aa r X i v : . [ m a t h . L O ] M a y REMARKS ON AFFINE COMPLETE DISTRIBUTIVE LATTICES
DOMINIC VAN DER ZYPEN
Abstract.
We characterise the Priestley spaces corresponding to affine complete boundeddistributive lattices. Moreover we prove that the class of affine complete bounded distributivelattices is closed under products and free products. We show that every (not necessarilybounded) distributive lattice can be embedded in an affine complete one and that Q ∩ [0 , Affine complete lattices A k -ary function f on a bounded distributive lattice L is called compatible if for any congru-ence θ on L and ( a i , b i ) ∈ θ , ( i = 1 , ..., k ) we always have ( f ( a , ..., a k ) , f ( b , ....b k )) ∈ θ . It iseasy to see that the projections pr i : L k → L are compatible. With induction on polynomialcomplexity one shows that every polynomial function is compatible (see [4]). A lattice L iscalled affine complete , if conversely every compatible function on L is a polynomial.G. Gr¨atzer [2] gave an intrinsic characterization of bounded distributive lattices that are affinecomplete: Theorem 1.1. ([2])
A bounded distributive lattice is affine complete if and only if it does notcontain a proper interval that is a Boolean lattice in the induced order.
Note that in particular, no finite bounded distributive lattice L is affine complete: Let x ∈ L be an element distinct from 1. Then x has an upper neighbor, ie, there exists y ∈ L such that[ x, y ] = { x, y } which is isomorphic to the 2-element Boolean lattice. Example 1.2.
The bounded distributive lattices [0 , and [0 , × [0 , are affine complete.Proof. First, take any x < y in [0 , a = x + y ∈ [ x, y ] has no complement a ′ in [ x, y ]: Otherwise we would have a ∧ a ′ = x which would imply a ′ = x , but then a ∨ a ′ = a = y .So [ x, y ] is not Boolean, whence [0 ,
1] has no proper Boolean interval.Secondly, let ( x , x ) < ( y , y ) ∈ [0 , × [0 , x + y , x + y ∈ [( x , x ) , ( y , y )]does not have a complement in [( x , x ) , ( y , y )]. Thus, [0 , × [0 ,
1] has no proper Booleaninterval and is therefore affine complete. (cid:3) The author is grateful for financial support from the Swiss National Science Foundation.AMS subject classification (2000): 06D50 (06D99)Keywords: distributive lattice, affine complete, Priestley spaces. Priestley duality
In [5], Priestley proved that the category D of bounded distributive lattices with (0 , P of compact totally order-disconnectedspaces (henceforth referred to as Priestley spaces ) with order-preserving continuous maps aredually equivalent. (A compact totally order-disconnected space ( X ; τ, ≤ ) is a poset ( X ; ≤ )endowed with a compact topology τ such that, for x , y ∈ X , whenever x y , then thereexists a clopen decreasing set U such that x ∈ U and y U .) The functor D : D → P assignsto each object L of D a Priestley space ( D ( L ); τ ( L ) , ⊆ ), where D ( L ) is the set of all primeideals of L and τ ( L ) is a suitably defined topology (the details of which will not be requiredhere). The functor E : P → D assigns to each Priestley space X the lattice ( E ( X ); ∪ , ∩ , ∅ , X ),where E ( X ) is the set of all clopen decreasing sets of X .Priestley duality therefore provides us with a “dictionary” between the world of boundeddistributive lattices and a certain category of ordered topological spaces. This is interesting inparticular because free products of lattices are “translated” into products of Priestley spaces.We will use this fact for showing that the class of affine complete bounded distributive latticesis closed under free products.3. Affine complete Priestley spaces
The aim of this section is to characterize the Priestley spaces corresponding to affine completedistributive (0,1)-lattices. Such spaces will be called affine complete Priestley spaces . In otherwords, a Priestley space X is affine complete iff E ( X ) is affine complete.The following theorem provides a rather straightforward translation of the algebraic conceptof affine completeness in order-topological terms. Theorem 3.1.
Let X be a Priestley space. Then the following statements are equivalent: (1) E ( X ) is affine complete. (2) If U ⊆ V are clopen down-sets and U = V , then the subposet V \ U of X is not anantichain, i.e. V \ U contains a pair of distinct comparable elements.Proof. (1) = ⇒ (2). Suppose V \ U is an antichain. Let C ∈ [ U, V ] ⊆ E ( X ). Take C ′ = U ∪ ( V \ C ). Claim: C ′ is a clopen down-set of X .It is clear that C ′ is a clopen subset of X since V \ C = V ∩ ( X \ C ). Now, let c ∈ C ′ andassume x < c . Then if c ∈ U , we are done, since U is a down-set. Assume c ∈ V \ U . Since V is a down-set, we get x ∈ V , and the fact that V \ U is an antichain tells us that x cannot be amember of V \ U . Therefore x ∈ U ⊆ C ′ which proves that C ′ is indeed a (clopen) down-set.Moreover, C ′ is the complement of C in [ U, V ], i.e. C ∩ C ′ = U and C ∪ C ′ = V . Because C was arbitrary, we see that [ U, V ] is a proper Boolean interval of E ( X ), whence E ( X ) is notaffine complete.(2) = ⇒ (1). Suppose U ⊆ V are distinct clopen down-sets. By assumption, there areelements x, y ∈ V \ U such that x < y . There is a clopen down-set A with x ∈ A and y / ∈ A .Consider B = ( A ∩ V ) ∪ U . So B ∈ [ U, V ] and y / ∈ B . Now we show that B has no complement EMARKS ON AFFINE COMPLETE DISTRIBUTIVE LATTICES 3 in [
U, V ]: Take any C ∈ [ U, V ] with C ∪ B = V . Then y ∈ C , but since C is a down-set, wehave x ∈ C , thus x ∈ ( B ∩ C ) \ U and B ∩ C = U . So whatever C we pick, C is no complementfor B , i.e. B is not complemented, and consequently [ U, V ] is not Boolean. It follows that noproper interval of E ( X ) is Boolean. (cid:3) We can formulate the above result in a more concise way:
Corollary 3.2.
A Priestley space X is affine complete if and only if each nonempty open setcontains two distinct comparable points.Proof. It follows directly from theorem 3.1 that if each nonempty open set contains twodistinct points that are comparable, then X is affine complete.Conversely, suppose that U is a nonempty open set which is an antichain, then there existopen down-sets C , C such that ∅ 6 = C ∩ ( X \ C ) ⊆ U . Then [ C ∩ C , C ] is a proper intervalsuch that C \ ( C ∩ C ) = C ∩ ( X \ C ) is an antichain (as a subset of the antichain U ). Thustheorem 3.1 implies that X is not affine complete. (cid:3) Note that the proof works exactly the same way if each occurrence of “open” is replaced by“clopen” (basically because each Priestley space is zero-dimensional). So we can state as well:A Priestley space X is affine complete if and only if each nonempty clopen set contains twodistinct comparable points.4. Products of affine complete lattices
We prove in this section that arbitrary products of affine complete lattices are affine complete.We don’t need Priestley duality to do this. Priestley duals of affine complete lattices, i.e. affinecomplete Priestley spaces, will come into play when we consider coproducts of affine completelattices.
Theorem 4.1. If ( L i ) i ∈ I is a family of (bounded) affine complete lattices, then Π i ∈ I L i isaffine complete.Proof. We prove the contrapositive of the theorem. Suppose that Π i ∈ I L i is not affine com-plete. Then it contains a proper interval [ ξ, η ] that is Boolean. There exists some k ∈ K suchthat ξ ( k ) < η ( k ). We claim that [ ξ ( k ) , η ( k )] ⊆ L k is a Boolean interval. Set x = ξ ( k ) , y = η ( k ). Suppose l ∈ [ x, y ] and define λ ∈ Π i ∈ I L i by λ ( i ) = ( l if i = kξ ( i ) if i = k Because [ ξ, η ] is Boolean, there exists λ ′ ∈ Π i ∈ I L i such that λ ∧ λ ′ = ξ and λ ∨ λ ′ = η . Thusit is easy to see that l ′ := λ ′ ( k ) is the complement of l ∈ [ x, y ]. Therefore, [ x, y ] is a properBoolean interval of L k and whence L k is not affine complete. (cid:3) Example 4.2.
Theorem 4.1 implies that [0 , N is affine complete. DOMINIC VAN DER ZYPEN Free products of affine complete lattices
Now we turn our attention to free products of affine complete bounded distributive lattices;we prove they are complete. A convenient way to obtain this result is to dualise the probleminto the category of Priestley spaces. Free products (that is, coproducts) in D correspondto products in P and vice versa; this is stated in the following proposition in a more generalway. Proposition 5.1. [3]
Let A and B be categories, and assume that F : A → B and G : B → A are contravariant functors that form a dual equivalence. Then: (1) If A is a product of a family of objects ( A i ) i ∈ I of A , then F ( A ) is a coproduct of ( F ( A i )) i ∈ I . (2) If A is a coproduct of a family of objects ( A i ) i ∈ I of A , then F ( A ) is a product of ( F ( A i )) i ∈ I . Moreover we have shown that affine complete lattices correspond to affine complete spacesunder the Priestley duality.
Theorem 5.2. If ( X i ) i ∈ I is a family of affine complete Priestley spaces, then Π i ∈ I X i is affinecomplete.Proof. Suppose that X i is affine complete for every i ∈ I . It suffices to show that everynonempty subset V of Π i ∈ I X i of the form V = π − i ( U ) ∪ ... ∪ π − i r ( U r )contains two distinct comparable elements (where U k ⊆ X i k open, nonempty). Take U . Itcontains elements a < b , because X i is affine complete. Now pick ξ ∈ V . Define ξ , ξ ∈ V by ξ ( i ) = ( ξ ( i ) if i = i a if i = i and ξ ( i ) = ( ξ ( i ) if i = i b if i = i Clearly, ξ , ξ are distinct comparable elements of V . (cid:3) Applying the Priestley duality now yields:
Corollary 5.3.
The class of (bounded) affine complete lattices is closed under free products. Embedding lattices in affine complete lattices
First we will stay away from affine completeness in the worst possible way: we will embedeach L into a powerset of some set, which, being Boolean, is as affine incomplete as it gets. Lemma 6.1.
Let L be a distributive lattice ( L need not be bounded). There is a set X and alattice embedding j : L ֒ → P ( X ) where P ( X ) is the powerset of the set X . EMARKS ON AFFINE COMPLETE DISTRIBUTIVE LATTICES 5
Proof.
First, endow L with a smallest element and a greatest element. Call this new boundeddistributive lattice L . By Priestley duality, there is a Priestley space ( X, τ, ≤ ) such that thelattice E ( X ) of clopen down-sets is isomorphic to L . Since E ( X ) is a sublattice of P ( X ),we are done. (cid:3) Next, we will embed that powerset in an affine complete lattice.
Lemma 6.2.
Let X be a set and let Q = { q ∈ Q ; 0 ≤ q ≤ } . Then there is a latticeembedding j : P ( X ) ֒ → Q X . Moreover, Q is affine complete.Proof. Set j : S χ S ∈ Q X for every S ⊆ X , where χ S is defined by χ S ( x ) = ( x ∈ S x / ∈ S It is easy to see that j is a lattice embedding. Next, we claim that Q is affine complete. Takeany x < y in Q . Then the element a = x + y ∈ [ x, y ] has no complement a ′ in [ x, y ]: Otherwisewe would have a ∧ a ′ = x which would imply a ′ = x , but then a ∨ a ′ = a = y . So [ x, y ] is notBoolean, whence Q has no proper Boolean interval. Therefore, Q is affine complete.Moreover, by 4.1, Q X is affine complete which concludes the proof. (cid:3) Lemmas 6.1 and 6.2 now imply:
Corollary 6.3.
Every distributive lattice (not necessarily bounded) can be embedded in abounded affine complete lattice.
Admittedly, the construction provided by 6.1 and 6.2 is highly non-unique and has no mini-mality properties.7. Q as initial object in the category of affine complete lattices The aim of this section is to show that the lattice Q = Q ∩ [0 ,
1] can be embedded into eachaffine complete lattice, which amounts to saying that Q is an initial object of the categoryof affine complete lattices (with (0,1)-homomorphisms, i.e. a full subcategory of the categorybounded distributive lattices). The key will be the notion of a dense chain. Definition 7.1.
A chain ( X, ≤ ) is called dense if for all x < y ∈ X there is z ∈ X with x < z < y . The first tool we need here is a well known result of model theory. It states that the theoryof dense linear orders is complete and has ( Q , ≤ ) as prime model. We will state this result ina more primitive way and prove it. Proposition 7.2. If ( X, ≤ ) is a bounded dense chain, there is a (0 , -embedding ϕ : Q ֒ → X. DOMINIC VAN DER ZYPEN
Proof.
Let a : ω → Q \{ , } be a bijection. We will write a k instead of a ( k ) to simplifynotation and will inductively build a subset f ⊆ ( Q \{ , } ) × ( X \{ X , X } )that’s an injective function from Q \{ , } to X \{ X , X } which is even order-preserving. n = 0: Choose b ∈ X \{ X , X } and set f := { ( a , b ) } . n → n + 1: Assume that f n has been defined in a way that for all k, l ∈ { , ..., n } the relation a k ≤ a l implies f n ( a k ) ≤ f n ( a l ) and that f n is an injective function from { a , ..., a n } to X \{ X , X } . Now consider the element a n +1 ∈ Q \{ , } . Case 1: a n +1 ≥ a i for all i ∈ { , ..., n } . Then, since X is dense, there is b n +1 ∈ X such that1 X > b n +1 ≥ f n ( a i ) for all i ∈ { , ..., n } . So, f n +1 := f n ∪ { ( a n +1 , b n +1 ) } is an injective order-preserving function that continues f n . Case 2: a n +1 ≤ a i for all i ∈ { , ..., n } . Proceed similarly as in Case 1. Case 3:
There are k, l ∈ { , ..., n } such that a k < a n +1 < a l . We may assume that thereis no k ′ ∈ { , ..., n } with a k < a k ′ < a n +1 and likewise that there is no l ′ ∈ { , ..., n } with a n +1 < a l ′ < a l . Consider b k = f n ( a k ) and b l = f n ( a l ). Since f n is order-preserving andinjective by assumption, we get b k < b l . Because X is dense, there is an element b n +1 suchthat b k < b n +1 < b l . Then f n +1 := f n ∪ { ( a n +1 , b n +1 ) } is easily seen to be an injective order-preserving map that continues f n .Now, it is easy to see that f := [ n ∈ ω f n is an injective order-preserving function from Q \{ , } to ( X \{ X , X } which is even order-preserving. So ϕ := f ∪ { (0 , X ) , (1 , X ) } is an order embedding from Q to X . (cid:3) Proposition 7.3.
Let L be a bounded affine complete distributive lattice. Then a) There is a maximal chain C ⊆ L , i.e., a chain that is not properly contained in anotherchain in L . b) If C is a maximal chain of L then C is dense.Proof. a ) is a standard application of Zorn’s Lemma: If K is a set of chains of L such thatfor any C , C ∈ K we either have C ⊆ C or C ⊇ C , then S K is easily checked to be achain in L : Let x, y ∈ S K , then there are members C, D containing x, y respectively; nowsince K is a chain with respect to ⊆ , at least one of the statements x, y ∈ C or x, y ∈ D holds.Since C, D are chains in L , either statement leads us to x ≤ L y or x ≥ L y . So K is boundedin the poset of all chains of L , thus Zorn’s Lemma implies that there is a maximal chain.As for b ), assume that C is a maximal chain such that x < y ∈ C but there is no z ∈ C with x < z < y . Now if there were no z in the whole lattice L such that x < z < y , then[ x, y ] = { x, y } is a proper Boolean interval of L which implies that L is not affine complete, EMARKS ON AFFINE COMPLETE DISTRIBUTIVE LATTICES 7 leading to a contradiction. Thus there is such a z , whence C ∪{ z } is a chain of L that properlycontains C , contradicting the maximality of C . (cid:3) Now the propositions 7.2 and 7.3 directly imply the following.
Theorem 7.4. If L is an affine complete lattice, then there exists a (0,1)-embedding ϕ : Q ֒ → L. Proof.
Pick any maximal chain C in L . Note that by maximality of C we have 0 , ∈ C since C ∪ { , } is a chain. So the inclusion map ι : C ֒ → L is a (0 , Q to C provided by proposition 7.3. Composing these two, we get a(0,1)-embedding from Q to L . (cid:3) Open questions
In chapter 6 we showed that ever bounded distributive lattice can be extended to an affinecomplete lattice. This was achieved by making use of Q which happens to be embeddablein any affine complete lattice, ie, the “smallest” affine complete lattice. Now the question is:Is the construction carried out in chapter 6 in some way canonical? For an arbitrary lattice L , does its ’affine hull’ have any interesting universal properties? References [1] B.A.Davey and H.A.Priestley,
Lattices and Order , Cambridge University Press, 1990.[2] G. Gr¨atzer,
Boolean functions on distributive lattices , Universal Algebra and Applications, vol. , BanachCenter Publications, Warsaw, 97-104.[3] S.MacLane, Categories for the working mathematician , Springer Verlag; 2nd edition (1998).[4] M.Ploˇsˇcica,
Affine Complete Distributive Lattices , Order (1994), 385-390.[5] H.A.Priestley, Representation of distributive lattices by means of ordered Stone spaces , Bull. London Math.Soc. 2 (1970), 186–190.[6] H.A.Priestley,
Ordered topological spaces and the representation of distributive lattices
Proc. London Math.Soc. (3) (1972), 507–530.[7] D.van der Zypen, Aspects of Priestley Duality , PhD thesis, University of Bern, 2004.