λ ( f , . . . , f m )( x ) = λ ( f ( x ) , . . . , f m ( x )) . It is clear that F ( X ) is a partially ordered algebra of the same type as A . Furthermore, if a ∈ A , then λ ( f , . . . , f m ) − ( a ) = ⋃{ f − ( b ) ∩ ⋯ ∩ f − m ( b n ) ∶ λ ( b , . . . , b m ) = a } . From this and
F C ( X ) = C ( X, A disc ) (see Notation 2.1(2)) it follows that λ ( f , . . . , f m ) ∈ F C ( X ) for each f , . . . , f m ∈ F C ( X ) . Thus, F C ( X ) is a subalgebra of F ( X ) . Definition 3.1.
For each m -ary operation λ on A , define the m -ary operation λ on F N ( X ) by λ ( f , . . . , f m ) = λ ( f , . . . , f m ) for all f , . . . , f n ∈ F N ( X ) .This makes F N ( X ) a partially ordered algebra of the same type as A , and F C ( X ) is asubalgebra of F N ( X ) . Alternatively, F N ( X ) can be viewed as a direct limit of the F C ( U ) ,where U ranges over the directed set I of dense open subsets of X , and the operations on F N ( X ) then are those induced by the pointwise operations on the F C ( U ) . Theorem 3.2makes this explicit. We use the fact that the direct limit of the directed system { F C ( U ) ∶ U ∈ I } can be described as the set of all pairs ( f, U ) with U ∈ I and f ∈ F C ( U ) , and where ( f, U ) = ( g, V ) whenever there is W ∈ I with W ⊆ U ∩ V and f ∣ W = g ∣ W (see [1, Sec. 1]).Since each F C ( U ) is an algebra of the same type as A , the direct limit is also an algebra ofthe same type as A . Theorem 3.2.
The algebra
F N ( X ) is isomorphic to the direct limit L of the algebras F C ( U ) as U ranges over all open dense subsets of X . G. Bezhanishvili, V. Marra, P. J. Morandi, B. Olberding
Proof.
For each open dense subset U of X , we have a map F C ( U ) → F N ( X ) , given by f ↦ f . These then induce a map α ∶ L → F N ( X ) , given by α ( f, U ) = f . This map is welldefined because if ( f, U ) = ( g, V ) , then f and g are normal functions extending f (and g ) on a dense open set W ⊆ U ∩ V . Thus, by Proposition 2.10, f = g . To see that α isa homomorphism, let λ be an m -ary operation on A and let g , . . . , g m ∈ L . We may find asingle open dense set U for which g i = ( f i , U ) for some f i ∈ F C ( U ) . Then α ( λ ( g , . . . , g m )) = α (( λ ( f , . . . , f m ) , U )) = λ ( f , . . . , f m ) . On the other hand, λ ( α ( g ) , . . . , α ( g m )) = λ ( f , . . . , f m ) . Both functions λ ( f , . . . , f m ) and λ ( f , . . . , f m ) are normal functions on X and restrictto λ ( f , . . . , f m ) on U . Thus, by Proposition 2.10, they are equal. This proves that α is a homomorphism. It is 1-1 because if ( f, U ) , ( g, V ) are in L and f = g , then byProposition 2.7, there is an open dense set W with f continuous on W . By replacing W with W ∩ U ∩ V , we may assume W ⊆ U ∩ V . By Proposition 2.10, f ∣ W = f ∣ W = g ∣ W = g ∣ W .Finally, α is onto because if h ∈ F N ( X ) , then by Proposition 2.7, h is continuous on anopen dense set U , so h = α ( h ∣ U , U ) by Proposition 2.10. Consequently, L and F N ( X ) areisomorphic as algebras. (cid:3) We have noted that
F N ( X ) is an algebra of the same type as A . The de Vries power of A by X is then the algebra F N ( X ) equipped with a canonically chosen proximity relation;that the relation indeed behaves like a proximity is proved in Theorem 3.4. Definition 3.3. (1) The de Vries power of the totally ordered algebra A by X is the algebra F N ( X ) with the relation ≺ X defined by f ≺ X g if Cl ( f − ( ↑ a )) ⊆ g − ( ↑ a ) for each a ∈ A. In other words, f ≺ X g provided f − ( ↑ a ) ≺ g − ( ↑ a ) in the de Vries algebra of regularopen subsets of X .(2) If ( B, ≺ ) is the de Vries algebra whose de Vries dual is X , then we call ( F N ( X ) , ≺ X ) the de Vries power of A by ( B, ≺ ) .As the next theorem shows, ≺ X satisfies typical axioms for a proximity relation. Theorem 3.4.
The relation ≺ X on F N ( X ) has the following properties. (1) f ≺ X g implies f ≤ g . (2) f ≤ g ≺ X h ≤ k implies f ≺ X k . (3) f ≺ X g, h implies f ≺ X g ∧ h . (4) f, g ≺ X h implies f ∨ g ≺ X h . (5) f ≺ X g implies there is h ∈ F N ( X ) with f ≺ X h ≺ X g . (6) f ≺ X f iff f ∈ F C ( X ) .Proof. (1) Let f ≺ X g . If a ∈ A , then f − ( ↑ a ) ≺ g − ( ↑ a ) , so f − ( ↑ a ) ⊆ g − ( ↑ a ) . Thus, byLemma 2.3, f ≤ g .(2) Let f ≤ g ≺ X h ≤ k and let a ∈ A . Then f − ( ↑ a ) ⊆ g − ( ↑ a ) ≺ h − ( ↑ a ) ⊆ k − ( ↑ a ) , so f − ( ↑ a ) ≺ k − ( ↑ a ) . Thus, f ≺ X k . e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 9 (3) Let f ≺ X g, h , and let a ∈ A . Then f − ( ↑ a ) ≺ g − ( ↑ a ) , h − ( ↑ a ) , so f − ( ↑ a ) ≺ g − ( ↑ a ) ∩ h − ( ↑ a ) . By Proposition 2.11, g − ( ↑ a ) ∩ h − ( ↑ a ) = ( g ∧ h ) − ( ↑ a ) . Therefore, f − ( ↑ a ) ≺ ( g ∧ h ) − ( ↑ a ) . Thus, f ≺ X g ∧ h .(4) Let f, g ≺ X h , and let a ∈ A . Then f − ( ↑ a ) , g − ( ↑ a ) ≺ h − ( ↑ a ) , so f − ( ↑ a ) ∨ g − ( ↑ a ) ≺ h − ( ↑ a ) . Since
Int ( Cl ( sup ( f, g ) − ( ↑ a ))) = Int ( Cl ( f − ( ↑ a ) ∪ g − ( ↑ a ))) = f − ( ↑ a ) ∨ g − ( ↑ a ) , we have Int ( Cl ( sup ( f, g ) − ( ↑ a ))) ≺ h − ( ↑ a ) . By Proposition 2.11, f ∨ g = sup ( f, g ) . Thus,by Definition 2.8, ( f ∨ g ) − ( ↑ a ) = Int ( Cl ( sup ( f, g ) − ( ↑ a ))) , so ( f ∨ g ) − ( ↑ a ) ≺ h − ( ↑ a ) , whichimplies that f ∨ g ≺ X h .(5) Let f ≺ X g and let the values of f and g be among a < ⋯ < a n . We have f − ( ↑ a i ) ≺ g − ( ↑ a i ) for each i , so there is a regular open set U i with f − ( ↑ a i ) ≺ U i ≺ g − ( ↑ a i ) . Set V i = U ∩ ⋯ ∩ U i for 0 ≤ i ≤ n and V n + = ∅ . Then the V i are decreasing regular open sets and f − ( ↑ a i ) ≺ V i ≺ g − ( ↑ a i ) for each i . If we define h by h ( x ) = a i provided x ∈ V i − V i + , then h ∈ F N ( X ) and f ≺ X h ≺ X g .(6) Let a < ⋯ < a n be the values of f . Then f ≺ X f iff Cl ( f − ( ↑ a i )) ⊆ f − ( ↑ a i ) for each i ,which happens iff f − ( ↑ a i ) is clopen for each i . This is clearly equivalent to f ∈ F C ( X ) . (cid:3) We next show that the notion of a de Vries power of a totally ordered algebra encompassesthat of a Boolean power.
Theorem 3.5. If X is a Stone space, then the Boolean power of a totally ordered algebra A by X is the subalgebra { f ∈ F N ( X ) ∶ f ≺ X f } of F N ( X ) . Thus, every Boolean power of atotally ordered algebra can be expressed as a subalgebra of a de Vries power of the algebra.Proof. As follows from [5, Prop. 5.4], the Boolean power C ( X, A disc ) is equal to F C ( X ) . ByTheorem 3.4(6), F C ( X ) = { f ∈ F N ( X ) ∶ f ≺ X f } . As we noted after Definition 3.1, F C ( X ) is a subalgebra of F N ( X ) . Therefore, the Boolean power C ( X, A disc ) is the subalgebra { f ∈ F N ( X ) ∶ f ≺ X f } of F N ( X ) . (cid:3) Corollary 3.6.
The Boolean power of a totally ordered algebra A by a Stone space X coin-cides with the de Vries power of A by X iff X is extremally disconnected.Proof. It is well known that X is extremally disconnected iff regular opens of X coincidewith clopens of X . This is clearly equivalent to F C ( X ) = F N ( X ) . The result follows. (cid:3) Of course, the proximity axioms in Theorem 3.4 ignore the algebraic structure of
F N ( X ) induced by that of A . Such axioms will depend on the behavior of the operations on A withrespect to the total order of A , and the interplay between the operations and the proximitycan be subtle. In what follows, we axiomatize de Vries powers of a totally ordered integraldomain, thus generalizing the axiomatization of Boolean powers of a totally ordered domaingiven in [5]. This includes the axiomatization of de Vries powers of such classic algebras as Z , Q , and R , thus generalizing the results of [16, 10, 6]. It would be of interest to axiomatizede Vries powers of other algebras as well.4. de Vries powers of totally ordered domains In this section X continues to denote a compact Hausdorff space, but we assume nowthat A is a totally ordered integral domain. Theorem 3.4 indicates how the relation ≺ X on F N ( X ) behaves with respect to the lattice structure of F N ( X ) . In this section we describe the algebraic structure of F N ( X ) and show how the ring operations on F N ( X ) induced bythose of the totally ordered domain A interact with the relation ≺ X .Recall that a ring with a partial order ≤ is an ℓ -ring (lattice-ordered ring) provided (i) itis a lattice, (ii) a ≤ b implies a + c ≤ b + c for each c , and (iii) 0 ≤ a, b implies 0 ≤ ab . An ℓ -ringis totally ordered if the order is a total order, and it is an f -ring if it is a subdirect productof totally ordered rings. It is well known (see, e.g., [8, Ch. XVII, Corollary to Thm. 8]) thatan ℓ -ring is an f -ring iff a ∧ b = c ≥ ac ∧ b = S is an ℓ -algebra if it is an ℓ -ring, an A -algebra (with A as above), and a ∈ A , s ∈ S with 0 ≤ a, s imply that 0 ≤ as . An ℓ -algebra S is an f -algebra provided the ring S is an f -ring. If S = { } , then we call S a trivial f -algebra. If S is a nontrivial torsion-free f -algebra, then a ↦ a ⋅ A into S , and without loss of generality, we view A as asubalgebra of S . Notation 4.1.
For a torsion-free f -algebra S over A , we denote the image a ⋅ a ∈ A in S by a . When S is nontrivial, then since S is torsion-free, we may in fact identify a with itsimage in S . However, when S is trivial, then for each a ∈ A , we have a = S under ourconvention. Since we will mostly be dealing with nontrivial algebras, this will cause littleconfusion. Definition 4.2.
Let S be a torsion-free f -algebra over A . A binary relation ≺ on S is a proximity if the following axioms are satisfied:(P1) 0 ≺ ≺ s ≺ t implies s ≤ t .(P3) s ≤ t ≺ r ≤ u implies s ≺ u .(P4) s ≺ t, r implies s ≺ t ∧ r .(P5) s ≺ t implies − t ≺ − s .(P6) s ≺ t and r ≺ u imply s + r ≺ t + u .(P7) s ≺ t implies as ≺ at for each 0 < a ∈ A , and as ≺ at for some 0 < a ∈ A implies s ≺ t .(P8) s, t, r, u ≥ s ≺ t and r ≺ u imply sr ≺ tu .(P9) s ≺ t implies there is r ∈ S with s ≺ r ≺ t .(P10) s > < t ∈ S with t ≺ s .A pair ( S, ≺ ) is a proximity A -algebra if S is a torsion-free f -algebra over A and ≺ is aproximity on S . If S is a Specker A -algebra, then we call ( S, ≺ ) a proximity Specker A -algebra . Remark 4.3. (1) It is an easy consequence of the axioms that s ≺ t and r ≺ u imply s ∧ r ≺ t ∧ u and s ∨ r ≺ t ∨ u . Also, it follows from (P1), (P7), and (P5) that for each a ∈ A , we have a ≺ a .(2) In “good” cases, one implication of axiom (P7), that as ≺ at for some 0 < a ∈ A implies s ≺ t , is superfluous. For example, if A is a field and as ≺ at for some 0 < a ∈ A , thenby the other implication of axiom (P7), we obtain a − as ≺ a − at , so s ≺ t . It is alsosuperfluous in some other cases, but we leave the details out because in what followswe will use axiom (P7) in its full strength.(3) By [5, Thm. 5.1], each Specker A -algebra has a unique partial order making it intoan f -algebra. Since A is a domain, it is a torsion-free f -algebra, so proximity Specker A -algebras are well defined. e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 11 We show in Theorem 4.10 that not only is
F N ( X ) a proximity A -algebra, but it has theparticularly transparent algebraic structure of a Baer Specker A -algebra, a notion we recallnow. Definition 4.4. A Baer ring S is a commutative ring such that the annihilator of eachsubset of S is generated as an ideal by an idempotent of the ring (see, e.g., [14, p. 260]). ASpecker algebra S over the domain A is a Baer Specker A -algebra provided S is a Baer ring,and a proximity Specker A -algebra ( S, ≺ ) is a proximity Baer Specker A -algebra provided S is a Baer Specker A -algebra.A Specker A -algebra is a Baer Specker A -algebra iff the Boolean algebra Id ( S ) of idempo-tents of S is a complete Boolean algebra [5, Thm. 4.3]. To prove that F N ( X ) is a proximityBaer Specker A -algebra, we rely on the following lemma, which can be viewed as a descriptionof the operations on F N ( X ) that are lifted from those of the domain A .We note that the operations on A are lifted to F ( X ) pointwise, while the operationson F N ( X ) are normalizations of the operations on F ( X ) . In particular, F N ( X ) is not asubalgebra of F ( X ) . For f, g ∈ F N ( X ) , we denote the sum, product, and join of f and g in F N ( X ) by f + g , f g , and f ∨ g , respectively. So f + g is the normalization of the pointwisesum, f g is the normalization of the pointwise product, and f ∨ g is the normalization of thepointwise join of f and g . On the other hand, as was shown in Proposition 2.11, f ∧ g is thepointwise meet of f and g . Lemma 4.5.
Let f, g ∈ F N ( X ) and let a ∈ A . (1) ( f + g ) − ( ↑ a ) = ⋁{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b + c ≥ a } . (2) If f, g ≥ , then ( f g ) − ( ↑ a ) = ⋁{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b, c ≥ , bc ≥ a } . (3) If < c ∈ A , then ( cf ) − ( ↑ a ) = f − ( ↑ b ) , where b is the smallest value of f for which bc ≥ a . Furthermore, cf is the pointwise scalar product of c and f . (4) If < c ∈ A , then ( cf ) − ( ↑ ( ca )) = f − ( ↑ a ) . (5) ( − f ) − ( ↑ a ) = ¬ f − ( ↑ b ) , where b is the smallest value of f larger than − a (provided itexists, otherwise b is any element of A larger than − a ).Proof. (1) Let h be the pointwise sum of f and g . Then f + g is the normalization of h , and f + g is determined by the formula ( f + g ) − ( ↑ a ) = Int ( Cl ( h − ( ↑ a ))) for all a ∈ A . We claim that ( f + g ) − ( ↑ a ) = ⋁{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b + c ≥ a } , where the join is in RO( X ) . To see this, we first point out that h − ( ↑ a ) = ⋃{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b + c ≥ a } . Therefore, ( f + g ) − ( ↑ a ) = Int ( Cl ( h − ( ↑ a ))) = Int ( Cl (⋃{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b + c ≥ a })) = ⋁{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b + c ≥ a } . (2) Suppose that f, g ≥
0. Let h be the pointwise product of f and g . Then f g is thenormalization of h . Since f, g ≥ h − ( ↑ a ) = ⋃{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b, c ≥ , bc ≥ a } . Therefore, ( f g ) − ( ↑ a ) = Int ( Cl (⋃{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b, c ≥ , bc ≥ a })) = ⋁{ f − ( ↑ b ) ∩ g − ( ↑ c ) ∶ b, c ≥ , bc ≥ a } . (3) Let h be the pointwise scalar product of c and f . Then cf is the normalization of h .Observe that h − ( ↑ a ) = ⋃{ f − ( ↑ b ) ∶ cb ≥ a } . Because f is finitely valued, there is a smallest value b of f for which cb ≥ a . The formulaabove then shows that h − ( ↑ a ) = f − ( ↑ b ) . This implies that h − ( ↑ a ) is regular open, so h isnormal. Thus, cf = h is the pointwise scalar product of c and f .(4) This follows from (3) since A being a totally ordered domain and 0 < c imply ca ≤ cb iff a ≤ b .(5) Let h be the pointwise negative of f (that is, h ( x ) = − f ( x ) for each x ∈ X ), and let − f be the normalization of h . Since f is finitely valued, we have h − ( ↑ a ) = { x ∈ X ∶ a ≤ h ( x )} = { x ∈ X ∶ f ( x ) ≤ − a } = X − f − ( ↑ b ) , where b is the smallest value of f larger than − a (provided it exists, otherwise b can be anyelement of A larger than − a ). Therefore, ( − f ) − ( ↑ a ) = Int ( Cl ( h − ( ↑ a ))) = Int ( Cl ( X − f − ( ↑ b ))) = Int ( X − f − ( ↑ b )) = ¬ f − ( ↑ b ) . (cid:3) Remark 4.6.
The operations of addition, multiplication, and join in
F N ( X ) are in generalnot pointwise. In fact, any one of these operations is pointwise iff X is extremally discon-nected. One implication follows from Corollary 3.6. To see the converse, say for addition,let U be a regular open subset of X that is not clopen. Then the pointwise sum of χ U and χ ¬ U is χ U ∪¬ U , and since U ∪ ¬ U ≠ X , we see that χ U ∪¬ U ≠
1. On the other hand, since U ∪ ¬ U is dense in X , we have χ U + χ ¬ U = ( χ U ∪¬ U ) = χ X = F N ( X ) . Remark 4.7.
By Remark 4.6, for f, g ∈ F N ( X ) , the sum f + g in F N ( X ) need not bethe pointwise sum. In spite of this, if one of f, g is a constant function, then f + g is thepointwise sum. The same is true for join and multiplication by a positive scalar. That scalarmultiplication by a positive scalar is pointwise was pointed out in Lemma 4.5(3). Moregenerally, these facts are immediate consequences of the following facts about normalization.Let f ∈ F ( X ) and a ∈ A . For notational convenience, let + refer to the pointwise sum in F ( X ) . Then ( a + f ) = a + f , sup ( a, f ) = a ∨ f . In addition, if 0 ≤ a and ⋅ refers to pointwise multiplication, then ( a ⋅ f ) = a ⋅ f . The arguments for each of these statements are similar, so we give the proof for the first.For each b ∈ A we have [( a + f ) ] − ( ↑ b ) = Int ( Cl (( a + f ) − ( ↑ b ))) = Int ( Cl ( f − ( ↑ ( b − a )))) = ( f ) − ( ↑ ( b − a )) = ( a + f ) − ( ↑ b ) . Thus, ( a + f ) = a + f . e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 13 In contrast, scalar multiplication by a negative scalar is not pointwise. To see this, let U be regular open that is not clopen, and let f = χ U . Then ( − ) f = ( − ) χ U = − + χ X − U , where − + χ X − U is the pointwise sum. Therefore, ( − ) f is not normal because (( − ) f ) − ( ↑ ) = X − U is not regular open.Finally, we point out that by Remark 2.6, if a < ⋯ < a n and X = U ⊃ U ⊃ ⋯ ⊃ U n ⊃ ∅ areregular open, then the pointwise sum a + ∑ ni = ( a i − a i − ) χ U i is normal. Thus, the sum of a and the ( a i − a i − ) χ U i in F N ( X ) is the same as the sum in F ( X ) .It is well known that in a commutative ring S , the set Id ( S ) of idempotents of S forms aBoolean algebra with respect to the operations s ∧ t = st, s ∨ t = s + t − st, ¬ s = − s . Lemma 4.8.
F N ( X ) is a commutative ring with 1, and f ∈ F N ( X ) is an idempotent iff f = χ U for some regular open U . Moreover, the map U ↦ χ U is a Boolean isomorphismbetween RO( X ) and Id ( F N ( X )) .Proof. Observe that
F C ( U ) is a commutative ring with 1 for each open dense subset U of X . Therefore, so is the direct limit of the F C ( U ) . Now apply Theorem 3.2 to conclude that F N ( X ) is a commutative ring with 1.Let U ∈ RO( X ) . Then χ U is idempotent in F ( X ) , and since it is a normal function, itis idempotent in F N ( X ) . Conversely, suppose that f ∈ Id ( F N ( X )) . If h is the pointwisesquare of f , then f = h . By Remark 2.9(2), Im ( h ) ⊆ Im ( h ) . This implies that Im ( f ) = Im ( f ) ⊆ Im ( h ) . Moreover, Im ( h ) = { a ∶ a ∈ Im ( f )} . Let a < ⋯ < a n be the values of f .The values of h are a , ⋯ , a n . Because Im ( f ) ⊆ Im ( h ) , we have Im ( f ) = Im ( h ) . This thenyields a i = a i for each i . Thus, a i ∈ { , } . Consequently, if U = f − ( ) , then f = χ U . ByRemark 2.9(2) and the fact that f is normal, we obtain χ U = f = f = χ Int ( Cl ( U )) , whichshows that U is regular open in X .It is clear that χ U ∩ V = χ U ∧ χ V . Also, if h is the pointwise negation of χ U , then in F ( X ) wehave h = − + χ X − U . Therefore, by Remark 4.7, in F N ( X ) we have − χ U = h = − + χ Int ( X − U ) = − + χ ¬ U . Thus, in F N ( X ) we have ¬ χ U = − χ U = + ( − + χ ¬ U ) = χ ¬ U . This yields that U ↦ χ U is a Boolean isomorphism between RO( X ) and Id ( F N ( X )) . (cid:3) In order to prove that de Vries powers of A are proximity Baer Specker A -algebras, we needthe following lemma, which will also be used in later sections. We recall [5, Thm. 5.1] thata Specker A -algebra S has a unique partial order ≤ for which S is a torsion-free f -algebraover A . Lemma 4.9.
Let S be a torsion-free f -algebra over A . (1) If e ∈ Id ( S ) , then ≤ e ≤ . (2) The restriction of ≤ to Id ( S ) is the Boolean ordering on Id ( S ) . (3) If e ∈ Id ( S ) and a ∈ A with ≤ a ≤ , then a ∧ e = ae . (4) If e ∈ Id ( S ) and a ∈ A with ≤ a , then ae ∧ = e . (5) If ≠ e ∈ Id ( S ) and a ∈ A with ae ≥ , then a ≥ . (6) Let ≠ e, k ∈ Id ( S ) and < a, b ∈ A . Then ae ≤ bk iff a ≤ b and e ≤ k .Proof. (1) Let e ∈ Id ( S ) . Then e = e is a square in S . Since S is an f -ring, squares in S are nonnegative [8, Ch. XVII, Lem. 2]. This forces e ≥
0. The same argument shows that1 − e ≥
0, so e ≤ (2) Let e, k ∈ Id ( S ) . We must show that e ≤ k iff ek = e . If ek = e , then by (1), e = ek ≤ ⋅ k = k . Conversely, suppose that e ≤ k . By (1), 0 ≤ − k . Therefore, 0 ≤ e ( − k ) ≤ k ( − k ) = e ( − k ) =
0, so ek = e .(3) Let e ∈ Id ( S ) and a ∈ A with 0 ≤ a ≤
1. We have e ∧ ( − e ) =
0. Then, since 1 − a ≥ S is an f -ring, ( − a ) e ∧ ( − e ) =
0. Using again that S is an f -ring and a ≥
0, weobtain ( − a ) e ∧ a ( − e ) =
0. Therefore, ( e − ae ) ∧ ( a − ae ) =
0. As S is an ℓ -ring, we have ( r + t ) ∧ ( s + t ) = ( r ∧ s ) + t for each r, s, t ∈ S . This implies ( e ∧ a ) − ae =
0, so a ∧ e = ae .(4) Let e ∈ Id ( S ) and a ∈ A with 1 ≤ a . We have ( ae ∧ ) − e = ( a − ) e ∧ ( − e ) . Now, since e ∧ ( − e ) = a − ≥
0, and S is an f -ring, ( a − ) e ∧ ( − e ) =
0. Thus, ( ae ∧ ) − e =
0, so ae ∧ = e .(5) Let 0 ≠ e ∈ Id ( S ) and a ∈ A with ae ≥
0. By (1), e ≥
0. If a / ≥
0, then as A is totallyordered, a <
0. Therefore, − a >
0, so − ae ≥
0. Thus, since ae, − ae ≥
0, we see that ae =
0. As e ≠ S is torsion-free, we conclude that a =
0, a contradiction. Consequently, a ≥ ae ≤ bk . Then 0 ≤ ae ( − k ) ≤ bk ( − k ) =
0. Therefore, ae ( − k ) =
0. As a ≠ S is torsion-free, e ( − k ) =
0, so e = ek .This, by (2), implies that e ≤ k . Next, ae ≤ bk implies ae ≤ bek , so ae ≤ be . Therefore, ( b − a ) e ≥
0. Thus, by (5), b − a ≥
0, so a ≤ b . (cid:3) Theorem 4.10.
The de Vries power ( F N ( X ) , ≺ X ) of A is a proximity Baer Specker A -algebra.Proof. We first show that
F N ( X ) is a Baer Specker f -algebra. If f ∈ F N ( X ) , thenLemma 2.2(1) and Remark 2.6 show that f is a linear combination of idempotents. Tosee that F N ( X ) is torsion-free over A , if f ∈ F N ( X ) and 0 ≠ a ∈ A with af =
0, we may as-sume that a >
0. By Lemma 4.5(3), af is the pointwise scalar product. Therefore, af ( x ) = x ∈ X . Since A is a domain, this forces f ( x ) = x ∈ X . Thus, f =
0, andso
F N ( X ) is a Specker A -algebra. Next, by Lemma 4.8, Id ( F N ( X )) is isomorphic to thecomplete Boolean algebra RO( X ) , so F N ( X ) is a Baer ring by [5, Thm. 4.3]. Finally, tosee that F N ( X ) is an f -algebra with respect to the pointwise order ≤ , since F N ( X ) is aSpecker A -algebra, by [5, Thm. 5.1], it has a unique partial order ≤ ′ which makes it into an f -algebra. We show that ≤ ′ is the same as the pointwise order ≤ on F N ( X ) . Since F N ( X ) is an f -algebra with respect to ≤ ′ , squares are nonnegative [8, Ch. XVII, Lem. 2]. Therefore,idempotents in F N ( X ) are nonnegative. Let f ∈ F N ( X ) and let a < ⋯ < a n be the valuesof f . Set U i = f − ( ↑ a i ) for each i . Then each U i is regular open and f = a + ∑ ni = ( a i − a i − ) χ U i .Since a i − a i − > χ U i ≥ ′
0, we have 0 ≤ ′ ∑ ni = ( a i − a i − ) χ U i . Thus, if 0 ≤ a , then 0 ≤ ′ f .Conversely, suppose that 0 ≤ ′ f . If f = a , then 0 ≤ a . Suppose that n ≥
1. Then U is aproper regular open set. Therefore, ¬ U ≠ ∅ , so χ ¬ U is a nonzero idempotent in F N ( X ) .Since it is nonnegative, we get 0 ≤ ′ f χ ¬ U = a χ ¬ U . Consequently, by Lemma 4.9(5), a ≥ ≤ ′ f iff 0 ≤ a . On the other hand, it is clear for the pointwise order ≤ that 0 ≤ f iff0 ≤ a as a is the smallest value of f . Thus, ≤ ′ is equal to ≤ , and so F N ( X ) is an f -algebrawith respect to the pointwise order ≤ .It remains to show that ≺ X is a proximity in the sense of Definition 4.2. That axioms(P1), (P2), (P3), (P4), and (P9) hold follows from Theorem 3.4.(P5) Suppose that f ≺ X g . Then f − ( ↑ a ) ≺ g − ( ↑ a ) for each a ∈ A , so ¬ g − ( ↑ a ) ≺ ¬ f − ( ↑ a ) .Therefore, by Lemma 4.5(5), ( − g ) − ( ↑ a ) ≺ ( − f ) − ( ↑ a ) . Thus, − g ≺ X − f .(P6) Suppose that f ≺ X h and g ≺ X k . Then, for each b ∈ A , we have f − ( ↑ b ) ≺ h − ( ↑ b ) and g − ( ↑ b ) ≺ k − ( ↑ b ) . Therefore, by Lemma 4.5(1) and the fact that the join in question e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 15 involves only finitely many regular open sets, ( f + g ) − ( ↑ a ) ≺ ( h + k ) − ( ↑ a ) for each a ∈ A .Thus, f + g ≺ X h + k .(P7) Suppose that f ≺ X g and 0 < c ∈ A . Lemma 4.5(3) then implies that ( cf ) − ( ↑ a ) ≺ ( cg ) − ( ↑ a ) for each a ∈ A . Thus, cf ≺ X cg . Conversely, suppose that c > cf ≺ cg . Then,for each b ∈ A , we have ( cf ) − ( ↑ b ) ≺ ( cg ) − ( ↑ b ) . Setting b = ca and applying Lemma 4.5(4)yields f − ( ↑ a ) ≺ g − ( ↑ a ) for each a ∈ A . Thus, f ≺ X g .(P8) Suppose that f, g, h, k ≥ f ≺ X h , and g ≺ X k . Lemma 4.5(2) and the fact thatthe join in question involves only finitely many regular open sets then give ( f g ) − ( ↑ a ) ≺ ( hk ) − ( ↑ a ) for each a ∈ A . Thus, f g ≺ X hk .(P10) Let 0 < g . Then X = g − ( ↑ ) and there is b > g − ( ↑ b ) ≠ ∅ . Let a < ⋯ < a n be the values of g . We have 0 ≤ a and at least one of the a i satisfies a i >
0. For each i > U i with ∅ ≠ U i ≺ g − ( ↑ a i ) , and set U = X . Since g − ( ↑ a ) = X , wehave U i ≺ g − ( ↑ a i ) for each i . Set V i = U ∩ ⋯ ∩ U i for 0 ≤ i ≤ n and V n + = ∅ . Define f by f ( x ) = a i if x ∈ V i − V i + . Then f ∈ F N ( X ) and f ≺ X g . Furthermore, 0 < f since each valueof f is at least 0 and one value is greater than 0. (cid:3) In Corollary 5.6 we prove the converse, that every proximity Baer Specker A -algebra isof the form ( F N ( X ) , ≺ X ) for an appropriate choice of X . This is accomplished through amore nuanced investigation of proximity Specker A -algebras.5. Proximity Specker algebras
In this section we continue to assume that A is a totally ordered domain; however, we nolonger assume that X is a fixed compact Hausdorff space. In the last section, we establishedthat when X is a compact Hausdorff space, then ( F N ( X ) , ≺ X ) is a proximity Baer Specker A -algebra. In this section we show that every proximity Baer Specker A -algebra is of the form ( F N ( X ) , ≺ X ) , for some compact Hausdorff space X , and we prove a uniqueness statementfor the proximity on F N ( X ) . In fact, these results can be framed in the more general contextof proximity Specker A -algebras.Let S be a Specker A -algebra. We call a set E of nonzero idempotents of S orthogonal if ek = e ≠ k in E . We say that s ∈ S is in orthogonal form provided s = ∑ ni = a i e i ,where the a i ∈ A are distinct and the e i are orthogonal. If in addition ⋁ ni = e i =
1, then we saythat s is in full orthogonal form . By [5, Lem. 2.1], each s ∈ S has a unique full orthogonaldecomposition.We say that s ∈ S is in decreasing form if s = a + ∑ ni = b i k i , where each b i > = k > k > ⋯ > k n >
0. There is a close connection between orthogonal and decreasing decompositions.To see this, write s = ∑ ni = a i e i in full orthogonal form, and suppose a < ⋯ < a n . Since the e i are orthogonal, e i + ⋯ + e n = e i ∨ ⋯ ∨ e n . Therefore, s = n ∑ i = a i e i = a ( e + ⋯ + e n ) + ( a − a )( e + ⋯ + e n ) + ⋯ + ( a n − a n − ) e n = a + n ∑ i = ( a i − a i − )( e i ∨ ⋯ ∨ e n ) = a + n ∑ i = ( a i − a i − ) k i , where k i = e i ∨ ⋯ ∨ e n . This writes s in decreasing form. Conversely, if s = a + ∑ ni = b i k i is indecreasing form, then we can recover the full orthogonal decomposition of s as follows: s = a + b ( k − k ) + ( b + b )( k − k ) + ⋯ + ( b + ⋯ + b n − )( k n − − k n ) + ( b + ⋯ + b n ) k n = a ( k ∧ ¬ k ) + ( a + b )( k ∧ ¬ k ) + ( a + b + b )( k ∧ ¬ k ) + ⋯+ ( a + n − ∑ i = b i ) ( k n − ∧ ¬ k n ) + ( a + n ∑ i = b n ) k n . Set e i = k i ∧ ¬ k i + and for i ≥
1, set a i = a + ∑ ij = b j . Then a < ⋯ < a n and s = ∑ ni = a i e i is infull orthogonal form. Since s has a unique full orthogonal decomposition, we see from theabove correspondence that s also has a unique decreasing decomposition. Proposition 5.1. If ( S, ≺ ) is a proximity Specker A -algebra, then ≺ restricts to a proximityon Id ( S ) .Proof. Axioms (DV1)–(DV4) are obvious. To verify (DV5), let e, k ∈ Id ( S ) with e ≺ k . By(P5), − k ≺ − e , so (P1) and (P6) yield 1 − k ≺ − e . But 1 − k = ¬ k and 1 − e = ¬ e . Thus, ¬ k ≺ ¬ e , and (DV5) is satisfied. To verify (DV6), let e, k ∈ Id ( S ) with e ≺ k . By (P9), thereis s ∈ S with e ≺ s ≺ k . Write s = ∑ ni = a i e i in orthogonal form with each a i ≠
0. Since the e i are orthogonal, se i = a i e i for each i . By Lemma 4.9(1), e i ≥
0, so as s ≥
0, we have se i ≥ a i e i ≥ i . This, by Lemma 4.9(5), yields a i >
0. Since a i e i ≤ s for each i ,by (P3), a i e i ≺ k . Then a i e i ≤ k by (P2), so a i ≤ e i ≺
1. Therefore, by (P7), a i e i ≺ a i . Thus, by (P4) andLemma 4.9(3), a i e i ≺ a i ∧ k = a i k . Then (P7) yields e i ≺ k . This implies l ∶ = e ∨ ⋯ ∨ e n ≺ k .Finally, if a = ∑ ni = a i , then s ≤ a ( e ∨ ⋯ ∨ e n ) = al , so e ≺ al by (P3). Then e ≤ al by (P2),so 1 ≤ a by Lemma 4.9(6). Therefore, as e ≺ e ≺ al ∧ = l by (P4) and Lemma 4.9(4). We thus have an idempotent l with e ≺ l ≺ k , so(DV6) is satisfied. To verify (DV7), let k be a nonzero idempotent in S . By (P11), there is1 < s ∈ S with s ≺ k . Write s = ∑ ni = a i e i as before. Then a e ≤ s , so a e ≺ k by (P3), and thesame argument as above yields e ≺ k . Therefore, (DV7) is satisfied. Thus, the restriction of ≺ to Id ( S ) is a proximity on Id ( S ) . (cid:3) We next use Proposition 5.1 to establish a representation theorem for an arbitrary prox-imity Specker A -algebra ( S, ≺ ) by showing that ( S, ≺ ) embeds into ( F N ( X ) , ≺ X ) for anappropriate choice of compact Hausdorff space X . Specifically, X is the space of ends of ( Id ( S ) , ≺ ) , which we next recall. Let B be a Boolean algebra and let ≺ be a proximity on B . For E ⊆ B , let ↡ E = { a ∈ B ∶ a ≺ e for some e ∈ E } , and define ↟ E dually. We call anideal I of B round if I = ↡ I . Dually, we call a filter F of B round if ↟ F = F . The dualcompact Hausdorff space of ( B, ≺ ) can be constructed either by means of maximal roundideals or maximal round filters of ( B, ≺ ) . In fact, there is a bijection between maximal roundfilters and maximal round ideals given by F ↦ { b ∶ ¬ b ∈ F } . De Vries preferred to work withmaximal round filters. We will instead work with maximal round ideals. Our choice is mo-tivated by their close connection to minimal prime ideals of a Specker A -algebra, which willbe discussed in Section 7. Because maximal round filters are called ends in the literature,we will use the same term for maximal round ideals.Let X be the set of ends of ( B, ≺ ) . For a ∈ B , let ζ ( a ) = { x ∈ X ∶ a ∈ x } . Define a topologyon X by letting ζ [ B ] = { ζ ( a ) ∶ a ∈ B } be a basis for the topology. The bijection above is e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 17 a homeomorphism between X and the space of maximal round filters, topologized by thebasis consisting of ξ ( a ) = { F ∶ a ∈ F } for a ∈ B . By [11, Ch. I.3], the space of maximal roundfilters is compact Hausdorff. Thus, X is compact Hausdorff.Adopting [11, Def. I.3.7], we call a subset T of a proximity Specker A -algebra ( S, ≺ ) dense if for each s, r ∈ S with s ≺ r , there is t ∈ T with s ≺ t ≺ r . Theorem 5.2.
Let ( S, ≺ ) be a proximity Specker A -algebra, and let X be the space of endsof ( Id ( S ) , ≺ ) . Then there is an ℓ -algebra embedding η ∶ S → F N ( X ) such that η [ S ] is densein F N ( X ) and s ≺ t iff η ( s ) ≺ X η ( t ) for all s, t ∈ S .Proof. Let B = Id ( S ) . By Proposition 5.1, the restriction of ≺ is a proximity on B . Let X be the space of ends of ( B, ≺ ) . Then ζ ∶ B → RO( X ) is an embedding [11, Ch. I.3]. We thushave a map σ ∶ B → Id ( F N ( X )) defined by σ ( e ) = χ ζ ( e ) . This is a Boolean homomorphismsince σ ( e ∧ k ) = χ ζ ( e ∧ k ) = χ ζ ( e )∩ ζ ( k ) = χ ζ ( e ) ∧ χ ζ ( k ) = σ ( e ) ∧ σ ( k ) and σ ( ¬ e ) = χ ζ (¬ e ) = χ ¬ ζ ( e ) = ¬ χ ζ ( e ) = ¬ σ ( e ) . Since S is a Specker A -algebra, by [5, Sec. 2] there is a uniquely determined A -algebra homo-morphism η ∶ S → F N ( X ) extending σ . By [5, Cor. 5.3], η is an ℓ -algebra homomorphism.To see that η is 1-1, let s ≠
0. As noted in the beginning of the section, we may write s indecreasing form s = a + n ∑ i = ( a i − a i − ) e i , with a < ⋯ < a n in A and 1 = e > e > ⋯ > e n > B . Then η ( s ) = a + ∑ ni = ( a i − a i − ) χ ζ ( e i ) .Therefore, η ( s )( x ) = a i provided x ∈ ζ ( e i ) − ζ ( e i + ) . If s = a , then as s ≠
0, we have a ≠
0, so η ( s ) ≠
0. Otherwise n >
0, so e ≠
0. Thus, if x ∈ ζ ( e ) , then η ( s )( x ) ≥ a and if x ∈ X − ζ ( e ) ,then η ( s )( x ) = a . Since a < a , we see that η ( s ) ≠ η [ S ] is dense in F N ( X ) . Let f, g ∈ F N ( X ) with f ≺ X g . Suppose a < ⋯ < a n contain all the values of f and g . Set U i = f − ( ↑ a i ) and V i = f − ( ↑ a i ) . From f ≺ X g it follows that U i ≺ V i in RO( X ) . By [11, Thm. I.3.9], ζ [ B ] is dense in RO( X ) .Therefore, for each i there is e i ∈ B with U i ≺ ζ ( e i ) ≺ V i , and as in the proof of Theorem 3.4(5),we may assume that the e i are decreasing. Set s = a + ∑ ni = ( a i − a i − ) e i and h = η ( s ) . Then h = a + ∑ ni = ( a i − a i − ) χ ζ ( e i ) . Also, by Lemma 2.2(1) and Remark 2.6, f = a + ∑ ni = ( a i − a i − ) χ U i and g = a + ∑ ni = ( a i − a i − ) χ V i . Since U i ≺ ζ ( e i ) ≺ V i for each i , we see that f ≺ X h ≺ X g .Thus, η [ B ] is dense in F N ( X ) .It remains to show that s ≺ t in S iff η ( s ) ≺ X η ( t ) . For this, we need the following claim. Claim 5.3.
Let s ∈ S and set f = η ( s ) . For each a ∈ A , we have f − ( ↑ a ) ∈ ζ [ B ] . Proof of Claim:
Write s = a + ∑ ni = ( a i − a i − ) e i in decreasing form. Then f = a + ∑ ni = ( a i − a i − ) χ ζ ( e i ) . Let a ∈ A . Then f − ( ↑ a ) is either empty or equal to f − ( ↑ a i ) for some i . As f − ( ↑ a i ) = ζ ( e i ) ∈ ζ [ B ] , the claim is proved. (cid:3) Now, let s, t ∈ S and set f = η ( s ) and g = η ( t ) . Suppose a < ⋯ < a n contain all the valuesof f and g . Set U i = f − ( ↑ a i ) and V i = g − ( ↑ a i ) . By Claim 5.3, U i , V i ∈ ζ [ B ] . Write U i = ζ ( e i ) and V i = ζ ( k i ) with e i , k i ∈ B .First suppose that s ≺ t . Then [( s − a i ) ∨ ] ∧ ( a i − a i − ) ≺ [( t − a i ) ∨ ] ∧ ( a i − a i − ) . We have η ([( s − a i ) ∨ ] ∧ ( a i − a i − )) = [( f − a i ) ∨ ] ∧ ( a i − a i − ) and η ([( t − a i ) ∨ ] ∧ ( a i − a i − )) = [( g − a i ) ∨ ] ∧ ( a i − a i − ) . It is easy to see that [( f − a i ) ∨ ] ∧ ( a i − a i − ) = ( a i − a i − ) χ U i and [( g − a i ) ∨ ] ∧ ( a i − a i − ) = ( a i − a i − ) χ V i . Because η is 1-1, [( s − a i ) ∨ ] ∧ ( a i − a i − ) = ( a i − a i − ) e i and [( t − a i ) ∨ ] ∧ ( a i − a i − ) = ( a i − a i − ) k i . Since the first is proximal to the second, we get e i ≺ k i , and as ζ preserves proximity [11,Ch. I.3], U i ≺ V i . Because this is true for each i , we conclude that f ≺ X g .Conversely, suppose that f ≺ X g . Then U i ≺ V i for each i . Since ζ reflects proximity [11,Ch. I.3], e i ≺ k i for each i . By Lemma 2.2(1), Remark 2.6, and the injectivity of η , we maywrite s = a + ∑ ni = ( a i − a i − ) e i and t = a + ∑ ni = ( a i − a i − ) k i . From this we conclude that s ≺ t . (cid:3) Corollary 5.4. If S is a proximity Specker A -algebra, then any two proximities on S thatrestrict to the same proximity on Id ( S ) are equal.Proof. Let ≺ and ≺ ′ be two proximities on S that restrict to the same proximity on Id ( S ) .Let X be the space of ends of ( Id ( S ) , ≺ ) . By Theorem 5.2, there is an ℓ -algebra embedding η ∶ S → F N ( X ) such that s ≺ t iff η ( s ) ≺ X η ( t ) for all s, t ∈ S . Another application ofTheorem 5.2 to ( S, ≺ ′ ) shows that s ≺ ′ t iff η ( s ) ≺ X η ( t ) for all s, t ∈ S . Thus, for all s, t ∈ S ,we have s ≺ t iff s ≺ ′ t , and hence ≺ and ≺ ′ are equal. (cid:3) Corollary 5.5. If S is a Specker A -algebra, then each proximity on Id ( S ) extends to aunique proximity on S . Consequently, there is a 1-1 correspondence between proximities on S and Id ( S ) .Proof. Let X be the space of ends of ( Id ( S ) , ≺ ) . As we saw in the proof of Theorem 5.2,there is an ℓ -algebra embedding η ∶ S → F N ( X ) . For s, t ∈ S , define s ≺ t iff η ( s ) ≺ X η ( t ) .By Theorem 4.10, ≺ X is a proximity on F N ( X ) . Therefore, ≺ satisfies (P1) through (P8).For (P9), let s, t ∈ S with s ≺ t . Set f = η ( s ) and g = η ( t ) . Then f ≺ X g . Since η [ S ] is densein F N ( X ) , there is r ∈ S such that f ≺ X η ( r ) ≺ X g . This implies s ≺ r ≺ t , as required. For(P10), let 0 < s . Write s = ∑ ni = a i e i in orthogonal form. Since s >
0, some a i >
0. As e i ≠ k ∈ Id ( S ) with 0 ≠ k ≺ e i . Because S is torsion-free, 0 < a i k ∈ S and a i k ≺ a i e i ≤ s .Thus, 0 < a i k ≺ s . Consequently, ≺ is a proximity on S , and it follows from Corollary 5.4that it is the unique proximity extending ≺ on Id ( S ) . (cid:3) Corollary 5.6.
Let ( S, ≺ ) be a proximity f -algebra over A . Then there is an ℓ -algebraisomorphism between S and F N ( X ) , for some compact Hausdorff space X , that preservesand reflects the proximity iff ( S, ≺ ) is a proximity Baer Specker A -algebra.Proof. That ( F N ( X ) , ≺ X ) is a proximity Baer Specker A -algebra follows from Theorem 4.10.Conversely, suppose that ( S, ≺ ) is a proximity Baer Specker A -algebra. By Theorem 5.2,there is an ℓ -algebra embedding η ∶ S → F N ( X ) such that s ≺ t iff η ( s ) ≺ X η ( t ) for all s, t ∈ S .Since S is a Baer Specker A -algebra, Id ( S ) is a complete Boolean algebra [5, Thm. 4.3].Therefore, ( Id ( S ) , ≺ ) is a de Vries algebra, hence Id ( S ) is isomorphic to RO( X ) [11, Ch. I.4].Thus, each f ∈ F N ( X ) can be written in decreasing form f = a + ∑ ni = ( a i − a i − ) χ ζ ( e i ) , and e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 19 setting s = a + ∑ ni = ( a i − a i − ) e i yields s ∈ S such that η ( s ) = f . Consequently, η is an ℓ -algebra isomorphism such that s ≺ t iff η ( s ) ≺ X η ( t ) for all s, t ∈ S . (cid:3) Proximity morphisms and continuous maps
In this section we show that a continuous map between compact Hausdorff spaces givesrise to what we term a proximity morphism between the corresponding proximity Specker A -algebras. We also characterize proximity morphisms between proximity Specker A -algebras ( S, ≺ ) and ( T, ≺ ) by means of de Vries morphisms from ( Id ( S ) , ≺ ) to ( Id ( T ) , ≺ ) , and bymeans of continuous maps between the corresponding dual compact Hausdorff spaces.Let ϕ ∶ X → Y be a continuous map between compact Hausdorff spaces. By de Vries duality[11], ̂ ϕ ∶ RO( Y ) → RO( X ) , given by ̂ ϕ ( U ) = Int ( Cl ( ϕ − ( U ))) , is a de Vries morphism.Define ϕ + ∶ F ( Y ) → F ( X ) by ϕ + ( f ) = f ○ ϕ . X ϕ / / f ○ ϕ ❅❅❅❅❅❅❅❅ Y f (cid:15) (cid:15) A It is straightforward to see that ϕ + is an A -algebra homomorphism. Next define ϕ ∗ ∶ F N ( Y ) → F N ( X ) by ϕ ∗ ( f ) = ( ϕ + ( f )) = ( f ○ ϕ ) . By the definition of normalization, ϕ ∗ ( f ) − ( ↑ a ) = (( f ○ ϕ ) ) − ( ↑ a ) = Int ( Cl ( ϕ − ( f − ( ↑ a )))) = ̂ ϕ ( f − ( ↑ a )) . In order to see what properties ϕ ∗ satisfies, we require two lemmas. The following is theproximity Specker analogue of a well-known fact about proximity Boolean algebras. Lemma 6.1.
Let ( S, ≺ ) be a proximity Specker A -algebra. Then each s ∈ S is the least upperbound of { t ∈ S ∶ t ≺ s } .Proof. We first show that if E is a subset of Id ( S ) and e ∈ Id ( S ) is the join of E in Id ( S ) ,then e is the join of E in S . For, since the partial order on S restricts to the usual Booleanorder on Id ( S ) (Lemma 4.9(2)), e ∈ S is an upper bound of E in S . Suppose s ∈ S is anotherupper bound of E . As s ∧ e is also an upper bound of E in S , without loss of generalitywe may assume that s ≤ e . Then, for each k ∈ E , we have k ≤ s ≤ e . Consider the map η ∶ S → F N ( X ) of Theorem 5.2. We have η ( k ) ≤ η ( s ) ≤ η ( e ) and η ( k ) , η ( e ) are idempotentsin F N ( X ) . By Lemma 4.8, there exist regular open sets U and V such that η ( k ) = χ U and η ( e ) = χ V . Therefore, η ( s ) must be the characteristic function of some subset between U and V . By normality and Remark 2.9(2), it must be the characteristic function of a regularopen set, and so η ( s ) is an idempotent in F N ( X ) . As η is an ℓ -algebra embedding, s is anidempotent in S . Thus, s = e , and so e is the join of E in S .Next let s ∈ S and write s = a + ∑ ni = b i e i in decreasing form with b , . . . , b n >
0. It is clearthat s is an upper bound of { t ∈ S ∶ t ≺ s } . If E i is the set of idempotents k i with k i ≺ e i ,then by the argument above, e i is the join of E i in S . By [15, §
2, Thms. 2.3 and 2.6], s = a + n ∑ i = b i e i = a + n ∑ i = b i (⋁ E i ) = ⋁{ a + n ∑ i = b i k i ∶ k i ∈ E i } . Since a + ∑ ni = b i k i ≺ s , it follows that s = ⋁{ t ∈ S ∶ t ≺ s } . (cid:3) Lemma 6.2.
Let ϕ ∶ X → Y be continuous, f ∈ F N ( Y ) , and a < ⋯ < a n be the values of f .Write f = a + ∑ ni = ( a i − a i − ) χ U i in decreasing form, where U i = f − ( ↑ a i ) are regular open.Then ϕ ∗ ( f ) = a + ∑ ni = ( a i − a i − ) ϕ ∗ ( χ U i ) . Proof.
By Lemma 2.2(1), f ( ϕ ( x )) = a i iff x ∈ ϕ − ( U i ) − ϕ − ( U i + ) for each x ∈ X . Therefore,in F ( X ) we have ϕ + ( f ) = f ○ ϕ = a + ∑ ni = ( a i − a i − ) χ ϕ − ( U i ) . Thus, by Remark 2.9(2), wehave ϕ ∗ ( f ) = ( f ○ ϕ ) = a + ∑ ni = ( a i − a i − ) χ ̂ ϕ ( U i ) . Since ϕ ∗ ( χ U i ) = ̂ ϕ ( U i ) , we conclude that ϕ ∗ ( f ) = a + ∑ ni = ( a i − a i − ) ϕ ∗ ( χ U i ) . (cid:3) The next proposition will motivate the definition of a proximity morphism.
Proposition 6.3.
Let ϕ ∶ X → Y be a continuous map between compact Hausdorff spaces.Then ϕ ∗ ∶ F N ( Y ) → F N ( X ) satisfies the following properties for each f, g ∈ F N ( Y ) and a ∈ A . (1) ϕ ∗ ( ) = . (2) ϕ ∗ ( f ∧ g ) = ϕ ∗ ( f ) ∧ ϕ ∗ ( g ) . (3) f ≺ Y g implies − ϕ ∗ ( − f ) ≺ X ϕ ∗ ( g ) . (4) ϕ ∗ ( f ) is the least upper bound of { ϕ ∗ ( g ) ∶ g ≺ Y f } . (5) ϕ ∗ ( f + a ) = ϕ ∗ ( f ) + a . (6) If a is positive, then ϕ ∗ ( af ) = aϕ ∗ ( f ) . (7) ϕ ∗ ( f ∨ a ) = ϕ ∗ ( f ) ∨ a .Proof. (1) ϕ ∗ ( ) = ( ϕ + ( )) = = ϕ ∗ ( a ) = a for each a ∈ A .(2) Let f, g ∈ F N ( Y ) . Recalling that meet in F N ( Y ) is pointwise, we see that ϕ ∗ ( f ∧ g ) = (( f ∧ g ) ○ ϕ ) = (( f ○ ϕ ) ∧ ( g ○ ϕ )) and ϕ ∗ ( f ) ∧ ϕ ∗ ( g ) = ( f ○ ϕ ) ∧ ( g ○ ϕ ) . Therefore, itsuffices to prove that ( h ∧ k ) = h ∧ k for each h, k ∈ F ( Y ) . Let a ∈ A . Then ( h ∧ k ) − ( ↑ a ) = ( h ) − ( ↑ a ) ∩ ( k ) − ( ↑ a ) = Int ( Cl ( h − ( ↑ a ))) ∩ Int ( Cl ( k − ( ↑ a ))) = Int ( Cl ( h − ( ↑ a ) ∩ k − ( ↑ a ))) = Int ( Cl ( h ∧ k ) − ( ↑ a )) = (( h ∧ k ) ) − ( ↑ a ) . Thus, ϕ ∗ ( f ∧ g ) = ϕ ∗ ( f ) ∧ ϕ ∗ ( g ) .(3) Let f, g ∈ F N ( Y ) with f ≺ Y g . Suppose the values of f and g are among a < ⋯ < a n .Then f − ( ↑ a i ) ≺ g − ( ↑ a i ) for each i . Therefore, since ̂ ϕ is a de Vries morphism, by (M3), ¬ ̂ ϕ ( ¬ f − ( ↑ a i )) ≺ ̂ ϕ ( g − ( ↑ a i )) for each i . By Lemma 2.2(1) and Remark 2.6, write f = a + ∑ ni = ( a i − a i − ) χ U i and g = a + ∑ ni = ( a i − a i − ) χ V i in decreasing form, where U i = f − ( ↑ a i ) and V i = g − ( ↑ a i ) are regular open and the sums are pointwise. Set b i = a i − a i − and b = ∑ ni = b i .We have − f = − a + n ∑ i = − b i χ U i = − a + b − b + n ∑ i = − b i χ U i = − a − b + n ∑ i = b i ( − χ U i ) = − ( a + b ) + n ∑ i = b i χ ¬ U i . e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 21 This writes − f in decreasing form since ¬ U n ⊇ ⋯ ⊇ ¬ U . Therefore, by Lemma 6.2, ϕ ∗ ( − f ) = − ( a + b ) + ∑ ni = b i ϕ ∗ ( χ ¬ U i ) . Thus, − ϕ ∗ ( − f ) = ( a + b ) + n ∑ i = − b i ϕ ∗ ( χ ¬ U i ) = ( a + b ) − b + b + n ∑ i = − b i ϕ ∗ ( χ ¬ U i ) = a + n ∑ i = b i ( − ϕ ∗ ( χ ¬ U i )) = a + n ∑ i = b i ( ¬ ϕ ∗ ( χ ¬ U i )) . Since ¬ ̂ ϕ ( ¬ U i ) ≺ ̂ ϕ ( V i ) , we have ¬ ϕ ∗ ( χ ¬ U i ) ≺ X ϕ ∗ ( χ V i ) , yielding − ϕ ∗ ( − f ) ≺ X ϕ ∗ ( g ) , asrequired.(4) Let f ∈ F N ( Y ) . We claim that ϕ ∗ ( f ) − ( ↑ a ) = ⋁{ ϕ ∗ ( g ) − ( ↑ a ) ∶ g ≺ Y f } . From theconnection between ϕ ∗ and ̂ ϕ , this amounts to proving ̂ ϕ ( f − ( ↑ a )) = ⋁{ ̂ ϕ ( g − ( ↑ a )) ∶ g ≺ Y f } .That ̂ ϕ ( f − ( ↑ a )) is an upper bound of { ̂ ϕ ( g − ( ↑ a )) ∶ g ≺ Y f } is clear. Conversely, firstsuppose that 0 ≤ f . Then 0 ≺ Y f . If a ≤
0, then ̂ ϕ ( f − ( ↑ a )) = X , and as 0 − ( ↑ a ) = X , theclaim is true in this case. Now suppose that a ≥
0. Let U ∈ RO( X ) with U ≺ f − ( ↑ a ) . Then h ∶ = aχ U satisfies h ≺ Y f and h − ( ↑ a ) = U . Because ̂ ϕ ( f ) − ( ↑ a ) = ⋁{ ̂ ϕ ( U ) ∶ U ≺ f − ( ↑ a )} ,the claim holds for f ≥
0. For an arbitrary f , since f is finitely valued, there is b ∈ A with0 ≤ f + b . By Remark 4.7, f + b is pointwise, so the case just done gives f − ( ↑ a ) = ( f + b ) − ( ↑ ( a + b )) = ⋁{ h − ( ↑ ( a + b )) ∶ h ≺ Y f + b } = ⋁{( h − b ) − ( ↑ a ) ∶ h ≺ Y f + b } = ⋁{ g − ( ↑ a ) ∶ g ≺ Y f } . The last equality follows since b ≺ b for all b ∈ A (Remark 4.3).For proving (5), (6), and (7) we use Remark 4.7 which gives that addition and join by ascalar and multiplication by a positive scalar are pointwise.(5) Let f ∈ F N ( Y ) and a ∈ A . Then ϕ ∗ ( f + a ) = (( f + a ) ○ ϕ ) = ( f ○ ϕ + a ) = ( f ○ ϕ ) + a = ϕ ∗ ( f ) + a. (6) Let f ∈ F N ( Y ) and a ∈ A be positive. Then ϕ ∗ ( af ) = (( af ) ○ ϕ ) = ( a ( f ○ ϕ )) = a ( f ○ ϕ ) = aϕ ∗ ( f ) . (7) Let f ∈ F N ( Y ) and a ∈ A . Then ϕ ∗ ( f ∨ a ) = (( f ∨ a ) ○ ϕ ) = (( f ○ ϕ ) ∨ a ) = ( f ○ ϕ ) ∨ a = ϕ ∗ ( f ) ∨ a. (cid:3) Proposition 6.3 motivates the following definition.
Definition 6.4.
Let ( S, ≺ ) and ( T, ≺ ) be proximity f -algebras over A . A map α ∶ S → T isa proximity morphism provided for each s, t ∈ S and a ∈ A , we have:(1) α ( ) = α ( s ∧ t ) = α ( s ) ∧ α ( t ) .(3) s ≺ t implies − α ( − s ) ≺ α ( t ) .(4) α ( s ) is the least upper bound of { α ( t ) ∶ t ≺ s } .(5) α ( s + a ) = α ( s ) + a . (6) If a is positive, then α ( as ) = aα ( s ) .(7) α ( s ∨ a ) = α ( s ) ∨ a . Remark 6.5. (1) It follows from (1) and (5) that α ( a ) = a for each a ∈ A . In particular, α ( ) =
1, so if T is nontrivial, then 0 ≠ T , and hence α is nonzero.(2) It follows from (2) that α is order preserving. Also, for s ∈ S and a ∈ A , we have α ( s ∧ a ) = α ( s ) ∧ α ( a ) = α ( s ) ∧ a . Proposition 6.6.
Let ( S, ≺ ) and ( T, ≺ ) be proximity Specker A -algebras and let α ∶ S → T be a proximity morphism. Then α ( Id ( S )) ⊆ Id ( T ) and α ∣ Id ( S ) ∶ Id ( S ) → Id ( T ) is a de Vriesmorphism.Proof. If T is trivial, there is nothing to verify, so assume that T is nontrivial. Let X be thespace of ends of Id ( T ) , and let η T ∶ T → F N ( X ) be the ℓ -algebra embedding of Theorem 5.2.Suppose e ∈ Id ( S ) . Because α is order preserving with α ( ) = α ( ) =
1, we see that0 ≤ α ( e ) ≤
1. Take x ∈ X and set a = η T ( α ( e ))( x ) . Then 0 ≤ a ≤
1. By Lemma 4.9(3), a ∧ e = ae , so a ∧ α ( e ) = α ( a ∧ e ) = α ( ae ) = aα ( e ) . Therefore, a ∧ η T ( α ( e )) = aη T ( α ( e )) . Evaluating at x yields a = a . Thus, as A is a domain, a = η T ( α ( e ))( x ) ∈ { , } . This shows that η T ( α ( e )) ∈ Id ( F N ( X )) . Since η T is an A -algebrahomomorphism, this implies that α ( e ) ∈ Id ( T ) . It follows that α ∣ Id ( S ) ∶ Id ( S ) → Id ( T ) is welldefined. It is also clear that α ∣ Id ( S ) satisfies (M1) and (M2). Suppose that e, k ∈ Id ( S ) with e ≺ k . Then ¬ α ( ¬ e ) = − α ( − e ) = − [ + α ( − e )] = − α ( − e ) . Because − α ( − e ) ≺ α ( k ) , weconclude that ¬ α ( ¬ e ) ≺ α ( k ) . Therefore, α ∣ Id ( S ) satisfies (M3). Let k ∈ Id ( S ) . Then α ( k ) is the least upper bound of { α ( s ) ∶ s ∈ S, s ≺ k } . Suppose that 0 ≤ s ≺ k . Write s = ∑ ni = a i e i in orthogonal form with each a i ≠
0. The proof of Proposition 5.1 then yields 0 < a i ≤ e i ≺ k for each i . Consequently, s ≤ e ∨ ⋯ ∨ e n ≺ k . Since α ( s ) ≤ α ( e ∨ ⋯ ∨ e n ) , we see that α ( k ) = ⋁{ α ( e ) ∶ e ∈ Id ( S ) , e ≺ k } . Thus, α ∣ Id ( S ) satisfies (M4). (cid:3) The next theorem, which is the main result of this section, characterizes proximity mor-phisms.
Theorem 6.7.
Suppose that ( S, ≺ ) and ( T, ≺ ) are proximity Specker A -algebras and α ∶ S → T is a map. Let X be the space of ends of ( Id ( T ) , ≺ ) and Y be the space of ends of ( Id ( S ) , ≺ ) .Then the following conditions are equivalent. (1) α is a proximity morphism. (2) The restriction α ∣ Id ( S ) ∶ Id ( S ) → Id ( T ) is a well-defined de Vries morphism, and if s = a + ∑ ni = b i e i is in decreasing form with b , . . . , b n > , then α ( s ) = a + ∑ ni = b i α ( e i ) . (3) There exists a continuous map ϕ ∶ X → Y such that the following diagram commutes. S α / / η S (cid:15) (cid:15) T η T (cid:15) (cid:15) F N ( Y ) ϕ ∗ / / F N ( X ) e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 23 Proof. (1) ⇒ (2): By Proposition 6.6, α ∣ Id ( S ) is a well-defined de Vries morphism. Let a i = a + ∑ ij = b j . Then a < ⋯ < a n and s − a i − = ( a + b e + ⋯ + b n e n ) − a i − = ( b i e i + ⋯ + b n e n ) − ( a i − − a − b e − ⋯ − b i − e i − ) = ( b i e i + ⋯ + b n e n ) − ( a + b + ⋯ + b i − − a − b e − ⋯ − b i − e i − ) = ( b i e i + ⋯ + b n e n ) − ( b ( − e ) + ⋯ + b i − ( − e i − )) = ( b i e i + ⋯ + b n e n ) − ( b ¬ e + ⋯ + b i − ¬ e i − ) . This exhibits s − a i − as the difference of two elements greater than or equal to 0, and weclaim their meet is 0. To see this, as the e i are decreasing, we have0 ≤ ( b i e i + ⋯ + b n e n ) ∧ ( b ¬ e + ⋯ + b i − ¬ e i − ) ≤ ( b i e i + ⋯ + b n e i ) ∧ ( b ¬ e i − + ⋯ + b i − ¬ e i − ) = ( b i + ⋯ + b n ) e i ∧ ( b + ⋯ + b i − ) ¬ e i − , which is 0 by using the f -ring identity twice along with e i ∧ ¬ e i − =
0. Therefore, by [8,Ch. XIII, Lem. 4], ( s − a i − ) ∨ = b i e i + ⋯ + b n e n . As was shown in the proof of Theorem 5.2, [( s − a i − ) ∨ ] ∧ b i = b i e i . Claim 6.8.
Let s ∈ S and let a, b ∈ A with a < b . Then ( s ∧ b ) − ( s ∧ a ) = (( s − a ) ∨ ) ∧ ( b − a ) . Proof.
We have ( s ∧ b ) − ( s ∧ a ) = (( s ∧ b ) − a ) − (( s ∧ a ) − a ) = (( s − a ) ∧ ( b − a )) − (( s − a ) ∧ ( a − a )) = (( s − a ) ∧ ( b − a )) − (( s − a ) ∧ ) . Write t = ( s − a ) ∧ ( b − a ) . Then ( s ∧ b ) − ( s ∧ a ) = t − ( t ∧ ) because 0 ≤ b − a . By [8, Ch. XIII,Thm. 7], t = ( t ∨ ) + ( t ∧ ) . Therefore, t − ( t ∧ ) = t ∨
0. Thus, ( s ∧ b ) − ( s ∧ a ) = t ∨ = (( s − a ) ∧ ( b − a )) ∨ = (( s − a ) ∨ ) ∧ ( b − a ) since S is a distributive lattice ([8, Ch. XIII,Thm. 4]). (cid:3) As α ([( s − a i − ) ∨ ] ∧ b i ) = [( α ( s ) − a i − ) ∨ ] ∧ b i and α ( b i e i ) = b i α ( e i ) , we obtain b i α ( e i ) = [( α ( s ) − a i − ) ∨ ] ∧ b i . Since b i = a i − a i − , by Claim 6.8, [( α ( s ) − a i − ) ∨ ] ∧ b i = ( α ( s ) ∧ a i ) − ( α ( s ) ∧ a i − ) . As a ≤ s ≤ a n , we have a ≤ α ( s ) ≤ a n . Consequently, α ( s ) − a = ( α ( s ) ∧ a n ) − ( α ( s ) ∧ a ) = n ∑ i = (( α ( s ) ∧ a i ) − ( α ( s ) ∧ a i − )) = n ∑ i = b i α ( e i ) . Adding a to both sides of the equation finishes the proof.(2) ⇒ (3): Let ϕ ∶ X → Y be the dual of the de Vries morphism α ∣ Id ( S ) . First let e ∈ Id ( S ) .By [11, Ch. I.6], the following diagram commutes.Id ( S ) α ∣ Id ( S ) / / ζ S (cid:15) (cid:15) Id ( T ) ζ T (cid:15) (cid:15) RO( Y ) ̂ ϕ / / RO( X ) We have ϕ ∗ ( χ U ) = χ ̂ ϕ ( U ) . This implies ϕ ∗ ( η S ( e )) = η T ( α ( e )) for each e ∈ Id ( S ) . Now,let s ∈ S be arbitrary. Write s = a + ∑ ni = b i e i in decreasing form. Then, since η S , η T are A -algebra homomorphisms, we have η T ( α ( s )) = η T ( a + n ∑ i = b i α ( e i )) = a + n ∑ i = b i η T ( α ( e i )) = a + n ∑ i = b i ϕ ∗ ( η S ( e i )) = ϕ ∗ ( η S ( s )) . Here the first equality follows from (2) and the last equality from Lemma 6.2. This yieldscommutativity of the diagram in the statement of (3).(3) ⇒ (1): By Proposition 6.3, ϕ ∗ is a proximity morphism. Because η S and η T are ℓ -algebraembeddings which preserve and reflect proximity, all the proximity morphism axioms butthe fourth are clearly true for α . To prove axiom (4), let s ∈ S . Since ϕ ∗ is a proximitymorphism, ϕ ∗ ( η S ( s )) = ⋁{ ϕ ∗ ( f ) ∶ f ≺ Y η S ( s )} . Because η S [ S ] is dense in F N ( Y ) and η S reflects proximity, we have ϕ ∗ ( η S ( s )) = ⋁{ ϕ ∗ ( η S ( u )) ∶ η S ( u ) ≺ Y η S ( s )} = ⋁{ ϕ ∗ ( η S ( u )) ∶ u ≺ s } . By (3) this yields η T ( α ( s )) = ⋁{ η T ( α ( u )) ∶ u ≺ s } . Now, since η T is order reflecting, α ( s ) is an upper bound of { α ( u ) ∶ u ≺ s } . To see it is the least upper bound, let t ∈ T satisfy α ( u ) ≤ t for each u ≺ s . Then η T ( α ( u )) ≤ η T ( t ) . Therefore, η T ( α ( s )) ≤ η T ( t ) . Thus, α ( s ) ≤ t . Consequently, α ( s ) = ⋁{ α ( u ) ∶ u ≺ s } . This finishes the proof that α is a proximitymorphism. (cid:3) Corollary 6.9.
Let ( S, ≺ ) and ( T, ≺ ) be proximity Specker A -algebras. If σ ∶ Id ( S ) → Id ( T ) is a de Vries morphism, then there is a unique proximity morphism α ∶ S → T with α ∣ Id ( S ) = σ .Proof. As we pointed out in the beginning of Section 5, each s ∈ S has a unique decreasingdecomposition s = a + ∑ ni = b i e i , where b , . . . , b n > > e > ⋯ > e n >
0. Define α ∶ S → T by α ( s ) = a + ∑ ni = b i σ ( e i ) . Since the decreasing decomposition is unique, α is well defined,and it follows from the definition that α ∣ Id ( S ) = σ . If α ′ is another proximity morphismextending σ , then Theorem 6.7 implies that α ′ ( s ) = a + ∑ ni = b i σ ( e i ) = α ( s ) , and thus α isthe unique proximity morphism extending σ . (cid:3) Corollary 6.10.
With the notation of Theorem 6.7 and with S and T Baer, there are 1-1correspondences between the proximity morphisms S → T , the de Vries morphisms Id ( S ) → Id ( T ) , and the continuous maps X → Y .Proof. By Corollary 6.9, α ↦ α ∣ Id ( S ) is a 1-1 correspondence between the proximity mor-phisms S → T and the de Vries morphisms Id ( S ) → Id ( T ) . By [11, Ch. I.6], there is a 1-1correspondence between the de Vries morphisms Id ( S ) → Id ( T ) and the continuous maps X → Y . (cid:3) The space of ends of a proximity Specker algebra
As we have seen, for a proximity Specker A -algebra ( S, ≺ ) , the space of ends of ( Id ( S ) , ≺ ) is useful for representing S as a ring of normal functions. In this section we pursue thisfurther by developing the notion of ends for a proximity Specker A -algebra ( S, ≺ ) .For a continuous map ϕ ∶ X → Y between compact Hausdorff spaces, we recall fromthe previous section that the de Vries morphism ̂ ϕ ∶ RO( Y ) → RO( X ) , given by ̂ ϕ ( U ) = e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 25 Int ( Cl ( ϕ − ( U ))) , and the proximity morphism ϕ ∗ ∶ F N ( Y ) → F N ( X ) , given by ϕ ∗ ( f ) = ( f ○ ϕ ) , are connected by the formula ϕ ∗ ( f ) − ( ↑ a ) = ̂ ϕ ( f − ( ↑ a )) . We also recall that since ̂ ϕ is a de Vries morphism, it has the property that whenever e i ≺ k i for 1 ≤ i ≤ n , then ̂ ϕ ( e ∨ ⋯ ∨ e n ) ≺ ̂ ϕ ( k ) ∨ ⋯ ∨ ̂ ϕ ( k n ) [3, Lem. 2.2]. Proposition 7.1. (1)
Let ϕ ∶ X → Y be a continuous map between compact Hausdorff spaces and let f, g, h, k ∈ F N ( Y ) . If f ≺ Y h and g ≺ Y k , then ϕ ∗ ( f + g ) ≺ X ϕ ∗ ( h ) + ϕ ∗ ( k ) . (2) Let α ∶ S → T be a proximity morphism between proximity Specker A -algebras and let s, t, u, v ∈ S . If s ≺ u and t ≺ v , then α ( s + t ) ≺ α ( u ) + α ( v ) .Proof. (1) It is sufficient to prove that ϕ ∗ ( f + g ) − ( ↑ a ) ≺ ( ϕ ∗ ( h ) + ϕ ∗ ( k )) − ( ↑ a ) for each a ∈ A .By Lemma 4.5(1), ϕ ∗ ( f + g ) − ( ↑ a ) = ̂ ϕ (( f + g ) − ( ↑ a )) = ̂ ϕ ( ⋁ b + c ≥ a f − ( ↑ b ) ∩ g − ( ↑ c )) . On the other hand, from the same lemma, we have ( ϕ ∗ ( h ) + ϕ ∗ ( k )) − ( ↑ a ) = ⋁ b + c ≥ a ϕ ∗ ( h ) − ( ↑ b ) ∩ ϕ ∗ ( k ) − ( ↑ c ) = ⋁ b + c ≥ a ̂ ϕ ( h − )( ↑ b ) ∩ ̂ ϕ ( k − )( ↑ c ) . Because f ≺ Y h and g ≺ Y k , for each b, c ∈ A , we have f − ( ↑ b ) ≺ h − ( ↑ b ) and g − ( ↑ c ) ≺ k − ( ↑ c ) .Therefore, f − ( ↑ b ) ∩ g − ( ↑ c ) ≺ h − ( ↑ b ) ∩ k − ( ↑ c ) . Thus, since ̂ ϕ is a de Vries morphism andthe joins in question are finite joins, ̂ ϕ ( ⋁ b + c ≥ a f − ( ↑ b ) ∩ g − ( ↑ c ))) ≺ ⋁ b + c ≥ a ̂ ϕ ( h − ( ↑ b ) ∩ k − ( ↑ c )) = ⋁ b + c ≥ a ̂ ϕ ( h − )( ↑ b ) ∩ ̂ ϕ ( k − ( ↑ c )) . Consequently, ϕ ∗ ( f + g ) ≺ X ϕ ∗ ( h ) + ϕ ∗ ( k ) .(2) Consider the maps η S ∶ S → F N ( Y ) and η T ∶ T → F N ( X ) provided by Theorem 5.2.By Theorem 6.7, ϕ ∗ ○ η S = η T ○ α . Since η S preserves proximity, η S ( s ) ≺ Y η S ( u ) and η S ( t ) ≺ Y η S ( v ) . As η S , η T preserve addition, (1) yields η T ( α ( s + t )) = ϕ ∗ ( η S ( s + t )) = ϕ ∗ ( η S ( s ) + η S ( t )) ≺ X ϕ ∗ ( η S ( u )) + ϕ ∗ ( η S ( v )) = η T ( α ( u )) + η T ( α ( v )) = η T ( α ( u ) + α ( v )) . Thus, since η T reflects proximity, α ( s + t ) ≺ α ( u ) + α ( v ) . (cid:3) We recall that if S is an ℓ -ring, then the absolute value of s ∈ S is defined as ∣ s ∣ = s ∨ ( − s ) .For each s, t ∈ S , we have ∣ s + t ∣ ≤ ∣ s ∣ + ∣ t ∣ and ∣ st ∣ ≤ ∣ s ∣ ⋅ ∣ t ∣ ; moreover, if S is an f -ring, then ∣ st ∣ = ∣ s ∣ ⋅ ∣ t ∣ (see, e.g., [8, Ch. XVII]). We also recall that an ideal I of S is an ℓ -ideal provided ∣ s ∣ ≤ ∣ t ∣ and t ∈ I imply s ∈ I for all s, t ∈ S . An ℓ -ideal I is proper if I ≠ S .Let ( S, ≺ ) be a proximity Specker A -algebra. For A ⊆ S , set ↡ A = { s ∈ S ∶ ∣ s ∣ ≺ a for some a ∈ A } . Definition 7.2.
We call an ℓ -ideal I of a proximity Specker A -algebra ( S, ≺ ) a round ideal provided I = S or ↡ I = I and I ∩ A =
0. We call I an end provided I is maximal amongproper round ideals of ( S, ≺ ) . Lemma 7.3.
Let α ∶ S → T be a proximity morphism between proximity Specker A -algebraswith T nontrivial. If I is an ℓ -ideal of T with I ∩ A = , then ↡ α − ( I ) is a proper round idealof S .Proof. Set J = ↡ α − ( I ) , and let s, t ∈ J . Then there are u, v ∈ S with ∣ s ∣ ≺ u , ∣ t ∣ ≺ v , and α ( u ) , α ( v ) ∈ I . Therefore, there are u ′ , v ′ with ∣ s ∣ ≺ u ′ ≺ u and ∣ t ∣ ≺ v ′ ≺ v . We have ∣ s ± t ∣ ≤ ∣ s ∣ + ∣ t ∣ ≺ u ′ + v ′ . Since α is order preserving, 0 ≤ α ( u ′ + v ′ ) . By Proposition 7.1(2), α ( u ′ + v ′ ) ≺ α ( u ) + α ( v ) ∈ I . As I is an ℓ -ideal, α ( u ′ + v ′ ) ∈ I . This yields u ′ + v ′ ∈ α − ( I ) , so s ± t ∈ J . Next, let s ∈ J and t ∈ S . Then ∣ s ∣ ≺ u for some u ∈ S with α ( u ) ∈ I . Since S is aSpecker A -algebra, write t = ∑ ni = a i e i in orthogonal form, and observe that ∣ t ∣ = ∣ n ∑ i = a i e i ∣ ≤ n ∑ i = ∣ a i e i ∣ = n ∑ i = ∣ a i ∣∣ e i ∣ ≤ n ∑ i = ∣ a i ∣ , where the last inequality follows from Lemma 4.9(1). Therefore, there is a ∈ A with ∣ t ∣ ≤ a .Then ∣ st ∣ = ∣ s ∣ ⋅ ∣ t ∣ ≤ a ∣ s ∣ ≺ au . As α ( au ) = aα ( u ) ∈ I , we see that st ∈ J . Thus, J is an ideal of S . To see it is an ℓ -ideal, let t ∈ J and s ∈ S satisfy ∣ s ∣ ≤ ∣ t ∣ . Then there is u ∈ S with ∣ t ∣ ≺ u and α ( u ) ∈ I . Therefore, ∣ s ∣ ≺ u , and so s ∈ J . We have thus proved that J is an ℓ -ideal of S .Next, take s ∈ J . Then ∣ s ∣ ≺ u for some u ∈ S with α ( u ) ∈ I . There is t ∈ S with ∣ s ∣ ≺ t ≺ u .This implies t ∈ J , so s ∈ ↡ J . Thus, ↡ J = J . To see that J ∩ A =
0, if a ∈ J ∩ A , then ∣ a ∣ ≺ u for some u ∈ S with α ( u ) ∈ I . Therefore, 0 ≤ ∣ a ∣ = α (∣ a ∣) ≤ α ( u ) ∈ I , so ∣ a ∣ ∈ I since I is an ℓ -ideal. Because I ∩ A =
0, we get a =
0. Consequently, J ∩ A =
0, and so J is a proper roundideal of S . (cid:3) For a proximity morphism α ∶ S → T between proximity Specker A -algebras, define the kernel of α as ker ( α ) = ↡ α − ( ) . As noted in Remark 6.5(1), if T is nontrivial, then α is nonzero. If α =
0, it is clear thatker ( α ) = S . On the other hand, if α is nonzero, then α ( a ) = a for each a ∈ A . Proposition 7.4. (1)
Let α ∶ S → T be a proximity morphism with T nontrivial. Then ker ( α ) is a properround ideal of S . (2) If P is a minimal prime ideal of S , then ↡ P is a proper round ideal of S .Proof. The first statement follows from Lemma 7.3 since ( ) is an ℓ -ideal of T . For thesecond statement, it is sufficient to observe that if P is a minimal prime ideal of S , then P is an ℓ -ideal by [17, p. 196], and P ∩ A = (cid:3) In Theorem 7.6 we give several characterizations of ends of a proximity Specker A -algebra ( S, ≺ ) . For this we require the following lemma. We recall that by Proposition 5.1, for aproximity Specker A -algebra ( S, ≺ ) , the restriction of ≺ is a proximity on Id ( S ) . Lemma 7.5.
Let ( S, ≺ ) be a proximity Specker A -algebra. If s, t ∈ S , we may write s = a + ∑ ni = b i e i and t = a + ∑ ni = b i k i in compatible decreasing form with each b i > . Moreover, s ≤ t iff e i ≤ k i for each i , and s ≺ t iff e i ≺ k i for each i . Furthermore, ≤ s iff ≤ a .Proof. Let η ∶ S → F N ( X ) be the embedding of Theorem 5.2. By Lemmas 2.2 and 2.3and Remark 2.6, we may write η ( s ) = a + ∑ ni = b i χ U i and η ( t ) = a + ∑ ni = b i χ V i , where X = U ⊇ U ⊇ ⋯ ⊇ U n ⊇ ∅ and X = V ⊇ V ⊇ ⋯ ⊇ V n ⊇ ∅ . Since η ( s ) , η ( t ) ∈ F N ( X ) , the U i and V i are regular open subsets of X . Claim 5.3 shows that there are e i , k i ∈ Id ( S ) with e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 27 ζ ( e i ) = U i and ζ ( k i ) = V i . Consequently, since η is an ℓ -algebra embedding, s = a + ∑ ni = b i e i and t = a + ∑ ni = b i k i . Moreover, s ≤ t iff η ( s ) ≤ η ( t ) , which by Lemma 2.3 is equivalent to U i ⊆ V i for each i . Since ζ ∶ Id ( S ) → RO( X ) is an embedding, the last condition is equivalentto e i ≤ k i for each i . Moreover, s ≺ t iff η ( s ) ≺ X η ( t ) , which is equivalent to U i ≺ V i for each i . As ζ preserves and reflects proximity, U i ≺ V i iff e i ≺ k i for each i . Finally, as each b i > s = a + ∑ ni = b i e i implies that 0 ≤ s if 0 ≤ a . For the converse, if 0 ≤ s , then0 ≤ η ( s ) . By Lemma 2.2(1) and Remark 2.6, a is the smallest value of the function η ( s ) .Thus, 0 ≤ η ( s ) implies 0 ≤ a . This finishes the proof. (cid:3) Theorem 7.6.
Let ( S, ≺ ) be a nontrivial proximity Specker A -algebra and let B = Id ( S ) .For an ℓ -ideal I of S , the following conditions are equivalent. (1) I is an end of ( S, ≺ ) . (2) I is the kernel of a proximity morphism α ∶ S → A . (3) I = ↡ P for some minimal prime ideal P of S . (4) I ∩ B is an end of ( B, ≺ ) and I is the ideal of S generated by I ∩ B . (5) I is the ideal of S generated by some end of ( B, ≺ ) .Proof. (1) ⇒ (3): Let I be an end of ( S, ≺ ) . Then I ∩ A =
0. A Zorn’s lemma argument showsthere is a prime ideal P of S with I ⊆ P and P ∩ A =
0. By [5, Lem. 4.5], P is a minimalprime ideal of S . Because I ⊆ P and I is round, I ⊆ ↡ P . Maximality of I then forces I = ↡ P since ↡ P is a proper round ideal by Proposition 7.4(2).(2) ⇒ (3): Let I = ker ( α ) and let σ be the restriction of α to B . Note that Id ( A ) is thetwo-element Boolean algebra . By Proposition 6.6, σ ∶ B → is a de Vries morphism. Defineker ( σ ) = ↡ σ − ( ) . Obviously ker ( σ ) ⊆ ker ( α ) ∩ B . For the reverse inclusion, if e ∈ ker ( α ) ∩ B ,then there is s ∈ S with e ≺ s and α ( s ) =
0. By an argument similar to the one given in theproof of Proposition 5.1, we can find k ∈ B with e ≺ k and σ ( k ) =
0. Therefore, e ∈ ker ( σ ) , andso ker ( α ) ∩ B = ker ( σ ) . Thus, E ∶ = I ∩ B is an end of ( B, ≺ ) (see, e.g., [4, Rem. 4.21]). Thisimplies that there is a maximal ideal M of B such that E = ↡ M (see, e.g., [2, Sec. 3]). Let P be the ideal of S generated by M . By [5, Lem. 2.5 and Thm. 2.7], the Boolean homomorphism B → whose kernel is M extends to an A -algebra homomorphism β ∶ S → A , and its kernelcontains P since it contains M . Let β ( s ) = s = ∑ ni = a i e i in orthogonal form withthe a i ∈ A distinct and nonzero. Then se i = a i e i , so 0 = β ( se i ) = β ( a i e i ) = a i β ( e i ) . Since S is torsion-free, this implies β ( e i ) =
0. Therefore, e i ∈ M , showing that s ∈ P . Thus, P is thekernel of the onto ring homomorphism β ∶ S → A , and as A is an integral domain, P is aprime ideal. Since P ∩ A =
0, by [5, Lem. 4.5] P is a minimal prime.We wish to show I = ↡ P . Take s ∈ I , and first suppose s ≥
0. There is t ∈ S with s ≺ t and α ( t ) =
0. By Lemma 7.5, write s = a + ∑ ni = b i e i and t = a + ∑ ni = b i k i in compatible decreasingform with b i > e i ≺ k i for each i . By Theorem 6.7, 0 = α ( t ) = a + ∑ ni = b i σ ( k i ) . Since σ ( k i ) ∈ and the b i are positive, this forces a = σ ( k i ) =
0. Pick l i ∈ B with e i ≺ l i ≺ k i . By replacing l i by l ∧ ⋯ ∧ l i we may assume the l i form a decreasing set. Let r = ∑ ni = b i l i . Then s ≺ r ≺ t and l i ∈ ker ( σ ) ⊆ M . Consequently, r ∈ P , and so s ∈ ↡ P . Foran arbitrary s , the argument we just gave shows ∣ s ∣ ∈ ↡ P . Since P is a minimal prime, byProposition 7.4(2), ↡ P is a round ideal, so ↡ P is an ℓ -ideal, and hence s ∈ ↡ P . This implies I ⊆ ↡ P . For the reverse inclusion, let s ∈ ↡ P . Then ∣ s ∣ ≺ t for some t ∈ P . To show s ∈ I , itsuffices to show ∣ s ∣ ∈ I since by Proposition 7.4(1), I is an ℓ -ideal. Therefore, assume s ≥ r with s ≺ r ≺ t . Write s = a + ∑ ni = b i e i , r = a + ∑ ni = b i l i , and t = a + ∑ ni = b i k i incompatible decreasing form with e i ≺ l i ≺ k i for each i . Since 0 ≤ s , Lemma 7.5 implies 0 ≤ a . We show that each k i ∈ M . As t ∈ P and P is generated by M , we can write t = ∑ mj = c j p j for some p j ∈ M . Set c = ∑ mj = ∣ c j ∣ and p = p ∨ ⋯ ∨ p m . Then cp ≥ t ≥ a = a ⋅ ≥
0. ByLemma 4.9(6), this forces a = p =
1. Since p ∈ M and 1 ∉ M , we see that a = cp ≥ t ≥ b k . Applying Lemma 4.9(6) again yields p ≥ k ≥ k i for each i . Since p j ∈ M for each j , we have p ∈ M , so each k i ∈ M . Now, l i ≺ k i and k i ∈ M yield l i ∈ ↡ M = E .Therefore, α ( r ) =
0, so as s ≺ r , we obtain s ∈ I . Thus, I = ↡ P .(3) ⇒ (4): Let I = ↡ P for some minimal prime ideal P of S , and set E = I ∩ B . Then E is anideal of B . Let M = P ∩ B . By [5, Prop. 3.11 and Thm. 4.6], M is a maximal ideal of B . Weshow E = ↡ M , which will prove that E is an end of ( B, ≺ ) . Let e ∈ E . Then e ∈ I , so there is s ∈ P with e ≺ s . An argument similar to the one given in the proof of Proposition 5.1 gives k ∈ M with e ≺ k . This implies e ∈ ↡ M , which yields E ⊆ ↡ M . For the converse, let e ∈ ↡ M .Then there is k ∈ M with e ≺ k . Therefore, k ∈ P , so e ∈ ↡ P = I , and hence e ∈ I ∩ B = E .This proves E = ↡ M , so E is an end of ( B, ≺ ) . We next prove that E generates I as an idealof S . One inclusion is obvious. For the reverse, let s ∈ I . Then there is t ∈ P with ∣ s ∣ ≺ t . By[8, Ch. XIII, Thm. 7], 0 ≤ s ∨ ≤ ∣ s ∣ ≺ t , so 0 ≤ s ∨ ≺ t . By writing s ∨ t in compatibledecreasing form, an argument similar to the one given in the proof of (1) ⇒ (3) yields that s ∨ E . Since − s ∈ I , we have that ( − s ) ∨ = − ( s ∧ ) is also inthis ideal. Because s = ( s ∨ ) + ( s ∧ ) (see, e.g., [8, Ch. XIII, Thm. 7]), we conclude that s lies in the ideal generated by E , and so I is generated by E .(4) ⇒ (5): This is obvious.(5) ⇒ (1): Let I be the ideal generated by an end E of B . Then I is the set of all A -linearcombinations of elements of E . Let M be a maximal ideal of B containing E . It followsfrom [5, Prop. 3.11 and Thm. 4.6] that M = P ∩ Id ( S ) for some minimal prime P . Then I ⊆ P . Therefore, I ∩ A ⊆ P ∩ A =
0. Let s ∈ I . Then s can be written as s = ∑ ni = a i e i witheach e i ∈ E . There are k i ∈ E with e i ≺ k i . If t = ∑ ni = ∣ a i ∣ k i , then t ∈ I and ∣ s ∣ ≤ ∑ ni = ∣ a i ∣ e i ≺ t .Therefore, ∣ s ∣ ≺ t . Thus, I is round. Finally, let J be an end of ( S, ≺ ) with I ⊆ J . Then E = I ∩ B ⊆ J ∩ B . Because J is an end and we have already proved (3) ⇒ (4), J ∩ B is anend of ( B, ≺ ) and generates J as an ideal. Maximality shows E = J ∩ B , so J is the idealgenerated by E . Thus, J = I , and so I is an end of ( S, ≺ ) .(5) ⇒ (2): Let E be an end of ( B, ≺ ) and suppose I is generated as an ideal by E . Let F = { b ∈ B ∶ ¬ b ∈ E } . As we pointed out in Section 5, F is a maximal round filter of ( B, ≺ ) ,so σ ∶ B → that sends the members of F to 1 and the rest of B to 0 is a de Vries morphism(see, e.g., [4, Rem. 4.21]). By Corollary 6.9, σ extends uniquely to a proximity morphism α ∶ S → A . We claim that I = ker ( α ) . To see this, let s ∈ I . Since I is an ℓ -deal, ∣ s ∣ ∈ I ,so ∣ s ∣ is a linear combination of idempotents from E . An argument similar to the one inthe proof of (1) ⇒ (3) showing that 0 ≤ t ∈ P has all the idempotents in its decreasing formin M yields that ∣ s ∣ can be written in decreasing form ∣ s ∣ = ∑ ni = b i e i with each b i > e i in E . Since E is a round ideal of ( B, ≺ ) , for each i there is k i ∈ E with e i ≺ k i . Set t = ∑ ni = b i k i . Then t ∈ I and ∣ s ∣ ≺ t . Moreover, α ( t ) = ∑ ni = b i σ ( k i ) =
0. Therefore, s ∈ ker ( α ) .Conversely, let s ∈ ker ( α ) . Then ∣ s ∣ ≺ t for some t with α ( t ) =
0. By Lemma 7.5, we maywrite ∣ s ∣ = ∑ ni = b i e i and t = ∑ ni = b i k i in compatible decreasing form with e i ≺ k i and all b i > = α ( t ) = ∑ ni = b i σ ( k i ) . Because σ ( k i ) ∈ { , } , the condition α ( t ) = σ ( k i ) = i . Thus, e i ∈ ker ( σ ) = E . So, ∣ s ∣ ∈ I , and hence s ∈ I since I is an ℓ -ideal. Thus, I = ker ( α ) . (cid:3) Remark 7.7.
A natural condition to add to the five equivalent conditions of Theorem 7.6would be that I = ↡ M for some maximal ℓ -ideal M of S . While this condition is not e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 29 equivalent to the others in general, we show that it is equivalent provided A is Archimedean ;meaning that for each a, b ∈ A , if na ≤ b for all n ∈ N , then a ≤
0. Indeed, we show that if A isArchimedean and S is a Specker A -algebra, then minimal primes in S coincide with maximal ℓ -ideals of S . First suppose that P is a minimal prime ideal of S . By [17, p. 196], P is an ℓ -ideal. By the proof of (1) ⇒ (3) of Theorem 7.6, S / P ≅ A . Since A is Archimedean, it issimple as an ℓ -algebra. Thus, P is maximal as an ℓ -ideal. Conversely, let M be a maximal ℓ -ideal. Then M ∩ A = ℓ -ideal of A and A is Archimedean,hence simple as an ℓ -algebra. A Zorn’s lemma argument then yields a prime ideal P ⊇ M with P ∩ A =
0. Since P ∩ A =
0, by [5, Lem. 4.5], P is a minimal prime ideal of S . Becauseit is a proper ℓ -ideal, maximality of M shows M = P . Thus, M is a minimal prime ideal of S . Definition 7.8.
Let ( S, ≺ ) be a proximity Specker A -algebra.(1) Let End ( S, ≺ ) be the space of ends of ( S, ≺ ) topologized by the basis { U ( s ) ∶ s ∈ S } ,where U ( s ) = { I ∶ s ∈ I } .(2) Let Hom ( S, A ) be the space of proximity morphisms from S to A topologized by thebasis { V ( s ) ∶ s ∈ S } , where V ( s ) = { α ∶ ∃ t with ∣ s ∣ ≺ t and α ( t ) = } . Remark 7.9. (1) That the sets U ( s ) , s ∈ S , form a basis for End ( S, ≺ ) is a consequence of the followingeasily verifiable facts: U ( s ) = U (∣ s ∣) ; U (∣ s ∣) ∩ U (∣ t ∣) = U (∣ s ∣ ∨ ∣ t ∣) ; U ( ) = ∅ ; and U ( ) = End ( S, ≺ ) .(2) The same formulas as in (1) hold for V ( s ) , s ∈ S . That V (∣ s ∣) ∩ V (∣ t ∣) = V (∣ s ∣ ∨ ∣ t ∣) isa consequence of the following lemma, which is parallel to [3, Lem. 2.2]. Lemma 7.10.
Let α ∶ S → T be a proximity morphism between proximity Specker A -algebras. (1) If s ∈ S , then α ( s ) ≤ − α ( − s ) . Therefore, if s ≺ t , then α ( s ) ≺ α ( t ) . (2) If s, t, u, v ∈ S with s ≺ u and t ≺ v , then α ( s ∨ t ) ≺ α ( u ) ∨ α ( v ) .Proof. (1) By Proposition 6.6, α ∣ Id ( S ) is a de Vries morphism. Thus, if e ∈ Id ( S ) , then α ( e ) ≤ ¬ α ( ¬ e ) . Let s ∈ S . We first assume s ≥
0. Write s = a + ∑ ni = b i e i in decreasing formwith b i >
0. By Theorem 6.7, α ( s ) = a + ∑ ni = b i α ( e i ) . Let b = ∑ ni = b i . As in the proof ofTheorem 6.3(3), we have − s = − a + n ∑ i = b i ( − e i ) = − a − b + b + n ∑ i = b i ( − e i ) = ( − a − b ) + n ∑ i = b i ( − e i ) = ( − a − b ) + n ∑ i = b i ¬ e i . Since ¬ e n > ⋯ > ¬ e , this writes − s in decreasing form, so α ( − s ) = ( − a − b ) + ∑ ni = b i α ( ¬ e i ) .Consequently, α ( − s ) = − a + ∑ ni = b i ( α ( ¬ e i ) − ) , so − α ( − s ) = a + ∑ ni = b i ( − α ( ¬ e i )) = a + ∑ ni = b i ( ¬ α ( ¬ e i )) . Finally, since α ( e i ) ≤ ¬ α ( ¬ e i ) for each i , we get α ( s ) ≤ − α ( − s ) since all b i are positive.For an arbitrary s ∈ S , since η S ( s ) is a finitely valued function, there is a ∈ A with η S ( s ) + a ≥
0. This yields s + a ≥ η S is an ℓ -algebra embedding. By the nonnegativecase, we have α ( a + s ) ≤ − α ( − ( a + s )) . This simplifies to a + α ( s ) ≤ a + ( − α ( − s )) . Consequently, α ( s ) ≤ − α ( − s ) . From this we conclude that if s ≺ t , then α ( s ) ≤ − α ( − s ) ≺ α ( t ) , so α ( s ) ≺ α ( t ) . (2) We have − α ( − s ) ≺ α ( u ) and − α ( − t ) ≺ α ( v ) , so − α ( − s ) ∨ − α ( − t ) ≺ α ( u ) ∨ α ( v ) . But, − α ( − s ) ∨ − α ( − t ) = − [ α ( − s ) ∧ α ( − t )] = − α ( − s ∧ − t ) = − α ( − ( s ∨ t )) . Thus, by (1), α ( s ∨ t ) ≤ − α ( − ( s ∨ t )) ≺ α ( u ) ∨ α ( v ) . (cid:3) We are ready to show that for a proximity Specker A -algebra ( S, ≺ ) , the 1-1 correspon-dences of Theorem 7.6 between the ends of ( S, ≺ ) , the ends of ( Id ( S ) , ≺ ) , and the proximitymorphisms S → A extend to the homeomorphisms of the corresponding spaces. Theorem 7.11.
Let ( S, ≺ ) be a proximity Specker A -algebra and let B = Id ( S ) . The spaces End ( S, ≺ ) and Hom ( S, A ) are homeomorphic to the compact Hausdorff space X of ends of ( B, ≺ ) .Proof. The theorem is clear if S is trivial. Suppose that S is nontrivial, and define ϕ ∶ End ( S, ≺ ) → X by ϕ ( I ) = I ∩ B for each I ∈ End ( S, ≺ ) . By Theorem 7.6, ϕ is a well-definedbijection. Since ϕ − ( ζ ( e )) = U ( e ) for each e ∈ B , we have that ϕ is continuous. To see that ϕ − is continuous, let s ∈ S and write s = ∑ ni = a i e i in orthogonal form with each a i ≠
0. Weshow that U ( s ) = U ( e ) ∩ ⋯ ∩ U ( e n ) . One inclusion is clear. For the other inclusion, let I bean end of ( S, ≺ ) and let s ∈ I . Then se i = a i e i , so a i e i ∈ I . Since I is an end, there is t ∈ I with ∣ a i e i ∣ ≺ t . By Theorem 7.6, I ∩ B is an end of ( B, ≺ ) and I is generated by I ∩ B . Therefore,we may write t = ∑ mj = b j k j with k j ∈ I ∩ B . If b = ∑ mj = ∣ b j ∣ and k = k ∨ ⋯ ∨ k m , then k ∈ I ∩ B and t ≤ bk . Thus, ∣ a i e i ∣ ≺ bk . This implies ∣ a i ∣ e i ≤ bk , so by Lemma 4.9(6), ∣ a i ∣ ≤ b and e i ≤ k ,yielding e i ∈ I ∩ B . Since this is true for each i , we conclude that I ∈ U ( e ) ∩ ⋯ ∩ U ( e n ) , so U ( s ) = U ( e ) ∩ ⋯ ∩ U ( e n ) . Therefore, ϕ ( U ( s )) = ϕ ( U ( e )) ∩ ⋯ ∩ ϕ ( U ( e n )) . Now, if e ∈ B ,then ϕ ( U ( e )) = { I ∩ B ∶ e ∈ I } = ζ ( e ) . Thus, ϕ ( U ( s )) is open in X . Consequently, ϕ is ahomeomorphism.Next, define τ ∶ Hom ( S, A ) → End ( S, ≺ ) by τ ( α ) = ker ( α ) . By Theorem 7.6, τ is a well-defined bijection. It is also easy to see that τ − ( U ( s )) = V ( s ) and τ ( V ( s )) = U ( s ) . Thus, Hom ( S, A ) and End ( S, ≺ ) are homeomorphic. (cid:3) Consequently, given a proximity Specker A -algebra ( S, ≺ ) , we can think of the dual com-pact Hausdorff space of ( S, ≺ ) as either the space of ends of ( S, ≺ ) , the space of ends of ( Id ( S ) , ≺ ) , or the space of proximity morphisms S → A .8. Categorical considerations
In this final section we show that the proximity Baer Specker A -algebras and proximitymorphisms between them form a category, which we denote by PBSp A . Using the resultsobtained in previous sections, we prove that PBSp A is dually equivalent to KHaus , thusproviding an analogue of de Vries duality for proximity Baer Specker A -algebras. As aconsequence, we obtain that PBSp A is equivalent to DeV . Proposition 8.1.
The proximity Baer Specker A -algebras and proximity morphisms form acategory PBSp A where the composition β ⋆ α of two proximity morphisms α ∶ S → S and β ∶ S → S is the unique proximity morphism extending the de Vries morphism β ∣ Id ( S ) ⋆ α ∣ Id ( S ) . e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 31 Proof.
It is easily seen that the identity map on a proximity Baer Specker A -algebra is aproximity morphism. By Proposition 6.6, if α and β are proximity morphisms, then theirrestrictions to the idempotents are de Vries morphisms. Therefore, β ∣ Id ( S ) ⋆ α ∣ Id ( S ) is a deVries morphism. Thus, by Corollary 6.9, β ⋆ α is a proximity morphism. We show that ⋆ is associative. Suppose that α ∶ S → S , α ∶ S → S , and α ∶ S → S are proximitymorphisms. Since the restrictions of α ⋆ ( α ⋆ α ) and ( α ⋆ α ) ⋆ α to the idempotentsare de Vries morphisms, we have that they are equal. Applying Corollary 6.9 again yields α ⋆ ( α ⋆ α ) = ( α ⋆ ) ⋆ α . Thus, the proximity Baer Specker A -algebras with proximitymorphisms form a category. (cid:3) Remark 8.2.
It would seem more natural to first introduce the category
PSp A of proximitySpecker A -algebras, and treat PBSp A as a full subcategory of PSp A . Similarly, it wouldseem more natural to first introduce the category PBA of proximity Boolean algebras, andtreat
DeV as a full subcategory of
PBA . However, if α ∶ B → B and β ∶ B → B areproximity morphisms between proximity Boolean algebras, then the formula ( β ⋆ α )( a ) = ⋁{ β ( α ( b )) ∶ b ≺ a } need not be well defined because, if C is not complete, then the join maynot exist. It is for this reason that we do not talk categorically about proximity Booleanalgebras and proximity Specker A -algebras.Next we show that although proximity morphisms are not A -algebra homomorphisms,proximity isomorphisms are A -algebra isomorphisms that preserve and reflect proximity.This is parallel to what happens in DeV [11, Ch. I.5].
Lemma 8.3.
Let ( S, ≺ ) , ( T, ≺ ) ∈ PBSp A and let α ∶ S → T be a proximity morphism. Then α is an isomorphism in PBSp A iff α is an A -algebra isomorphism such that s ≺ t in S iff α ( s ) ≺ α ( t ) in T .Proof. First suppose that α ∶ S → T is an A -algebra isomorphism such that s ≺ t iff α ( s ) ≺ α ( t ) . Since α is an A -algebra homomorphism, − α ( − s ) = α ( s ) . As s ≺ t implies α ( s ) ≺ α ( t ) , we obtain that s ≺ t implies − α ( − s ) ≺ α ( t ) . Consequently, to see that α is a proximity morphism, we only need to check that α ( t ) is the least upper bound of { α ( s ) ∶ s ≺ t } . By [5, Cor. 5.3], α is an ℓ -algebra isomorphism, so α is order preserving.Therefore, α ( t ) is an upper bound of { α ( s ) ∶ s ≺ t } . Let r be an upper bound of this set.Then α ( s ) ≤ r for all s with s ≺ t . As α is an A -algebra isomorphism, so is α − . Therefore, α − is order preserving, and α ( s ) ≤ r implies s ≤ α − ( r ) . By Lemma 6.1, t is the least upperbound of all elements proximal to it. Thus, t ≤ α − ( r ) , and so α ( t ) ≤ r . This proves that α ( t ) is the least upper bound of { α ( s ) ∶ s ≺ t } . It follows that α is a proximity morphism.The same argument shows that α − is a proximity morphism. To see that α − ⋆ α = id S ,let s ∈ S and write s = a + ∑ ni = b i e i in decreasing form. Then, using Theorem 6.7 and thefact that the restriction of α − ⋆ α to the idempotents is the identity on the idempotents, weobtain ( α − ⋆ α )( s ) = a + ∑ ni = b i ( α − ⋆ α )( e i ) = a + ∑ ni = b i e i = s . Therefore, α − ⋆ α = id S . Asimilar argument shows α ⋆ α − = id T . Thus, α is a proximity isomorphism.Next suppose that α ∶ S → T is a proximity isomorphism. Then there is a proximityisomorphism β ∶ T → S such that β ⋆ α is the identity on S and α ⋆ β is the identity on T . By the definition of ⋆ , we see that the restrictions of α and β to the idempotents areinverse de Vries isomorphisms, so α ∣ Id ( S ) and β ∣ Id ( T ) are inverse Boolean isomorphisms. By[5, Lem. 2.5 and Thm. 2.7], there is a unique A -algebra homomorphism α ′ ∶ S → T extending α ∣ Id ( S ) . We show that α ′ = α . Let s ∈ S and write s = a + ∑ ni = b i e i in decreasing form. ByTheorem 6.7, α ( s ) = a + ∑ ni = b i α ( e i ) . Since α ′ is an A -algebra homomorphism, we also have α ′ ( s ) = a + ∑ ni = b i α ( e i ) . Thus, α ′ = α , and so α is an A -algebra homomorphism. By thesame reasoning, β is an A -algebra homomorphism. It follows that β ○ α is the identity on S because it is an A -algebra endomorphism which is the identity on Id ( S ) , a generating setof S as an A -algebra. Similarly, α ○ β is the identity on T . Consequently, α is an A -algebraisomorphism whose inverse is β , and s ≺ t iff α ( s ) ≺ α ( t ) because both α and β preserveproximity. (cid:3) Next we construct contravariant functors ( − ) ∗ ∶ PBSp A → KHaus and ( − ) ∗ ∶ KHaus → PBSp A that yield a dual equivalence of PBSp A and KHaus .Define a contravariant functor ( − ) ∗ ∶ KHaus → PBSp A as follows. For X ∈ KHaus , let X ∗ = ( F N ( X, A ) , ≺ X ) be the de Vries power of A by X ; and for a continuous map ϕ ∶ X → Y ,let ϕ ∗ ∶ F N ( Y ) → F N ( X ) be the proximity morphism given by ϕ ∗ ( f ) = ( f ○ ϕ ) . ByTheorem 4.10 and Proposition 6.7, ( − ) ∗ ∶ KHaus → PBSp A is a well-defined contravariantfunctor.Define a contravariant functor ( − ) ∗ ∶ PBSp A → KHaus as follows. For ( S, ≺ ) ∈ PBSp A ,let S ∗ be the space of ends of ( S, ≺ ) . By Theorem 7.11, S ∗ ∈ KHaus . For a proximitymorphism α ∶ S → T , let α ∗ ∶ T ∗ → S ∗ be given by α ∗ ( I ) = ↡ α − ( I ) . That α ∗ is a well-definedcontinuous map is proved in the next lemma. Lemma 8.4.
Let α ∶ S → T be a proximity morphism between proximity Specker A -algebras ( S, ≺ ) and ( T, ≺ ) . Define α ∗ ∶ T ∗ → S ∗ by α ∗ ( I ) = ↡ α − ( I ) . Then α ∗ is a well-definedcontinuous map.Proof. If T is trivial, then there is nothing to prove. Suppose that T is nontrivial. Let I bean end of T . By Lemma 7.3, ↡ α − ( I ) is a proper round ideal of S . To see that ↡ α − ( I ) is anend, let J be an end of S containing ↡ α − ( I ) . By Theorem 7.6, J ∩ Id ( S ) is an end of Id ( S ) and J is generated by J ∩ Id ( S ) . The same reasoning gives I ∩ Id ( T ) is an end of Id ( T ) and I is generated by I ∩ Id ( T ) . We have ( ↡ α − ( I ∩ Id ( T ))) ∩ Id ( S ) ⊆ ↡ α − ( I ) ∩ Id ( S ) ⊆ J ∩ Id ( S ) . By Proposition 6.6, the restriction of α to Id ( S ) is a de Vries morphism. Therefore, ↡ α − ( I ∩ Id ( T )) ∩ Id ( S ) is an end of Id ( S ) , and so ↡ α − ( I ∩ Id ( T )) ∩ Id ( S ) = J ∩ Id ( S ) . This impliesthat J is generated by ↡ α − ( I ) ∩ Id ( S ) . Thus, J ⊆ α − ( I ) , yielding that α ∗ ( I ) = ↡ α − ( I ) isan end.To show that α ∗ is continuous, it is sufficient to see that α − ∗ ( U ( s )) = ⋃{ U ( α ( t )) ∶ ∣ s ∣ ≺ t } .Indeed, I ∈ α − ∗ ( U ( s )) iff s ∈ ↡ α − ( I ) , which happens iff there is t with ∣ s ∣ ≺ t and α ( t ) ∈ I ,which is true iff I ∈ ⋃{ U ( α ( t )) ∶ ∣ s ∣ ≺ t } . (cid:3) This implies that ( − ) ∗ ∶ PBSp A → KHaus is a well-defined contravariant functor.
Theorem 8.5.
The functors ( − ) ∗ and ( − ) ∗ yield a dual equivalence of PBSp A and KHaus .Proof.
By Corollary 6.10, for ( S, ≺ ) ∈ PBSp A and X ∈ KHaus , we have hom
PBSp A ( S, X ∗ ) ≃ hom KHaus ( X, S ∗ ) . It follows from the proof of Corollary 6.10 that the bijection is natural.Therefore, ( − ) ∗ and ( − ) ∗ define a contravariant adjunction between PBSp A and KHaus .Let ( S, ≺ ) ∈ PBSp A and let X be the space of ends of ( Id ( S ) , ≺ ) . By Theorem 7.11, S ∗ is homeomorphic to X , so ( S ∗ ) ∗ is isomorphic to ( F N ( X ) , ≺ ) . By Corollary 5.6 andLemma 8.3, the ℓ -algebra embedding η ∶ S → F N ( X ) of Theorem 5.2 is an isomorphism in PBSp A . Thus, the unit of the contravariant adjunction is an isomorphism. e Vries powers: A generalization of Boolean powers for compact Hausdorff spaces 33 Let X ∈ KHaus . By Theorem 4.10, X ∗ is a proximity Baer Specker A -algebra. ByLemma 4.8, RO( X ) is isomorphic to Id ( F N ( X )) . Therefore, by Theorem 7.11, ( X ∗ ) ∗ ishomeomorphic to the space of ends of RO( X ) . By de Vries duality, X is homeomorphicto the space of ends of RO( X ) , so X is homeomorphic to ( X ∗ ) ∗ . Thus, the counit of thecontravariant adjunction is an isomorphism. Consequently, PBSp A is dually equivalent to KHaus . (cid:3) Corollary 8.6. PBSp A is equivalent to DeV .Proof.
Combine Theorem 8.5 with de Vries duality. (cid:3)
Remark 8.7.
The equivalence of
PBSp A and DeV can be established directly through thecovariant functors I ∶ PBSp A → DeV and S ∶ DeV → PBSp A , where I sends a proximityBaer Specker A -algebra ( S, ≺ ) to the de Vries algebra of idempotents of ( S, ≺ ) , while S sendsa de Vries algebra ( B, ≺ ) to the de Vries power of A by ( B, ≺ ) . Remark 8.8.
Following [2, Def. 4.5], we call a de Vries algebra ( B, ≺ ) zero-dimensional provided a ≺ b implies that there is c ∈ B with c ≺ c and a ≺ c ≺ b . Let zDeV be the fullsubcategory of DeV whose objects are zero-dimensional de Vries algebras, and let
Stone bethe full subcategory of
KHaus whose objects are Stone spaces (zero-dimensional compactHausdorff spaces). By [2, Thm. 4.12], zDeV is dually equivalent to
Stone .Analogously, we call a proximity Specker A -algebra ( S, ≺ ) zero-dimensional provided s ≺ t implies that there is r ∈ S with r ≺ r and s ≺ r ≺ t . Let zPBSp A be the full subcategoryof PBSp A of zero-dimensional proximity Baer Specker A -algebras. It is a consequence ofTheorem 8.5, Corollary 8.6, and [2, Thm. 4.12] that zPBSp A is equivalent to zDeV and isdually equivalent to Stone . Thus, by [2, Thm. 4.9], zPBSp A is a coreflective subcategoryof BPSp A . Remark 8.9.
Following [2, Sec. 5], we call a de Vries algebra ( B, ≺ ) extremally disconnected provided a ≺ b iff a ≤ b . Let eDeV be the full subcategory of DeV whose objects areextremally disconnected de Vries algebras, and let ED be the full subcategory of KHaus whose objects are extremally disconnected compact Hausdorff spaces. Then eDeV is afull subcategory of zDeV , ED is a full subcategory of Stone , eDeV is isomorphic to thecategory cBA of complete Boolean algebras and Boolean homomorphisms, and eDeV isdually equivalent to ED [2, Sec. 6.2].Analogously, we call a proximity Specker A -algebra ( S, ≺ ) extremally disconnected pro-vided s ≺ t iff s ≤ t . Let ePBSp A be the full subcategory of PBSp A of extremally discon-nected proximity Baer Specker A -algebras. Then ePBSp A is a full subcategory of zPBSp A and is isomorphic to the category BSp A of Baer Specker A -algebras and A -algebra homo-morphisms. Thus, by [5, Thm. 4.7], ePBSp A is dually equivalent to ED . In addition, ePBSp A is equivalent to eDeV . Remark 8.10. (1) For a compact Hausdorff space X , we recall [13] that the Gleason cover Y of X isthe Stone space of RO ( X ) . Therefore, RO ( X ) is isomorphic to the Boolean algebra Clopen ( Y ) of clopen subsets of Y . Since up to isomorphism, F N ( Y ) is generated by RO ( X ) and F C ( Y ) is generated by Clopen ( Y ) , we obtain that F N ( X ) is isomorphicto F C ( Y ) . This yields an alternate representation of finitely valued normal functionson X . (2) For a de Vries algebra ( B, ≺ ) , we recall [2, Sec. 7] that the Stone space of B is theGleason cover of the space of ends of ( B, ≺ ) . Similarly, for a proximity Baer Specker A -algebra ( S, ≺ ) , the Gleason cover of the space of ends of ( S, ≺ ) can be constructedas the minimal prime spectrum of S . For an Archimedean A , the Gleason cover canalternately be constructed as the space of maximal ℓ -ideals of S . References
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