Set representations of abstract lattices

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Set representations of abstract lattices

• Zhao Dongsheng
• 2009.6

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Outline A. Set representation B. Topological
representation C. Representation as set of
lower semicontinuous functions D.
Representations as Scott closed sets E. Some
problems
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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, ?).
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Which lattices have a proper set representation?
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• 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.

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1. Representation as families of closed sets

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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.
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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?
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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 .
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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
Acl(x).
Theorem 2 For any sober space X, X is
homeomorphic to the space ?(C(X)), where
?An?(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), ? ) .
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Theorem 3 If X is Hausdorff space, then
for any space Y, C(X) C(Y) implies that X is
homeomorphic to Y.
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Let L be a C-lattice. For any base B? ?(L),
let C(B) ?an?(L) a L .
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(B) ) is
homeomorphic to X.
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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 .
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C. Representation as set of lower semicontinuous
functions
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Given a topological space X, let L(X) be the
set of all lower semi- continuous functions f
X ?R ( x f(x)gt 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.
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• An allowable R-action on a lattice M is a
  function
• G R M ? M
• such that
• Gr given by Gr (f) G(r, f) is an automorphism,
• G0 (f) f, Gr (f) gt f if r gt 0, Gr (f) lt f if r
  lt 0, and
• Gr Gs Grs .
• An ideal I of a lattice is closed iff for any
  A?I with
• sup A exists, then sup A is in I.

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• 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.

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• 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)A
• X is TD iff A r(X)

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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
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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.

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D. Representation as families of Scott closeds
sets
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• A subset F of a poset P is a Scott closed set if
  
• F is a lower set(F?Fx yx for some y in F)
  
• for any directed subset D, D ?F implies sup D
  is in F whenever sup D exists.
• G(P) denotes the set of all Scott closed
• sets of P, and s(P) denotes the set of all
• Scott open sets of P.
• (G(P) , ?)
• is a set lattice for each poset P.

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P is called a Scott representation of L if L is
isomorphic to (G(P) , ?)
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1. Which L has a Scott representation?
2. If P and Q both are Scott representations of
   L, how are they related?
3. Which P has the property that G(P) G(Q)
   implies P isomorphic Q?
4. Which P can be recovered from G(P) ?

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• Theorem 9
• A poset P is continuous iff G(P) is a
  completely distributive lattice.
• (2) If L is a completely distributive lattice,
  
• then ?(L) is a continuous dcpo and
• L G(?(L))
• (3) For any continuous dcpo P,
• P ?(G (L))

If P and Q are continuous dcpos , then
G(P) G(Q) implies PQ.
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Remark Every completely distributive lattice L
has a Scott representation. Every continuous
dcpo can be recovered from G(P)
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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).
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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, ?an k(L) is a
Scott closed set of K(L).
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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 G(P) ( Pk(G(P) ))
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A dcpo-completion of a poset P is a dcpo
E(P) together with a universal Scott continuous
mapping ? P ?E(P) from P to dcpo.
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• Theorem 11
• For any poset P, E(P) exists.
• P is algebraic iff E(P) is continuous.
• For any poset P, G(P) G(E(P))

If L has a Scott representation, then it has a
dcpo Scott representation
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E. Some problems

• Is it true that for any two dcpos P, Q,
• G(P) G(Q) implies PQ ?

2. Which dcpo P has the property that G(P)
G(Q) implies P Q?
3. Is it true that for any C-lattice L, there is
a dcpo P, such that (P, s(P)) is sober and P is
a Scott representation of L?
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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 LM also has
Scott representation?
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• 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.