1、Set representations of abstract latticesZhao Dongsheng2009.6OutlineA.Set representationB.Topological representationC.Representation as set of lower semicontinuous functionsD.Representations as Scott closed setsE.Some problemsA.Set representationA lattice is a set lattice if its elements are sets,its
2、 order relation is given by set inclusion,and it is closed under taking finite unions and intersections.A set representation of a lattice L is a pair(C,),f)where(C,)is a set lattice and f is an isomorphism from L to(C,).Which lattices have a proper set representation?Some classical results:1.Birkhof
3、f A finite lattice has a set lattice representation iff it is distributive.2.Stone Every Boolean algebra has a set lattice representation3.Priestly Every bounded distributive lattice has a set lattice representation.B.Representation as families of closed setsGiven a topological space X,let C(X)be th
4、e set of all closed sets of X.(C(X),)is a set lattice.If a lattice L is isomorphic to(C(X),)for atopological space X,then X is called a topological representation of L.A lattice that has a topological representation is called a C-lattices.QuestionsB-1)Which lattices have a topological representation
5、?B-2)Which spaces(X,C )can be reconstructed from the lattice(C ,)?B-3)Which space (X,C )have the property:for any space (Y,E),if (E,)is isomorphic to(C ,),then X is homeomorphic to Y?B-4)How to construct all topological representations of a given lattice L?An element r of a lattice L is an irreducib
6、le element if r=xy implies r=x or r=y.The set of all reducible elements of L is denoted by(L).Theorem 1(W.J.Thron)A lattice has a topological representation iff it is complete and distributive,and all irreducible elements form a (join)base.A topological space X is sober if for anyirreducible element
7、 A of(C(X),),there is a unique point x of X,such that A=cl(x).Theorem 2For any sober space X,X is homeomorphic tothe space (C(X),where A(C(X):A is from C(X)is the set of closed sets of(C(X).*Every sober space spaces X can be reconstructed from the lattice(C(X),).Theorem 3 If X is Hausdorff space,the
8、n for any space Y,C(X)C(Y)implies that X is homeomorphic to Y.(E,)(E,)(X,C )(Y,E)Theorem 4 Let L be a C-lattice.1)For any base B(L),(B,C(B)a topological representation of L,where C(B)is the set of all closed sets of B.2)For any topological representation X of L,there is a base B of L such that (B,C(
9、B)is homeomorphic to X.Let L be a C-lattice.For any base B(L),let C(B)=a(L):a L.Theorem 5(Blanksma)A space X has the property that C(X)C(Y)implies X is homeomorphic to Y iffX is both sober and TD.C.Representation as set of lower semicontinuous functionsGiven a topological space X,let L(X)be the set
10、of all lower semi-continuous functions f:X R (x:f(x)r is open for all r in R)(L(X),)is a lattice under the pointwise order:f g iff f(x)g(x)for all x in X.An allowable R-action on a lattice M is a function :R M M such that r given by r(f)=(r,f)is an automorphism,0(f)=f,r(f)f if r 0,r(f)f if r 0,and r
11、 s=r+s.An ideal I of a lattice is closed iff for any AI with sup A exists,then sup A is in I.Theorem 6(Thornton)A lattice M is isomorphic to L(X)for some topological space X iff M is a conditionally complete and distributive lattice which has an allowable R-action and an R-basis of closed prime idea
12、ls.A space X is a TP space if for each x in X,either x is a G set or x is a closed set.A space is a TD space if x is closed for each x.Let A*,r(X)and A denote the set of all complete irreducible closed sets,point closures and irreducible closed sets of X respectively.Then A*r(X)A X is sober iff r(X)
13、=A X is TD iff A*=r(X)Nel and Wilson 1972 introduced fc-spaces and observed that“the sober space and fc-spaces play roles in the theorey of T0-spaces analogous to the roles of compact and real compact spaces in the theory of Tychnoff spaces”-H.Herrlich and G.StreckerTheorem 7(Thornton)X has the prop
14、erty that L(X)isomorphic to L(Y)implies X is homeomorphic to Y iff X is an fc and TP space.Theorem 8(Thornton)Let X and Y be TP spaces.Then L(X)L(Y)iff X is homeomorphic to Y.TDsoberTPfcD.Representation as families of Scott closeds setsA subset F of a poset P is a Scott closed set if(i)F is a lower
15、set(F=F=x:yx for some y in F);(ii)for any directed subset D,D F implies sup D is in F whenever sup D exists.(P)denotes the set of all Scott closedsets of P,and (P)denotes the set of allScott open sets of P.(P),)is a set lattice for each poset P.P is called a Scott representation of L if L is isomorp
16、hic to (P),)(i)Which L has a Scott representation?(ii)If P and Q both are Scott representations of L,how are they related?(iii)Which P has the property that(P)(Q)implies P isomorphic Q?(iv)Which P can be recovered from(P)?Theorem 9(1)A poset P is continuous iff (P)is a completely distributive lattic
17、e.(2)If L is a completely distributive lattice,then (L)is a continuous dcpo and L(L)(3)For any continuous dcpo P,P(L)If P and Q are continuous dcpos,then (P)(Q)implies PQ.RemarkEvery completely distributive lattice L has a Scott representation.Every continuous dcpo can be recovered from(P)Let L be a
18、 complete lattice.For x,y in L,define x y iff for any Scott closed set D,y sup D implies x belongs to D.If x x,x is called a C-compact element.The set of C-compact elements is denoted by k(L).L is a C-prealgebraic lattice if k(L)is a join base of L.A C-prealgebraic lattice is C-algebraic if For any
19、a in L,a k(L)is a Scott closed set of K(L).Theorem 10 6(1)A lattice has a bounded complete Scott representation iff it is C-prealgebraic.(2)A lattice has a complete Scott representation iff it is C-algebraic.(3)If P is a bounded complete poset,then P can be recovered from(P)(Pk(P)A dcpo-completion o
20、f a poset P is a dcpo E(P)together with a universal Scott continuous mapping :P E(P)from P to dcpo.Theorem 11(1)For any poset P,E(P)exists.(2)P is algebraic iff E(P)is continuous.(3)For any poset P,(P)(E(P)If L has a Scott representation,then it has a dcpo Scott representationE.Some problems1.Is it
21、true that for any two dcpos P,Q,(P)(Q)implies PQ?2.Which dcpo P has the property that (P)(Q)implies P Q?3.Is it true that for any C-lattice L,there is a dcpo P,such that(P,(P)is sober and P is a Scott representation of L?4.As in the case of topological representations,we can also define a partial or
22、der on the set of Scott representations of a lattice.Which C-lattice have a maximal(minimal)Scott representation?5.If L and M have Scott representations,must LM also has Scott representation?References1.C.E.Aull and R.Lowen,Handbooks of the history of general topology,Kluwer Academic Publishers,1997
23、.2.W.J.Thron,Lattice equivalence of topological spaces,Duke Math.J.29(1962),671-680.3.D.Drake and W.J.Thron,On the representations of an abstract lattices as the family of closed sets of a topological space,Trans.Amer.Math.Soc.120(1965),57-71.4.M.C.Thornton,Topological spaces and lattices of lower semicontinuous functions,Trans.Amer.Math.Soc.181(1973),495-560.5.G.Gierz,K.H.Hoffmann,K.Keimel,J.D.Lawson,M.W.Mislove and D.S.Scott,Continuous Lattices and Domains,Cambridge University Press,2003.6.W.K.Ho and D.Zhao,Lattices of Scott closed sets,Comment.Math.Univ.Carolinae,50(2009),2:297-314.
