A Poset with Spectral Scott Topology

DOI10.1007/s11083-009-9129-5A Poset with Spectral Scott Topologyis a Quasialgebraic DomainTomoo YokoyamaReceived:23March2008/Accepted:17August2009©Springer Science+Business Media B.V.2009Abstract We prove that a spectral Scott space(i.e.a poset on which the Scott topology is spectral)is a quasialgebraic domain.Keywords Scott topology·Spectral topology·Quasialgebraic domain1IntroductionIn[13],Priestley asked whether there exists a non-quasicontinuous domain on which the Scott topology is spectral.(Notice that quasicontinuous domains are called quasicontinuous posets in[13].)We show that there are no such domains.Precisely, we state the following theorem.Theorem1.1Let(X,≤)be a poset.Suppose that the Scott topologyσ(X)is spectral. Then X is a quasialgebraic domain.In particular,X is a quasicontinuous domain.Since any spectral poset is down-complete and up-complete,the above result is a corollary of the following criterion.Proposition1.2Let(X,≤)be a down-complete poset.X is a quasialgebraic poset if and only ifσ(X)is compactly based(i.e.has a compact open basis).The referees pointed out that Proposition 1.2in turn is a special case of a more general result due to Erné[1],saying that any compact open set in The author is partially supported by the21st Century COE Program at Graduate School of Mathematical Sciences,the University of Tokyo.T.Yokoyama(B)Graduate School of Mathematical Sciences,University of Tokyo,Komaba Meguro,Tokyo153-8914,Japane-mail:yokoyama@ms.u-tokyo.ac.jpso-called monotone determined spaces (see [1])and,in particular,in Scott spaces,is already hypercompact (where a set is hypercompact [1]or finitely bottomed [7]if its saturation is finitely generated).Indeed Erné’s result and Corollary 7[1],saying that the Scott spaces of quasialgebraic posets are exactly the sober spaces with a basis of hypercompact open sets,imply Proposition 1.2.2PreliminariesA topology T on a set X is defined to be spectral if (X ,T )is homeomorphic to the set Spec R of its prime ideals with the Zariski topology for some commutative ring R with unit.In [6],Hochster proved that a topology T is spectral if and only if T is sober,compact,with a compact open basis closed under finite intersections.A poset (X ,≤)is said to be spectral or representable if there is a commutative ring R with unit such that X is order isomorphic to Spec R with the inclusion order.For an element x of a poset (X ,≤),↑x :={y ∈X |x ≤y }(resp.↓x :={y ∈X |y ≤x })is called the upset or the saturation (resp.the downset or the point closure)of x .For a subset Y ⊆X ,↑Y := y ∈Y ↑y (resp.↓Y := y ∈Y ↓y )is called the upset or the saturation (resp.the downset or the point closure)of Y .A subset Y ⊆X is called an upset (resp.downset)if Y =↑Y (resp.Y =↓Y ).An upset (resp.downset)Y ⊆X is said to be finitely generated if there is a finite subset F ⊆X such that Y =↑F (resp.Y =↓F ).A subset D of a poset X is (up-)directed (resp.down-directed)if every finite subset of D has an upper (resp.lower)bound in D .A poset X is said to be down-complete (resp.up-complete)if each down-directed (resp.up-directed)set S of X has S (resp. S ).An up-complete poset is also called a directed complete poset or a dcpo.Let (X ,T )be a topological space and ≤a partial ordering on X .The topology T is said to be ≤-compatible,if {x }=↓x ,for each x ∈X .This definition is in accordance with the terminology in [2]but not with some earlier papers at the borderline between order theory and theoretical computer science (cf.[9,10]).One can obviously see that (X ,≤)is spectral if and only if there exists an ≤-compatible spectral topology on X .A subset A of a poset X is Scott closed if it is a downset and is closed under directed sups (i.e.for any directed set D ⊆A , D ∈A whenever D exists).The complements of the Scott closed sets form a topology,called the Scott topology and denoted by σ(X ).We say that a nonempty family F of subsets of a poset X is directed if given F 1,F 2∈F ,there exists F ∈F such that F ⊆↑F 1∩↑F 2.For a dcpo X and non-empty subsets F and G of X ,we say that F approximates G (written by F G )if a directed set D ⊆X with D ∈↑G meets ↑F .A dcpo X is called a quasicontinuous domain if for all x ∈X ,fin (x )is directed and ↑x = F ∈fin (x )↑F ,where fin (x )={F |F is finite ,F {x }}.A dcpo X is called a quasialgebraic domain if for all x ∈X ,compfin (x )is directed and ↑x = F ∈compfin (x )↑F ,where compfin (x )={F |F is finite ,F F ,F x }.In [5],Heckmann showed that the following statements are equivalent for a dcpo X :(1)X is a quasicontinuous domain;(2)for all x ∈U ∈σ(X ),there is a nonemptyfinite F ⊆X such that x ∈int σ(X )↑F ⊆↑F ⊆U ,where the interior is taken in the Scott topology σ(X ).According to [11],a poset X is called a quasicontinuous poset (resp.quasialge-braic poset)if for all x ∈U ∈σ(X ),there is a nonempty finite F ⊆X such that x ∈int σ(X )↑F ⊆↑F ⊆U (resp.x ∈int σ(X )↑F =↑F ⊆U ).By the above equivalence,every quasicontinuous domain is a quasicontinuous poset.3Proofs of ResultsFirst we prove the following lemma.Lemma 3.1Let (X ,≤)be a down-complete poset.A Scott open subset U of X is Scott compact if and only if U is hypercompact.Notice that Erné’s aforementioned result (pointed out by the referees)implies that Lemma 3.1holds without down-completeness.Proof Let U be a Scott open subset of X .If U is hypercompact,then there is a finite subset G ⊆X such that U =int σ(X )↑G =↑G and so U is Scott compact.Conversely,suppose that U is Scott compact.Then U is dually Scott closed.(By dually Scott closed,we mean Scott closed on the underlying set with the opposite order.)Indeed,if D ⊆U would be a down-directed set with D /∈U ,then {X −↓d |d ∈D }would be an open cover with no finite subcover and so U is not Scott compact.Thus for any down-directed set S ⊆U , S ∈U .By Zorn’s lemma,↑Min U =U where Min U is the set of all minimal elements in U .Put F =X −U .Claim 1For V ⊆Min U,V ∪F is Scott closed.Indeed,obviously V ∪F is a downset.Notice that Min U is an antichain (i.e.any pair of distinct elements of U are incomparable).For any directed subset D ⊆V ∪F such that D exists,if there are x =y ∈D ∩V ,then there is an upper bound z ∈D of {x ,y }.Since x and y are incomparable,this implies that x ,y <z and so z /∈V ∪F .This is a contradiction to D ⊆V ∪F .Thus |D ∩V |≤1.If D ∩V is empty,then D ⊆F and so D ∈F ⊆V ∪F .Otherwise there is a unique element x ∈D ∩V .If there is an element y ∈D such that x and y are incomparable,then there is an upper bound z ∈D of {x ,y }such that x ,y <z .This contradicts D ⊆V ∪F .Thus any element of D is less than or equal to x .Hence D =x ∈V ⊆V ∪F .Therefore V ∪F is Scott closed.If Min U is infinite,then we can take a decreasing sequence {V i |V i ⊆Min U }i ∈Z >0of nonempty subsets such that ∩i ∈Z >0V i =∅.Then {X −(V i ∪F )}i ∈Z >0is an open cover of U which has no finite subcover.Thus U is not Scott compact.Thus Min U is finite.Since U is Scott open,U =↑Min U =int σ(X )(↑Min U ).Hence Min U is a desired finite subset.Proof of Proposition1.2Suppose that there is a compact open basis B of the Scott topologyσ(X).For x∈U∈σ(X),there is a neighborhood W∈B of x such that W⊆U.By Lemma3.1,there is afinite subset F⊆X such that W=intσ(X)↑F=↑F.Thus x∈W=intσ(X)↑F=↑F⊆U.Conversely,suppose that X is a quasialgebraic poset.Define B as follows:B ={↑F|F⊆X isfinite,↑F∈σ(X)}Then B is a compact open basis ofσ(X).We say that a poset X have property HC if↑H∩↑G is hypercompact for any hypercompact subsets H and G with H H and G G.Note that a upset U of a poset is Scott open if and only if U U.We also state a characterization of spectral posets with spectral Scott topology. Corollary3.2Let(X,≤)be a spectral poset.The following conditions are equivalent:1)The Scott topologyσ(X)is spectral.2)σ(X)is compact and compactly based,and X has property HC.In particular,if the poset X is a lattice,then the condition1)is equivalent to 2’)X is compactly based.Proof Let(X,≤)be a spectral poset.Notice thatσ(X)is compactly based in every cases.Since any spectral poset is down-complete and up-complete,Proposition1.2 implies that X is a quasialgebraic domain.In particular,X is a quasicontinuous domain.By Proposition4.4[3],X is sober.Suppose thatσ(X)is spectral.It suffices to show that X has property HC. Let H,G be hypercompact subsets with G G,H H.Then↑H and↑G are Scott compact open.Sinceσ(X)is spectral,↑H∩↑G is Scott compact open.By Proposition1.2,it is hypercompact with↑H∩↑G ↑H∩↑G.Therefore X has property HC.Suppose that2)holds.For any Scott compact open subsets H and G,we have that H H and G G.By Lemma3.1,H and G are hypercompact.By property HC,H∩G=↑H∩↑G is hypercompact.By Proposition1.2,it is Scott compact. Thereforeσ(X)is spectral.Suppose that the poset X is a lattice and2’)holds.Since any up-complete and down-complete lattice is a complete lattice,X is a complete lattice.Then clearly σ(X)is compact.Hence it suffices to show that X has property HC.For any hypercompact subsets H and G with G G and H H,since Min G and Min H arefinite,we have that{g∨h|g∈Min G,h∈Min H}isfinite.Hence↑H∩↑G=↑{g∨h|g∈Min G,h∈Min H}is hypercompact.Therefore X has property HC.Acknowledgements The author wishes to thank the anonymous referees for pointing out the results on[1]and for critical reading of the manuscript and the comments for improving the paper.References1.Erné,M.:Infinite distributive laws versus local connectedness and compactness properties.Topology Appl.156,2054–2069(2009)2.Gierz,G.,Hofmann,K.H.,Keimel,K.,Lawson,J.D.,Mislove,M.,Scott,D.S.:ContinuousLattices and Domains.Encyclopedia of Mathematics and its Applications,vol.93.Cambridge University Press,Cambridge(2003)3.Gierz,G.,Lawson,J.D.,Stralka,A.:Quasicontinuous posets.Houston J.Math.9,191–208(1983)4.He,W.,Jiang S.:Remarks on the sobriety of Scott topology and weak topology on posets.Comment.Math.Univ.Carolin.43(3),531–535(2002)5.Heckmann,R.:An upper power domain construction in terms of strongly compact sets.In:Lecture Notes in Computer Science,vol.598,pp.272–293.Springer,Berlin(1992)6.Hochster,M.:Prime ideal structure in commutative rings.Trans.Amer.Math.Soc.142,43–60(1969)7.Isbell,J.R.:Function spaces and adjoints.Math.Scand.36,317–339(1975)8.Kaplansky,A.:Commutative Rings,revised edition.The University of Chicago Press,Chicago(1974)9.Lewis,W.J.,Ohm,J.:The ordering of Spec(R).Canad.J.Math.28,820–835(1976)wson,J.D.:The upper interval topology,propertyM,and compactness.Electron.Notes Theor.Comput.Sci.13,158–172(1998)11.Mao,X.,Xu,L.:Quasicontinuity of posets via Scott topology and sobrification.Order23(4),359–369(2006)12.Priestley,H.A.:Representation of distributive lattices by means of ordered Stone spaces.Bull.London Math.Sot.2,186–190(1970)13.Priestley,H.A.:Intrinsic spectral topologies.Papers on General Topology and Applications.Papers from the8th summer conference at Queens College,New York,NY,USA,18–20June 1992.New York,NY:The New York Academy of Sciences.Ann.N.Y.Acad.Sci.728,78–95 (1994)。