Set representations of abstract
lattices
Zhao Dongsheng
2009.6
Outline
A. Set representation
B. Topological representation
C. Representation as set of lower semicontinuous
functions
D. Representations as Scott closed sets
E. Some problems
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, ⊆).
Which lattices have a proper set representation?
Some classical results:
1. 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.
B. Representation as families of closed sets
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.
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?
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 .
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), ⊆ ) .
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 , ⊆)
(Y, E )
(E, ⊆)
Let L be a C-lattice. For any base B⊆ ∆(L), let
C(B) = { ↓a∩∆(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.
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 .
C. Representation as set of lower
semicontinuous functions
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.
• 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
Γr Γs = 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.
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.
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)
Nel and Wilson [1972] introduced fcspaces 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
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.
TP
sober
fc
TD
D. Representation as families of Scott
closeds sets
A subset F of a poset P is a Scott closed set
if
(i)F is a lower set(F=↓F={x: y≤x for some y
in F}) ;
(ii) 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.
P is called a Scott representation of L if L is
isomorphic to (Γ(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 Γ(P) ≈ Γ(Q)
implies P isomorphic Q?
(iv) Which P can be recovered from Γ(P) ?
Theorem 9
(1)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.
Remark
Every completely distributive lattice L has a Scott
representation.
Every continuous dcpo can be recovered from Γ(P)
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).
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).
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) ))
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.
Theorem 11
(1) For any poset P, E(P) exists.
(2) P is algebraic iff E(P) is continuous.
(3) For any poset P, Γ(P) ≈Γ(E(P))
If L has a Scott representation, then it has a
dcpo Scott representation
E. Some problems
1. 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?
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?
References
1. C. E. Aull and R. Lowen, Handbooks of the history of general topology,
Kluwer Academic Publishers, 1997.
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.