Set representations of abstract lattices

1. Set representations of abstract lattices Zhao Dongsheng 2009.6

2. Outline A. Set representation B. Topological representation C. Representation as set of lower semicontinuous functions D. Representations as Scott closed sets E. Some problems

3. A. Set representation A lattice is a set lattice if its elements are sets , its 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, ⊆).

4. Which lattices have a proper set representation?

5. Some classical results: • Birkhoff • A finite lattice has a set lattice representation iff it is distributive. • 2. Stone • Every Boolean algebra has a set lattice representation • 3. Priestly • Every bounded distributive lattice has a set lattice representation.

6. Representation as families of closed sets

7. Given a topological space X, let C(X) be the set of all closed sets of X. (C(X), ⊆ ) is a set lattice. If a lattice L is isomorphic to (C(X), ⊆ ) for a topological space X, then X is called a topological representation of L. A lattice that has a topological representation is called a C- lattices.

8. Questions B-1) Which lattices have a topological representation? 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?

9. An element r of a lattice L is an irreducible element if r = x∨y 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 .

10. A topological space X is sober if for any irreducible element A of (C(X) , ⊆), there is a unique point x of X, such that A=cl({x}). Theorem 2 For any sober space X, X is homeomorphic to the 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), ⊆ ) .

11. Theorem 3 If X is Hausdorff space, then for any space Y, C(X) ≈C(Y) implies that X is homeomorphic to Y. (X, C ) (E , ⊆) (E, ⊆) (Y, E )

12. Let L be a C-lattice. For any base B⊆ ∆(L), let C(B) = { ↓a∩∆(L): a L }. Theorem 4Let 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(B) ) is homeomorphic to X.

13. Theorem 5 (Blanksma) A space X has the property that C(X) ≈C(Y) implies X is homeomorphic to Y iff X is both sober and TD .

14. C. Representation as set of lower semicontinuous functions

15. Given a topological space X, let L(X) be the set 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.

16. An allowable R-action on a lattice M is a function Γ: R × M → M such that Γr given by Γr (f) = Γ(r, f) is an automorphism, Γ0 (f) =f, Γr (f) > f if r > 0, Γr (f) < f if r < 0, and ΓrΓs = Γr+s . • An ideal I of a lattice is closed iff for any A⊆I with sup A exists, then sup A is in I.

17. 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 ideals.

18. A space X is a TP space if for each x in X, either {x} is a G𝛅 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)=A X is TD iff A*=r(X)

19. 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. Strecker

20. Theorem 7(Thornton) X has the property 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.

21. TP sober TD fc

22. D. Representation as families of Scott closeds sets

23. A subset F of a poset P is a Scott closed set if • F is a lower set(F=↓F={x: y≤x for some y in F}) ; • for any directed subset D, D ⊆F implies sup D is in F whenever sup D exists. • Γ(P) denotes the set of all Scott closed • sets of P, and σ(P) denotes the set of all • Scott open sets of P. • (Γ(P) , ⊆) • is a set lattice for each poset P.

24. P is called a Scott representation of L if L is isomorphic to (Γ(P) , ⊆)

25. Which L has a Scott representation? • If P and Q both are Scott representations of L, how are they related? • Which P has the property that Γ(P) ≈ Γ(Q) implies P isomorphic Q? • Which P can be recovered from Γ(P) ?

26. Theorem 9 • A poset P is continuous iff Γ(P) is a completely distributive lattice. • (2) If L is a completely distributive lattice, • then ∆(L) is a continuous dcpo and • L≈ Γ(∆(L)) • (3) For any continuous dcpo P, • P≈ ∆(Γ (L)) If P and Q are continuous dcpos , then Γ(P)≈ Γ(Q) implies P≈Q.

27. Remark Every completely distributive lattice L has a Scott representation. Every continuous dcpo can be recovered from Γ(P)

28. Let L be a 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).

29. 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 a in L, ↓a∩ k(L) is a Scott closed set of K(L).

30. 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 Γ(P) ( P≈k(Γ(P) ))

31. A dcpo-completion of a poset P is a dcpo E(P) together with a universal Scott continuous mapping ƞ: P →E(P) from P to dcpo.

32. Theorem 11 • For any poset P, E(P) exists. • P is algebraic iff E(P) is continuous. • For any poset P, Γ(P) ≈Γ(E(P)) If L has a Scott representation, then it has a dcpo Scott representation

33. E. Some problems • Is it true that for any two dcpos P, Q, • Γ(P)≈Γ(Q) implies P≈Q ? 2. Which dcpo P has the property that Γ(P) ≈ Γ(Q) implies P ≈ Q? 3.Is it true that for any C-lattice L, there is a dcpo P, such that (P, σ(P)) is sober and P is a Scott representation of L?

34. 4.As in the case of topological representations, we can also define a partial order 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 L×M also has Scott representation?

35. References • C. E. Aull and R. Lowen, Handbooks of the history of general topology, Kluwer Academic Publishers, 1997. • W.J. Thron, Lattice equivalence of topological spaces, Duke Math. J. 29 (1962), 671-680. • 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. • M. C. Thornton, Topological spaces and lattices of lower semicontinuous functions, Trans. Amer. Math. Soc. 181 (1973), 495-560. • 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. • W. K. Ho and D. Zhao, Lattices of Scott closed sets, Comment. Math. Univ. Carolinae , 50(2009) , 2 : 297-314.