Stone Duality for Skew Boolean Algebras with Intersections

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1 Stone Duality for Skew Boolean Algebras with Intersections Andrej Bauer Faculty of Mathematics and Physics University of Ljubljana Karin Cvetko-Vah Faculty of Mathematics and Physics University of Ljubljana May 30, 2011 Abstract We extend Stone duality between generalized Boolean algebras and Boolean saces, which are the zero-dimensional locally-comact Hausdorff saces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale mas between Boolean saces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morhisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section. 1 Introduction The fundamental examle of the kind of duality we are interested in was established by Marshall Stone [9, 10]: every Boolean algebra corresonds to a zero-dimensional comact Hausdorff sace, or a Stone sace for short, as well as to a Boolean ring, which is a commutative ring of idemotents with a unit. In modern language the duality is stated as equivalence of categories of Boolean algebras, Boolean rings, and Stone saces, where the later equivalence is contravariant. The duality has many generalizations, see [3]. Already in Stone s second aer [10, Theorem 8] we find an extension of duality to Boolean saces, which are the zero-dimensional locally comact Hausdorff saces. They corresond to commutative rings of idemotents, ossibly without a unit, or equivalently to generalized Boolean algebras, which are like Boolean algebras without a to element. Our contribution to the toic is a study of the non-commutative case. Among several variations of non-commutative Boolean algebras we are able to rovide duality for skew Boolean algebras with intersections (which we call skew algebras) because they have a well-behaved theory of ideals. The aer is organized as follows. In Section 2 we recall the necessary background material about skew Boolean algebras, Boolean saces, and étale mas. In Section 3 we sell out the well-known Stone duality for commtutative algebras. In Section 4 we establish the duality between right-handed skew algebras 1

2 and skew Boolean saces, which we then extend to the duality between skew algebras and rectangular skew Boolean saces. In Section 5 we further analyze the situation and consider several variations of the duality in which morhisms are restricted. In Section 6 we use the duality to construct a right-handed skew algebra without a lattice section. This answers negatively a hitherto oen question of existence of such algebras. Acknowledgment. We thank Jeff Egger, Mai Gehrke, Ganna Kudryavtseva, Jonathan Leech, and Alex Simson for discussing the toic with us and offering valuable advice. 2 Preliminary definitions In the first art of the section we review basic concets and notation regarding skew Boolean algebras. In the second art we recall some basic facts about Boolean saces and étale mas. 2.1 Skew Boolean algebras A skew lattice is an algebra (A,, ) with idemotent and associative binary oerations meet and join satisfying the absortion identities x (x y) = x = (y x) x and x (x y) = x = (y x) x. If one of the oerations is commutative then so is the other, in which case A is a lattice, see [5]. A skew lattice has two order structures. The natural artial order x y is defined by x y = y x = x, or equivalently x y = y x = y. The natural reorder x y is defined by x y x = x, or equivalently y x y = y. The oset refelection of the natural reorder is known as Green s relation D. By Leech s First Decomosition Theorem [5], D is the finest congruence for which A/D is a lattice. In other words, the functor A A/D is a reflection of skew lattices into ordinary lattices. We denote the D-equivalence class of a by D a. The reflection can be analysed further into its left- and right-handed arts. A skew lattice A is right-handed if it satisfies the identity x y x = y x, or equivalently x y x = x y. We define left-handed lattices analogously. Quotients by Green s congruence relations R and L, which are defined by x R y x y = y and y x = x, x L y x y = x and y x = y, rovide reflections of a skew lattice A into left-handed and right-handed skew lattices, resectively. By Leech s Second Decomosition Theorem [5] the square of canonical quotient mas A A/L A/R A/D is a ullback in the category of skew algebras. 2

3 A rectangular band (A, ) is an algebra with a binary oeration which is idemotent, associative, and it satisfies the rectangle identity x y z = x z. The name comes from the fact that every rectangular band is isomorhic to a cartesian roduct X Y with the oeration (x 1, y 1 ) (x 2, y 2 ) = (x 1, y 2 ). If A is non-emty the sets X and Y are unique u to bijection and can be taken to be A/L and A/R, resectively. Each rectangular band is a skew lattice for and the associated oeration defined as x y = y x. It turns out that the D-classes of a skew lattice form rectangular bands, with the oeration induced by the skew lattice. A skew Boolean algebra (A, 0,,, \) is a skew lattice which is meet-distributive, i.e., it satisfies the identities x (y z) = (x y) (x z) and (y z) x = (y x) (z x), has a zero 0, which is neutral for, and a relative comlement \ satisfying (x \ y) (x y x) = 0 and (x \ y) (x y x) = x. It follows from these requirement that the rincial subalgebras x A x of a skew Boolean algebra are Boolean algebras. For examle, the two identities for the relative comlement say that \ restricted to a rincial subalgebra acts as the comlement oeration. See Leech [6] for further details on skew Boolean alebras. We remark that a skew Boolean algebra with a to element 1 is degenerate in the sense that it is already a Boolean algebra. Also note that a skew Boolean algebra whose meet and join are commutative is the same thing as a generalized Boolean algebra. Often it is the case that any two elements x and y of a skew Boolean algebra A have the greatest lower bound x y with resect to the natural artial order. When this is the case, we call intersection and seak of a skew Boolean intersection algebra. Many examles of skew lattices occurring in nature osses intersections. The significance of skew Boolean intersection algebras is witnessed by the fact that they form a discriminator variety [1], and are therefore both congruence ermutable and congruence distributive. Moreover, results by Bignall and Leech [1] imly that every algebra A in a ointed discriminator variety is term equivalent to a right-handed skew Boolean intersection algebra whose congruences coincide with those of A. In contrast, it was observed already by Cornish [2] that the congruence lattices of skew Boolean algebras in general satisfy no articular lattice identity. Henceforth we shall consider exclusively skew Boolean intersection algebras, so we simly refer to them as skew algebras. A homomorhism of skew algebras reserves all the oerations, namely 0,,, \, and. Recall that an ideal, which we sometimes call -ideal, in a skew algebra A is a subset I A which is lower with resect to and is closed under finite joins, so in articular 0 I. An ideal P is rime if it is non-trivial and a b P imlies a P or b P. It can be shown easily that the rime ideals in A coincide with non-zero mas A 2 into the two-elments lattice 2 = {0, 1} which reserve 0,, and (but not necessarily ). Because 2 is commutative, such mas are in bijective corresondence with non-zero mas A/D 2. In other words, the assignment P P/D = {D a a P } 3

4 is a bijection from rime ideals in A to rime ideals in A/D. We write f : X Y to indicate that f is a artial ma from X to Y, defined on its domain dom(f) X. The restriction f D of f : X Y to D X is the ma f with the domain restricted to dom(f) D. We denote the set of all artial mas from X to Y by P(X, Y ). Leech s construction [6] shows how P(X, Y ) can be endowed with a right-handed skew algbera structure by setting 0 =, f g = g dom(f) dom(g), f g = f g dom(f) dom(g), f \ g = f dom(f) dom(g), f g = f g, where is set-theoretic difference, and the set-theoretic oerations on the righthand sides act on f and g viewed as functional relations. We generalize this construction to algebras which are not necessarily right-handed, because we will need one in Section 4.4. Given a subset D X, let P(X, Y ) D = {f : X Y dom(f) = D} be the set of those artial mas X Y whose domain is D. Suose we are given for each D X a binary oeration D on P(X, Y ) D. We say that the family { D D X} is coherent when it commutes with restrictions: for all E D X and f, g P(X, Y ) D it holds (f D g) E = f E E g E. (1) We usually omit the subscrit from D and write just. Theorem 2.1 Let ( D ) D X be a coherent family of rectangular bands. Then P(X, Y ) is a skew algebra for the following oerations, defined for f, g P(X, Y ) with dom(f) = F and dom(g) = G: 0 =, f g = f F G g F G, f g = (f F G ) (g G F ) (g f), f \ g = f F G, f g = f g. Proof. Let f, g, h P(X, Y ) be artial mas with domains dom(f) = F, dom(g) = G, and dom(h) = H, resectively. Observe that the oerations defined in the statement of the theorem all commute with restrictions, for examle (f g) D = f (F G) D g (G F ) D (f D F G g D F G ) = f (D F ) (D G) g (D G) (D F ) (f D F G g D F G ) = f D g D. Thus a good strategy for checking an identity is to do it by arts as follows. To check u = v it suffices to check u X = v X and u Y = v Y searately, rovided that dom(u) = dom(v) = X Y. Of course, this only works if X and Y are 4

5 suitably chosen so that the restrictions u X, u Y, u Y, and v Y simlify when we ush the restrictions by X and Y inwards. The following roerties are easily verified: 0 is neutral for, idemotency of and, associativity of. That comutes greatest lower bounds holds because the natural artial order on P(X, Y ) is subset inclusion of functions viewed as functional relations. It remains to check associativity of, meet distributivity, and the roerties of \. Associativity of is checked by arts. The domain of f (g h) and (f g) h is F G H, which is covered by the arts (F G) H, (F H) G, (G H) F, and F G H. On the first art we get and ((f g) h) (F G) H = (f F H g G H ) h = f F H g G H. (f (g h)) (F G) H = f F H (g G H h ) = f F H g G H. On (F H) G and (G H) F the calculation is similar, while on F G H both meets and joins turn into and the identity follows as well. Next we check that meet distributivity holds. The domain of f (g h) and (f g) (f h) is F (G H). It is covered by the arts (F G) H, (F H) G, and F G H. On the first art we get and (f (g h)) (F G) H = f (F G) H (g (F G) H h ) = (f g) (f h)) (F G) H = (f (F G) H g (F G) H ) (f (F G) H h ) = f (F G) H g (F G) H (f (F G) H g (F G) H ) 0 = f (F G) H g (F G) H. The calculation on (F H) G is similar, and on F G H it again trivializes because all oerations become. For the other half of meet distributivity, namely (f g) h = (f h) (g h) we use the arts (F H) G, (G H) F, and F G H. Finaly, \ satisfies the axioms of relative comlementation because and (f \ g) (f g f) = f F G f F G = = 0 (f \ g) (f g f) = f F G f F G = f F G f F G = f. 2.2 Boolean saces and étale mas We start by recaling several standard toological notions. A sace is zerodimensional if its cloens (sets which are both oen and closed) form a toological base. A Stone sace is a comact zero-dimensional Hausdorff sace, while 5

6 a Boolean sace is a locally comact zero-dimensional Hausdorff sace. We call a set which is comact and oen a coen. In a Boolean sace the coens form tological base. A artial ma f : X Y is said to be continuous when it is continuous as a ma defined on the subset dom(f) X with the induced toology. Unless noted otherwise, the domain of definition dom(f) is always going to be an oen subset of X. A continuous ma is roer if its inverse image ma takes comact subsets to comact subsets, while a artial continuous ma with an oen domain of definition is roer when the inverse image f 1 (K) of a comact subset K Y is comact in dom(f), or equivalently in X. An étale ma : E B, also known as local homeomorhism, is a continuous ma for which E has an oen cover such that for each U in the cover the restriction U : U (U) is a homeomorhism onto the image, and (U) is oen in B. We call E the total sace and B the base of the étale ma. The fiber above x B is the subsace E x = {y E (y) = x}. A section of is a continuous ma s : U E defined on a subset U B, usually oen, such that s = id U. Étale mas with a common base B form a category, even a toos, in which morhisms are commutative triangles f E E B where f is a continuous ma. It follows that f is an étale ma, see for examle [7, II.6]. One consequence of this is that a section s : U E of an étale ma : E B defined on an oen subset U B is itself an étale ma (because the inclusion U E is an étale ma). In articular, the image s(u) is oen in E and (images of) oen sections form a base for E. A coen section of : E B is a section s : U E defined on a coen subset U B. Its image s(u) is not only oen but also comact in E, and the restriction s(u) is a homeomorhism from s(u) onto U. Conversely, if S E is coen and S : S (S) is a homeomorhism onto (S) then ( S ) 1 : (S) S is a coen section. This is so because étale mas are oen. We therefore have two views of coen sections: as sections defined on coen subsets, and as those coen subsets of the total sace which cover each oint in the base at most once. In Section 4.4 we will need to know how to comute equalizers and coequalizers in the category of étale mas over a given base. Proosition 2.2 Let f and g be morhisms of étale mas, E f g E B The equalizer and coequalizer of f and g in the category of étale mas with base B are comuted as in the category of toological saces. Moreover, the quotient ma from E to the coequalizer is oen. 6

7 Proof. In the category of toological saces the equalizer of f and g is the subsace I = {x E f(x) = g(x)} with the subsace inclusion i : I E. For this to be an equalizer in the category of étale mas, i : I B must be étale, which is the case because I is an oen subsace of E. To see this, consider any x I. Because f and g are étale mas there is an oen section U E containing x such that f U : U f(u) and g U : U g(u) are homeomorhisms onto oen subsets of E. Thus the intersection f(u) g(u) is oen, from which it follows that U f 1 (g(u)) g 1 (f(u)) is an oen neighborhood of x contained in I. The coequalizer of f and g is comuted in toological saces as the quotient sace Q = E /R where R E E is the least equivalence relation generated by the relation S = {(f(x), g(x)) E E x E}. Because S is an oen subset of the fibered roduct E B E so is R, from which it follows that the canonical quotient ma q : E Q is oen. Furthermore, because f = = g the ma factors through q as = r q. To comlete the roof, we need to show that r is étale, but this is easy because we already know that q is oen. 3 Duality for commutative algebras Before embarking on duality for skew algebras we review the familiar commutative case. The sectrum St(A) of a Boolean algebra A is the Stone sace whose oints are the rime ideals of A. The elements a A corresond to the basic cloen sets N a = {P St(A) a P }. A homomorhism f : A A between Boolean algebras induces a continuous ma f : St(A ) St(A) that mas a rime ideal P to its reimage f (P ) = f 1 (P ). In the other direction the duality mas a Stone sace to the Boolean algebra of its cloen subsets, with the exected oerations of intersection and union. A short ath to Stone duality for generalized Boolean algebras goes through the observation that the category GBA of generalized Boolean algebras is equivalent to the slice category BA/2 of Boolean algebras over the initial algebra 2. By Stone duality for Boolean algebras GBA is then dual to ointed Stone saces and continuous mas which reserve the chosen oint. For our uroses it is more convenient to take yet another equivalent category, namely Stone saces without one oint, which are recisely the Boolean saces, and suitable artial mas between them. Let us describe the duality exlicitly. Starting from a generalized Boolean algebra A, we construct its sectrum St(A) as the Boolean sace of rime ideals. An element a A corresonds to the basic coen set N a = {P St(A) a P }. A homomorhism f : A A between generalized Boolean algebras induces a artial ma f : St(A ) St(A) that mas a rime ideal P to its reimage f (P ), rovided the reimage is not all of A. The domain of definition of f is oen, for if a f (P ) then f is defined on N f(a). The fact that the inverse image f 1 (N a ) of a basic coen is the basic coen N f(a) imlies that f is both continuous and roer. Indeed, f is roer because the inverse image f 1 (K) of a comact subset K St(A) is a closed subset of the comact set f 1 (N a1 ) f 1 (N an ) where N a1,..., N an is some finite cover of K by basic coen sets. In summary, the category of generalized Boolean algebras 7

8 is equivalent to the category of Boolean saces and roer continuous artial mas with oen domains of definition. 4 Duality for skew algebras Any attemt at extension of Stone duality naturally leads to consideration of rime ideals. Since the one-oint sace 1 corresonds to the initial Boolean algebra 2, a oint 1 X on the toological side corresonds to homomorhisms A 2 on the algebraic side. However, since such homomorhisms factor through the commutative reflection A A/D, they give us insufficient information about the non-commutative structure of A. We should therefore exect that on the toological side we have to look for structures that can be rich even though they have few global oints, while on the algebraic side we cannot afford to use ideals exclusively, but must also consider congruence relations. In the case of a commutative algebra the congruence relation generated by a rime ideal has just two equivalence classes. In contrast, the least congruence relation θ P whose zero-class contains the rime ideal P A in a skew algebra A generally has many equivalence classes, which ought to be accounted for on the toological side of duality. The following result of Bignall and Leech [1] characterizes the congruence relations generated by ideals in a skew algebra. Lemma 4.1 (Bignall & Leech) For an ideal I A in a skew algebra A let θ I be the least congruence relation on A whose zero-class contains I. Then for all x, y A, x θ I y if, and only if, (x \ (x y)) (y \ (x y)) I. In fact, the zero-class of θ I equals I. Consequently, the ideals of a skew algebra are in bijective corresondence with congruence relations. A consequence of Lemma 4.1 is that x θ f 1 (I) y is equivalent to f(x) θ I f(y), for any skew algebra homomorhism f : A A and any ideal I A. From this we obtain the following roerties of rime ideals. Lemma 4.2 Let P be a rime ideal in a skew algebra A and let x, y A. Then: 1. x P or y \ x P. 2. If x y and x P then x θ P y. 3. If x y then x P is equivalent to y θ P y \ x. Proof. 1. We have x (y \ x) = 0 P. Because P is a rime ideal it follows that x P or y \ x P. 2. We need to show that (x \ (x y)) (y \ (x y)) P. Since x y it follows that x y = x and thus we only need y \ x P, which follows from the first statement of the lemma. 3. The element x = y \ x is the comlement of x in the Boolean algebra y A y. Hence x y and y \ x = x. Thus Lemma 4.1 imlies that y θ P x is equivalent to (y \ x ) (x \ x ) = x P. 8

9 4.1 From algebras to saces Following our own advice that the toological side of duality should account for the equivalence classes of congruences, we define the skew sectrum Sk(A) of a skew algebra A to be the sace whose oints are airs (P, e), where P is a rime ideal in A and e is a non-zero equivalence class of θ P. Since every non-zero equivalence class equals [t] θp for some t P, a general element of the skew sectrum may be written as (P, [t] θp ). We write just [t] P. Thus, as a set the skew sectrum is Sk(A) = {[t] P P rime ideal in A and t P }, where [t] P = [u] Q when P = Q and t θ P u. The toology of Sk(A) is the one whose basic oen sets are of the form, for a A, M a = {[a] P a P }. These really form a basis because they are closed under intersections. Lemma 4.3 Let a, b A. Then M a M b = M a b. to Proof. For a rime ideal P and t A the statement [t] P M a M b amounts while [t] P M a b means (a P ) (b P ) (t θ P a) (t θ P b), (2) (a b P ) (t θ P a b). (3) If (2) holds then a b θ P a a = a θ P t which roves (3). To rove the converse, suose (3) holds. Because a b P, a b a and a b b, Lemma 4.2 imlies a θ P a b θ P b which suffices for (2). The skew sectrum on its own contains too little information to act as the dual. For examle, both Sk(2 2) and Sk(3) are the discrete sace on two oints. Recall that 3 is the right-handed skew algebra whose set of elements is {0, 1, 2} and whose D-classes are {0} and {1, 2}. One art of the missing information is rovided by the ma q A : Sk(A) St(A) defined by q A ([t] P ) = P/D, where we used the shorthand St(A) = St(A/D). Proosition 4.4 The ma q A : Sk(A) St(A) is onto and étale. Proof. Because rime ideals are non-trivial the ma q A is onto. To show that q A is continuous, we rove q 1 A (N a) = b a M b, (4) where we used the shorthand N a = N Da. For one inclusion, observe that b a imlies D b D a and hence q A (M b ) = N b N a. For the other inclusion, 9

10 suose q A ([t] P ) N a. Then t a t P because a P and t P, and by Lemma 4.2 we get t θ P t a t from which [t] P M t a t follows. Finally, q A is étale because its restriction to a basic oen set M a is a (continuous) bijection onto the basic oen set N a. Corollary 4.5 The skew sectrum Sk(A) is a Boolean sace. Proof. Local comactness and zero-dimensionality are lifted from St(A) to Sk(A) by the étale ma q A. To see that Sk(A) is Hausdorff, let [s] P and [t] Q in Sk(A) be two distinct oints in Sk(A). Consider the case P = Q. We claim that M s\(s t) and M t\(s t) are disjoint and are neighborhoods of s and t, resectively. Disjointness follows by Lemma 4.3 from the fact that (t \ (s t)) (s \ (s t)) = 0. From (t θ P s) we conclude by Lemma 4.1 that (s \ (s t)) (t \ (s t)) P, thus s \ (s t) P or t \ (s t) P. In either case s t P by (1) of Lemma 4.2, therefore s θ P s \ (s t) by (3) of Lemma 4.2, which roves [s] P M s\(s t), as claimed. We similarly show that [t] P M t\(s t). Consider the case P Q. Because St(A) is Hausdorff there exist disjoint basic oen nehigborhoods N a and N b of q A (P ) and q A (Q), resectively. Thus q 1 A (N a) and q 1 A (N b) are disjoint and are oen neighborhoods of [s] P and [t] Q, resectively. Because Boolean saces are zero-dimensional, the étale ma q A : Sk(A) St(A) has more sections that one would normally exect. Proosition 4.6 The étale ma q A : Sk(A) St(A) has a section above every coen set, assing through a rescribed oint above the coen set. Proof. More recisely, the roosition claims that given a coen set U St(A) and a oint x Sk(A) such that q A (x) U, there is a coen section above U containing x. For the roof we only need surjectivity of q A and zerodimensionality of St(A). By surjectivity the coen set U can be decomosed into airwise disjoint coen sets U 1,..., U n, each of which has a section s i. The sections can be glued together into a section s above U. To make sure that s asses through a rescribed oint x, just change it suitably on a small coen neighborhood of q A (x). Proosition 4.7 The coen sections of q A : Sk(A) St(A) are recisely the sets M a with a A. Proof. We already know that each M a is a coen section because q A mas it homeomorhically onto the coen set N a. Conversely, let V be a coen section in Sk(A). It is a finite union of basic coen sets V = M a1 M an. We show that V = M a1 a n. If n = 1 there is nothing to rove. Consider the case n = 2. The fact that V = M a1 M a2 is a section amounts to: for all rime ideals P in A, if a 1 P and a 2 P then a θ P b. Let us show that this imlies M a1 M a1 a 2. If a 1 P then a 1 a 2 P. Either a 2 P or a 2 P. In the first case a 2 θ P 0 and so a 1 θ P a 1 0θ P a 1 a 2. In the second case, 10

11 a 2 θ P a 1 and so a 1 θ P a 1 a 1 θ P a 1 a 2. We similarly show that M a2 M a1 a 2, from which we get M a1 M a2 M a1 a 2. To comlete the case n = 2 we still have to rove M a1 a 2 M a1 M a2. Suose a 1 a 2 P. Then either a 1 P or a 2 P. We consider the case a 1 P, the other one is similar. Either a 2 P or a 2 P. In the first case a 2 θ P 0 so a 1 θ P a 1 0 θ P a 1 a 2. In the second case the assumtion gives us a 1 θ P a 2 and so a 1 θ P a 1 a 1 θ P a 1 a 2. The cases n > 2 follow by reeated use of the case n = 2. Later on we will need to know how the ma a M a interacts with the oerations of A. For this urose we define the saturation oeration σ A (U) = q 1 A (q A(U)). Since q A is oen, the saturation of a coen is a cloen. Proosition 4.8 The assignment a M a is an order-isomorhism from a skew algebra A, ordered by the natural artial order, onto the coen sections of Sk(A), ordered by subset inclusion, satisfying the identities M a b a = σ A (M b ) M a and M a b a = M a (M b \ σ A (M a )). Proof. If M a = M b then a θ P b for all rime ideals P. Thus for any rime ideal P (a \ (a b)) (b \ (a b)) P, and so both a \ (a b) and b \ (a b) belong to P. Because the intersection of all rime ideals is {0}, it follows that a \ (a b) = 0 = b \ (a b). Therefore, D a = D a b = D b, which is only ossible if a = b. It follows that the assignment a M a is injective, while its surjectivity follows from Proosition 4.7. That a M a is an order isomorhism follows from the following chain of equivalences, where we use Lemma 4.3 and injectivity of a M a in the second and third ste, resectively: a b a b = a M a b = M a M a M b = M a M a M b. It remains to rove the two identites. For the first one, let [a b a] P M a b a. Then a b a P which imlies both b P (and thus [a b a] P σ A (M b )) and a P. We have a b a a and a b a θ P a follows by Lemma 4.2. Hence [a b a] P σ A (M b ) M a. To rove the converse, let [a] P σ A (M b ) M a. Hence a P and b P. So a b a P, a b a a and thus a b a θ P a by Lemma 4.2. To rove the second equality first assume that [a b a] P M a b a. Hence a b a P. If a P then a θ P a b a by Lemma 4.2 because always a a b a, and [a b a] P M a follows. If a P then b P follows from a b a P. Thus a b a θ P 0 b 0 θ P b and [a b a] P M b \σ A (M a ) follows. Finally, assume that [s] P M a (M b \σ A (M a )). Then either a P and [s] P = [a] P, or a P, b P and [s] P = [b] P. In the first case it follows that 11

12 a b a P and thus a θ P a b a by Lemma 4.2. Therefore [s] P = [a] P M a b a. In the second case it follows that a b a P and a b aθ P 0 b 0θ P b. Again, [s] P = [b] P M a b a. The attentive reader will oint out that the étale ma q A : Sk(A) St(A) cannot ossibly be the toological dual of the skew algebra A because A gives the same étale ma as its oosite algebra (in which the oerations and are the mirror version of those in A). Indeed, the étale ma rovides sufficient information only when A is right-handed (or left-handed), as will be shown in Section 4.3. We shall consider the general case in Section 4.4. Let f : A A be a homomorhism of skew algebras. As we exlained in Section 3, the induced homomorhism A/D A /D is dual to a roer artial ma f : St(A ) St(A) with an oen domain, which mas rime ideals to their inverse images, when defined. There is also a ma f : Sk(A ) Sk(A) of the same kind between the skew sectra. It is characterized by the requirement f ([f(a)] P ) = [a] f (P ), which uniquely determines the value of f, when defined, because a θ f (P ) b is equivalent to f(a) θ P f(b). If f ([f(a)] P ) is defined then f(a) P, hence f is defined on the basic coen set M f(a), which shows that the domain of definition of f is oen. Next, f is continuous and roer because f 1 (M a ) = M f(a) for all a A. The square Sk(A) f Sk(A ) (5) q A q A St(A) St(A ) f commutes. It is easy to see that the values of q A f and f q B coincide whenever they are both defined, so we only show that they have the same domain of definition. If f (P ) is defined then there is a A such that f(a) P, in which case f is defined at [f(a)] P and q B ([f(a)] P ) = P. Conversely, if f ([f(a)] P is defined then f (q B ([f(a)] P )) = f (P ) is defined because f(a) P. Lemma 4.9 The ma f is a bijection on fibers, i.e., given any P dom(f ), f mas Sk(B) P dom(f ) bijectively onto the fiber Sk(A) f (P ). Proof. Consider any P dom(f ). The Lemma states that f is a bijective ma between the sets {[f(a)] P f(a) P } and {[a] f (P ) a f (P )}. For injectivity, suose f ([f(a)] P ) = f ([f(a )] P ) where f(a) P and f(a ) P. By the definition of f it follows that [a] f (P ) = [a ] f (P ), which is equivalent to a θ f (P ) a, and hence f(a) θ P f(a ). This establishes the fact that [f(a)] P = [f(a )] P. For surjectivity, ick any [a] f (P ) where a f (P ). It follows that f(a) P, so f is defined at [f(a)] P and mas it to [a] f (P ). In view of the revious lemma one might contemlate turning f around, so that instead of a artial ma which is bijective on fibers we would get a total ma 12

13 which is injective on fibers. The trouble is that the inverted ma need not be continuous, so the toological nature of f must be obscured by a more comlex condition. 4.2 From saces to algebras As we already indicated, the original algebra A cannot always be reconstructed from q A : Sk(A) St(A). However, if A is right-handed then q A : Sk(A) St(A) carries all the information needed, so we consider this case first. We call a surjective étale ma : E B between Boolean saces a skew Boolean sace. The corresonding right-handed skew algebra A consists of coen sections of, i.e., an element of A is a coen subset S E such that S is injective. To describe the right-handed skew structure on A, recall the saturation oeration σ (S) = 1 ((S)), and define for S, R A 0 =, S R = σ (S) R, S R = S (R σ (S)), S \ R = S σ (R), S R = S R. It is clear that these oerations ma back into A. For examle, S R is a section, and it is coen because it is the intersection of two coen subsets of the Hausdorff sace E. It is not difficult to check that the above oerations form a skew algebra with bare hands. An alternative, more elegant way of establishing the skew structure is to view the elements of A as artial mas s, r : B E and exhibit A as a subalgebra of the right-handed skew algebra P(B, E) as described in Section 2.1. The construction of the skew algebra A induces the usual construction of a generalized Boolean algebra via the lattice reflection, as follows. Proosition 4.10 Let : E B be a skew Boolean sace and let B be the Boolean algebra of coen subsets of B. The ma A B defined by S (S) is the lattice reflection of A. Proof. By Proosition 4.6 the ma S (S) is surjective, and it is easily seen to be a lattice homomorhism. Thus we only have to check (S) = (R) is equivalent to S D R, where S and R are coen sections of. This follows from S R S R S = S σ (S) σ (R) S = S S σ (R) (S) (R). We turn attention to morhisms next. We already know that a homomorhism f : A A of skew algebras induces a commutative square (5) in which the horizontal artial mas are roer, continuous and have oen domains of 13

14 definition, and that the to ma is bijective on fibers by Lemma 4.9. So we define a morhism (g, h) between skew Boolean saces : E B and : E B to be a commutative diagram E g E (6) B h in which g and h are roer continuous artial mas with oen domains of definition. Furthermore, we require that g is a bijection on fibers, in the sense that g mas E x dom(g) bijectively onto E h(x) for every x dom(h). Lemma 4.11 If (g, h) is a morhism between skew Boolean saces : E B and : E B then (g 1 (S)) = h 1 ( (S)) for every coen section S in E. Proof. Let S be a coen section in E. If y (g 1 (S)) then there exists x g 1 (S) such that y = (x). Then h(y) = h((x)) = (g(x)) (S), so that y h 1 ( (S)) and (g 1 (S)) h 1 ( (S)) follows. On the other hand, if h(y) (S) then h(y) = (z) for some z S. Because g is surjective on fibers there exists x E y with the roerty g(x) = z. Now, y = (x) (g 1 (S)) and we get h 1 ( (S)) (g 1 (S)). We would like to construct a corresonding homomorhism (g, h) : A A. For a coen section S E, the inverse image g 1 (S) is coen in E because g is continuous and roer, and it is a section because g is injective on fibers. Thus we may define (g, h) (S) = g 1 (S). Let us show that g 1 commutes with the saturation oerations σ and σ. If S is a coen section in E, then (g 1 (S)) = h 1 ( (S)) by Lemma Therefore g 1 (σ (S)) = 1 (h 1 ( (S)) = 1 ((g 1 (S)) = σ (g 1 (S)), as claimed. The oerations on A and A are defined in terms of basic settheoretic oerations and the saturation mas σ and σ. Because g 1 commutes with all of them, (g, h) is an algebra homomorhism. The following is the counterart of Proosition 4.10 for morhisms. Proosition 4.12 Let (g, h) be a morhism between skew Boolean saces : E B and : E B. Its lattice reflection (g, h) /D : A /D A /D is isomorhic to the lattice homomorhism h : (B ) B defined by h (S) = h 1 (S). B Proof. By Proosition 4.10 the vertical mas in the diagram A (g,h) A B h (B ) 14

15 are lattice reflections. Because the right-hand vertical arrow is ei it suffices to show that the diagram commutes, which is just Lemma Duality for right-handed algebras Let us take stock of what we have done so far, in terms of category theory. On the algebraic side we have the category SkAlg of skew algebras and homomorhisms, as well as its reflective full subcategory SkAlg R on right-handed skew algebras. On the toological side we have the category SkS of skew Boolean saces. A morhism (g, h) : between skew Boolean saces : E B and : E B is a commutative square (6) with roer continuous artial mas g and h whose domains of definition are oen, and the to ma g is bijective on fibers. Admittedly, the morhisms in SkS are not very nice, but in Section 5 we show that they decomose nicely into artial identities and ullbacks. In Section 4.1 we defined a functor S : SkAlg o SkS which assigns to each skew algebra A a skew Boolean sace S(A) = q A : Sk(A) St(A). The functor takes a homomorhism f : A B to the corresonding morhism S(f) = (f, f ) : S(B) S(A). In Section 4.2 we defined a functor A : SkS o SkAlg R which mas an étale ma : E B between Boolean saces to a righthanded skew algebra A() = A, and a morhism (g, h) : as in (6) to a homomorhism A(g, h) = (g, h) : A( ) A(). We now work towards showing that S restricted to SkAlg R and A form a duality. For a skew algebra A define the ma φ A : A A(S(A)) by φ A (a) = M a. That φ A is an isomorhism of right-handed skew algebras follows from Lemma 4.3 and Proosition 4.8. By the lemma φ A reserves, it obviously reserves 0, and by the roosition it is a bijection which reserves the right-handed skew oerations and : M a b = M b a b = σ A (M a ) M b = M a M b M a b = M a b a = M a (M b \ σ A (M a )) = M a M b. Naturality of φ amounts to the identity f 1 (M a ) = M f(a), where f : A B is a homomorhism and a A. After unraveling the definition of f we see that the set on the left-hand side consists of elements [f(a)] P with f(a) P, which is just the descrition of the right-hand side. We have shown that φ is a natural isomorhism between the identity and A S. To establish the equivalence we also need an isomorhism ψ between a skew Boolean sace : E B and q A : Sk(A ) St(A ), natural in. It consists of 15

16 two homeomorhism (ψ ) = h and (ψ ) = g for which the following diagram commutes: g E Sk(A ) (7) B h q A St(A ) Because the to ma determines the bottom one, we consider g first. By Proosition 4.8 the ma S M S is an order isomorhism between A and the coen sections of Sk(A ). But since A is the set of coen sections of E, and the natural artial order in A coincides with the subset relation in E, the ma S M S mas the basis for E isomorhically onto the basis for Sk(A ). Consequently, the toologies of E and Sk(A ) are isomorhic as osets, too, and because E and Sk(A ) are sober saces they are homeomorhic. Exlicitly, the homeomorhism g : E Sk(A ) induced by the isomorhism S M S takes a oint y E to the unique oint g(y) Sk(A ) satisfying, for all coen sections S in E, y S g(y) M S. Similarly, the homeomorhism h : B St(A ) is characterized by the requirement, for all coen sections S in E, x (S) h(x) N S. It is not hard to verify that h(x) = {D R R A x (R)} and that g(x) = [S] h(x) for any S A such that x (S). We verify that (7) commutes by checking that the corresonding square of inverse image mas does. On one hand, starting with a coen section N R in St(A ), we have 1 (h 1 (N R )) = 1 (R) = {S (S) R}, where S in the union ranges over coen sections in E. On the other hand, g 1 (q 1 A (N R )) = g 1 ( {M S (S) R}) = {g 1 (M S ) (S) R} = {S (S) R}, where S again ranges over coen sections in E. Naturality of ψ involves the commutativity of a cube which we refer not to draw because six of its faces commute by definition and the two remaining faces are B (ψ) St(A ) E (ψ) Sk(A ) h (g,h) B (ψ ) St(A ) g E (ψ ) Sk(A ) ((g,h) ) 16

17 where : E B and : E B are skew Boolean saces and (g, h) is a morhism from to. We check commutativity of the right-hand square, the other one is similar. Again, we verify that the corresonding square of inverse image mas commutes. For any coen section M S in the lower-right corner we have and (ψ ) 1 g 1 ((ψ ) 1 (M S )) = g 1 (S ) (((g, h) ) 1 (M S )) = (ψ ) 1 (M (g,h) (S )) = (g, h) (S ) = g 1 (S ). We have roved the following main theorem. Theorem 4.13 The category of right-handed skew Boolean algebras with intersections is dual to the category of skew Boolean saces. Clearly, there is also duality between left-handed skew algebras and skew Boolean saces, simly because the categories of left-handed and the right-handed skew algebras are isomorhic. 4.4 Duality for skew algebras To see what is needed for duality in the case of a general skew algebra, consider what haens when we take a skew algebra A to its skew Boolean sace q A : Sk(A) St(A), and then the sace to the right-handed skew algebra A qa. By Proosition 4.8 the elements and the natural artial order do not change (u to isomorhism), but the oerations do. The new ones are exressed in terms of the original ones as x y = y x y and x y = x y x. If we want to recover and from and we need to break the symmetry that is resent in and by keeing around enough information about the original oerations of A. Recall that a rectangular band is a set with an oeration which is idemotent, associative and it satisfies the rectangular identity. We can similarly define rectangular bands in any category with finite roducts. For examle, a rectangular band in the category of étale mas over a given base sace B is an étale ma : E B together with a continuous ma : E B E E over B which satisfies the required identities fiber-wise. The skew Boolean sace q A : Sk(A) St(A) carries the structure of a rectangular band whose oeration : Sk(A) St(A) Sk(A) Sk(A) is defined by [a] P [b] P = [a b] P. Idemotency and associativity of follow immediately from the corresonding roerties of and the fact that θ P is a congruence. To see that the rectangular identity is satisfied, let a, b, c P. Because (A, ) forms a normal band, namely it satisfies the identity x y z w = x z y w, it follows that a b c a c and so a b c θ P a c by Lemma 4.2, which is equivalent to [a] P [b] P [c] P = [a] P [c] P. 17

18 We have to check that is continuous. Let ([a] P, [b] P ) be any oint of the domain of and suose [a b] P M c for some c A. We seek an oen neighborhood of ([a] P, [b] P ) which is maed into M c by. Because basic oen subsets of Sk(A) St(A) Sk(A) are of the form {([u] Q, [v] Q ) Q A rime ideal, u Q and v Q}, it suffices to find u, v A such that a θ P u and b θ P v, and for all rime ideals Q A, if u Q and v Q then u v θ Q c. We claim that u = (a b a) (c a) v = (b a b) (b c) satisfy these conditions. We note that u v c because u v = ((a b a) (c a)) ((b a b) (b c)) (c a) (b c) c, where we used the fact that is comatible with the natural artial order, which is the case because (A, ) is a normal band. Next observe that a b a θ P c a because [a b] P M c, hence u = (a b a) (c a) θ P (a b a) (a b a) = a b a θ P a, where the last ste follows from Lemma 4.2 and a, b P. We similarly show that v θ P b. If Q A is a rime ideal such that u Q and v Q, then u v Q and since u v c we get u v θ Q c, again by Lemma 4.2. The rectangular band structure on A is recisely what is needed for duality in the general case. Theorem 4.14 The category of skew Boolean algebras with intersections is dual to the category of rectangular skew Boolean saces. Proof. By rectangular skew Boolean sace ( : E B, ) we mean a rectangular band in the category of surjective étale mas over B. More recisely, it is a skew Boolean sace : E B together with a (not necessarily roer) continuous ma over B E B E E B which makes every fiber E x into a rectangular band. Notice that in general a rectangular skew Boolean sace is not a rectangular band in the category of skew Boolean saces because need not be roer. A morhism between ( : E B, ) and ( : E B, ) is a morhism of skew Boolean saces (g, h) : which commutes with the oerations on its domain of definition: dom(g) B dom(g) dom(g) g g g E B E E 18

19 Note that the commutativity of the square imlies that dom(g) is closed under, so dom(g) E x is a rectangular sub-band of E x at every x B. And since g is bijective on fibers, g x : dom(g) E x E h(x) is an isomorhism of rectangular bands for every x dom(h). We denote the category of rectangular skew Boolean saces and their morhisms by SkRS. The duality is witnessed by a air of contravariant functors S : SkAlg o SkRS and A : SkRS o SkAlg. The functor S mas a skew algebra A to the rectangular skew Boolean sace S(A) = ( : Sk(A) St(A), ), as described above. It takes a morhism f : A A to the morhism of skew Boolean saces S(f) = (f, f ), which commutes with because f commutes with. The functor A mas a rectangular skew Boolean sace ( : E B, ) to the skew algebra A(, ) whose elements are the coen sections of and the oerations are defined as follows: 0 =, S R = (S σ (R)) (σ (S) R), S R = (S σ (R)) (R σ (S)) (R S), S \ R = S σ (R), S R = S R, These form a skew algebra because they are restrictions of the oerations from Theorem 2.1. The functor A mas a morhism (g, h) : ( : E B, ) ( : E B, ) to the homomorhism A(g, h) = (g, h). We need to verify that A(g, h) reserves. Recall that (g, h) is just g 1 acting on coen sections. In Section 4.2 we checked that g 1 commutes with the saturation oerations. Because for every x dom(h) the ma g x : E x dom(g) E h(x) is an isomorhism of rectangular bands, it is not hard to see that g 1 commutes with. Therefore, g 1 commutes with all the oerations used to define the oerations on G(, ) and G(, ), so it is a homomorhism of skew algebras. It remains to be checked that S A and A S are naturally isomorhic to identity functors. Luckily, we can reuse a great deal of verification of duality for right-handed algebras from Section 4.3. The natural isomorhism φ from the identity to A S is defined as in the right-handed case: for a skew algebra A set φ A (a) = M a. Thus we already know that it is a bijection which reserves intersections and relative comlements, but we still have to check that it reserves meets and joins. It reserves meets because M a M b = M a b a M b a b = M a b where we used Proosition 4.8 in the first ste and the fact that [a b a] P [b a b] P = [a b] P in the last ste. With the hel of Proosition 4.8 it is not hard to verify that whenever a and b commute then M a b = M a M b = M b a, so φ A reserves commuting joins. But since for arbitrary a and b their join can be exressed as a commuting join a b = (a \ b) (b \ a) (b a), and we already know that φ A reserves \ and, it follows that φ A reserves joins. Naturality of φ A is checked as in the right-handed case. 19

20 The natural isomorhism ψ from the identity to S A is defined as in the right-handed case. Given a rectangular skew Boolean sace ( : E B, ), let ψ, be the morhism consisting of the two homeomorhisms (ψ, ) = h and (ψ, ) = g from diagram (7). All that we need to check in addition to what was already checked for ψ in Section 4.3 is that g reserves the rectangular band structure. For any b B, x, y E b and T A we have g(x y) M T x y T. On the other hand, if g(x) = [S] h(b) and g(y) = [R] h(b) then g(x) g(y) M T [S R] h(b) M T S R θ h(b) T and b (T ) x y T. We see that g(x) g(y) and g(x y) have the same neighborhoods, therefore they are equal. It may be argued that our duality has not gone all the way from algebra to geometry because a rectangular skew Boolean sace still carries the algebraic structure of a rectangular band. However, this is not really an honest algebraic structure, as can be susected from the fact that the category of non-emty rectangular bands is equivalent to the category of airs of sets. The equivalence takes a rectangular band (A, ) to the air of sets (A/R, A/L) where A/R and A/L are the quotients of A by Green s relations R and L, resectively. In the other direction, a air of sets (X, Y ) is maed to the rectangular band X Y with the oeration (x 1, y 1 ) (x 2, y 2 ) = (x 1, y 2 ). The analogous decomosition of rectangular skew Boolean saces yields the following variant of duality for skew algebras. Theorem 4.15 The category of skew Boolean algebras with intersections is dual to the category of airs of skew Boolean saces with common base. Proof. A air of skew Boolean saces with a common base is a diagram L E L B R E R (8) where L : E L B and R : E R B are skew Boolean saces. A morhism is a commutative diagram L E L B R E R (9) g L E L L h B R E R g R in which the left- and right-hand square are morhisms of skew Boolean saces (in the vertical direction). The diagrams are comosed in the obvious way and we clearly get a category. We establish the duality by showing that the category of airs of skew Boolean saces with common base is equivalent to the category of rectangular skew Boolean saces. The idea is to have equivalence functors work at the 20

21 level of fibers in the same way as the equivalence of non-emty rectangular bands and airs of sets. To convert a air of skew Boolean saces (8) into a rectangular Boolean sace we form the ullback E E R R E L L B to obtain a skew Boolean sace : E B. Concretely, the fiber E x over x B consists of airs (u, v) E L E R such that L (u) = x = R (v), and (u, v) = L (u) = R (v). The rectangular band oeration on : E B defined by (u 1, v 1 ) (u 2, v 2 ) = (u 1, v 2 ), obviously makes the fibers into rectangular bands. A morhism (9) corresonds to the morhism E g E B h B where g is the artial ma with domain dom(g L ) B dom(g R ) defined by g(x, y) = (g L (x), g R (y)). It clearly reserves. In the oosite direction we start with a rectangular skew Boolean sace ( : E B, ) and form a air of skew Boolean saces with a common base as follows. First construct the fiber-wise Green s relation L on E as the equalizer l id E B E L E B E E B E B B in the toos of étale mas with base B. In the above diagram : E B E E is the oeration associated with by x y = y x. Still in the toos, we form the coequalizer π 1 l L E π 2 l B q L L The quotient E L is Hausdorff because by Proosition 2.2 the ma q L is oen, and a air of oints in E may always be searated by cloen sections. We now have one of the skew Boolean saces L : E L B, and there is an analogous construction of R : E R B. On a single fiber E x over x B the functor just erforms the usual decomosition of the rectangular band E x into its left- and E L 21

22 right-handed factors E x /R x and E x /L x. This is so because by Proosition 2.2 equalizers and coequalizers of étale mas are comuted fiber-wise. A morhism (g, h) from ( : E B, ) to ( : E B, ), dislayed exlicitly as inclusions of the domains of definition and total mas, E dom(g) g E (10) B dom(h) h B corresonds to a morhism between airs of skew Boolean saces as described next. Consider the commutative diagram L (dom(g) B dom(g)) dom(g) B q L L q L (dom(g)) E L where the two arallel arrows into dom(g) are the restrictions of π 1 l and π 2 l to dom(g). By Proosition 2.2 the ma q L is oen, hence q L (dom(g)) is an oen subsace of E L. Moreover, because dom(g) is an oen subsace of E and q L is oen, it is not hard to check that the to row of the diagram is a coequalizer. Because g commutes with, the ma q L g factors through the coequalizer, dom(g) g E q L q L (dom(g)) g L E L q L We have obtained a artial ma g L : E L E L whose domain q L(dom(g)) is an oen subsace of E L. Also q L is roer because g is roer. We similarly obtain the right-handed version g R : E R E R. This gives us the desired morhism E L L R E R g L Bh E L L B R E R g R To see that the two functors just described form an equivalence, we use the fact that fiber-wise they corresond to the equivalence between non-emty rectangular bands and airs of non-emty sets. We omit the details. 5 Variations The morhisms between skew Boolean saces were determined by our taking all homomorhisms on the algebraic side of duality. In this section we consider 22

23 several variants in which the homomorhisms are restricted. We limit attention to the right-handed case, and ask the kind reader who will work out the general case to let us know whether there are any surrises. As a rearation we first show how morhisms of skew Boolean saces decomose into oen inclusions and ullbacks. Lemma 5.1 Suose : E B and : E B are skew Boolean saces and g : E E is a roer continuous ma such that g E E B commutes. If g is bijective on fibers then it is a homeomorhism. Proof. It is obvious that g is a bijection, so we only need to check that it is a closed ma. If K E is comact then the restriction g g 1 (K) : g 1 (K) K is a closed ma because it mas from the comact sace g 1 (K) to the Hausdorff sace E. Therefore, if F E is closed then g(f ) K = g g 1 (K) (F g 1 (K)) is closed in K for every comact K E. Because E is locally comact, it is comactly generated and we may conclude that g(f ) is closed. Lemma 5.2 Suose : E B and : E B are skew Boolean saces and g : E E is a roer continuous ma. A commutative square E g E B h B is a ullback if, and only if, g is bijective on fibers. Proof. It is easy to check that g is bijective on fibers if the square is a ullback. Conversely, suose g is bijective on fibers. We form the ullback of h and and obtain a factorization e, as in the diagram g E e P q E B h B The ma e is roer because g is roer. Indeed, if K P is comact then e 1 (K) is a closed subset of the comact subset g 1 (q(k)), therefore it is comact. Furthermore, e is bijective on fibers because g and q are. By Lemma 5.1 the ma e is a homeomorhism, therefore the outer square is a ullback. 23