Scott Closed Set Lattices
And Applications
1. Some preliminaries
2. Dcpo-completion
3. Equivalence between CONP and CDL
4. The Hoare power domain
5. Scott closed set lattices of complete
semilattices
6. Some problems and remarks for further research
1. Preliminaries
Definition (Poset)
A partially ordered set, or poset for short, is a non-empty
set P equipped with a binary relation 
which is
(i) reflexive ( for every x in P, x  x);
(ii) transitive ( x  y and y  z imply x z );
(iii) Antisymmetric ( x y and y  x imply x=y )
Definition
(1) Let A be a subset of a poset P. The supremum of A in P,
denoted by sup A or A is the least upper bound of A.
The infimum inf A or  A is the greatest lower bound of A.
(2) For any subset A of poset P, denote
 A={ x in P: x  a for some a in A} and
 A={ x in P: x  a for some a in A} .
A is called a lower set if A=  A. Upper sets are defined
dually.
(3) A poset is a complete lattice if every subset has a
supremum and infimum.
Definition
(1) A non-empty subset D of a poset P is a directed set if
every two elements of D has an upper bound in D.
(2) A poset P is called a directed complete poset, or dcpo,
if every directed subset of P has a supremum in P.
(3) A subset A of a poset P is called a Scott closed set if
i) A is a lower set; and
ii) for any directed set D  A, sup D is in A whenever
sup A exists.
The set of all Scott closed sets of P is denoted by C(P).
(4) The complements of Scott closed sets of P are called
Scott open sets. All Scott open sets of P form a
topology---Scott topology, denoted by  (P).
Remark
(1) A subset U is a Scott open set if U is an upper set and
for any directed set D,
sup D  U implies D  U is non-empty.
(2) For any x in P, ↓x is Scott closed.
Example
(1) A subset U of the poset R of real numbers is Scott open
iff U=R, or U=(a, + ).
(2) In the power set lattice ( (X), ) , a subset 
is Scott open if it is an upper set and for any A in 
there is a finite set B in  with B  A.
(3) Let P={[a,b]: a  b } be the set of all nonempty closed
intervals. With the relation  it is a dcpo. An upper set
U of P is Scott open iff for any [a,b] in U there is [c,d] in P
such that c<a, b<d.
(4) If P is a discrete poset, then every subset is Scott open
Definition
A mapping f : P→ Q is called Scott continuous if it
preserves the supremum of directed sets, that is for any
directed set of P, if sup D exists then
f(sup D)=sup f(D).
•f is Scott continuous iff it is continuous with respect to the
Scott topologies of P and Q
• Scott continuous mappings models computable functions
in a most general context
Definition Let P be a poset . We say that an element a is
way-below b ( or a is an approximation of b) , denoted by
a<< b,
if for any directed set D, y  sup D then x  d for some
d in D.
A poset P is called continuous if for any a in P,
(i) the set { x: x<< a } is a directed set and
(ii) a=sup{ x: x<< a}.
* A continuous dcpo is called a domain.
Theory of
Computation
General Topology
Domain Theory
Category Theory
and Logic
Analysis and
Algebra
Example
(1)In the poset ( (X), ), A<< B iff A is a finite subset of
B. Thus the poset is continuous.
(2)In (R, ≤ ) , x<< y iff x< y. So it is also continuous.
(3)The poset of all nonempty closed intervals of R is a
continuous dcpo.
(4)If X is a locally compact topological space, then the
lattice consisting of all open sets is a continuous poset
with respect to the inclusion relation.
Definition Let L be a complete lattice. An element a of L
is called long way-below element b, denoted by a◄ b if
for any subset B, b  sup B implies a x for some
x in B.
A complete lattice is completely distributive iff for any
element a in L,
a=sup { x: x ◄ a }
•a◄ b implies a << b.
•Every completely distributive lattice is continuous
Example
(1) In (  (X),  ), A ◄B if and only if A={a} where a is a
member of B.
Since for any A in (X),
A=sup{ {x}: x is in A }
=  { {x}: x is in A},
So  (X) is a completely distributive lattice.
(2) Every complete chain is completely distributive.
2. Dcpo-completion of posets
Definition
A D-completion of a poset P is a
dcpo A together with a Scott continuous mapping
 :P  A ,
P
such that for any Scott continuous mapping f:P → B into a
dcpo B there exists a unique Scott continuous mapping
h:A →B satisfying f  h  P .
P
P
A
h
f
B
Question:
Dose every poset has a D- completion?
What are the other connections of posets and their Dcompletions?
Definition
A subset E of a dcpo is a subdcpo if E is closed under
existing supremum of directed set.
For any subset X of P, let cld (X )
all subdcpo containing X.
• Every Scott closed set is a subdcpo
• All subdcpos form a co-topolgy
be the intersection of
Theorem
Let P be a poset. The smallest subdcpo of C(P)
containing {↓ x: x is an element of P} is a D-completion.
Let E(P) be the above dcpo. Define
P : P  E( P)
by
P ( x)  x, for all x P.
Then  P is Scott continuous.
E(P) can be constructed from {↓ x: x is an element of P}
Recursively. Then we can verify that E(P) with this  P
Is a D-completion of P
Proposition
If E(P) is a D-completion of poset P, then
C(P) is isomorphic to C(E(P)).
P
C(P)
E(P)
Scott
closed set
Posets
lattices
Theorem
A poset P is a continuous poset if and only if its
D-completion is a continuous dcpo.
dcpos
P
Continu
ous
posets
Contin
uous
dcpos
posets
3. Equivalence between categories CONDCP and CDL
CONP : the category of continuous dcpos and Scott
continuous mappings that preserve the relation <<.
CDL: the category of completely distributive lattices and
the mappings that preserve supremum of arbitrary
subsets and the relation 
L
P
?
Q
CONP
M
CDL
Lemma A dcpo P is continuous if and only if the lattice
Of Scott open sets is a completely distributive lattice.
This is one of the most important results in domain theory, which was proved
independently by K.Hofmmann and J.Lawson
Corollary A dcpo P is continuous if and only if the lattice
C(P) of Scott closed sets is a completely distributive
lattice.
Definition
An element x of a lattice L is called co-prime if for any y,z
x  y or x  z.
x y z
in L,
implies
•The set of all co-primes of L is denoted by Spec(L).
• For any complete lattice L, Spec(L) is a dcpo with respect to the
inheritated order
Lemma
(1) For any completely distributive lattice L, Spec(L) Is a
continuous poset, and
L
C(Spec(L)).
(2) For any continuous poset P,
P
Spec(C(P)).


C(-)
L
P
Spec(-)
Lemma
(1) For any morphism f: P
Q in CONP, the mapping
C(f): C(P)
C(Q) is a morphism in CDL, where
for any A in C(P), C(f)(A)=cl(f(A).
(1) For any morphism g: L
M in CDL, the restrict of g
g |Spec( L) : Spec( L)  Spec(M )
is a morphism in CONP.
Two functors
P
C
C(P)
C(f)
CONP
Q
Spec(L)
C(Q)
Spec
L
g |Spec( L)
CONP
Spec(M)
CDL
g
M
CDL
Definition ( Equivalence of Categories)
A functor S: A B is an equivalence of categories ( and
the categories A and B are equivalence ) if there is a
functor T: B
A such that there is a natural
isomorphism ST I: B B and TS
I : A A.


Lemma A functor S: A
B is an equivalence of
categories if S is full and faithfull, and each object b of B
is isomorphic to Sa for some a in A.
Theorem The functor C: CONP
CDL is an
equivalence of categories. Thus the two categories
CONP and CDL are equivalent.
Remark: Classically one was interested in the category CDL* of
completely distributive lattices and the complete homomorphisms
(mappings preserving arbitrary joins and meets). One can show,
however a mapping between CDLs is a complete homomorphism iff its
right adjoint is a morphism in CDL, thus CDL is dual to CDL* . The
equivalence between CONP and CDL* was proved independently by
K.Hofmann and J.Lawson. The result was later named as
Hofmann-Lawson Duality .
4. The Hoare power domain
In mathematics, one often needs to consider "power
structure" from a given structure.
• the power set of a set X,
• the lattice IdL(P) of all ideals of a poset P,
• the exponential space C(X) of a topological space X
( the set C(X) of all closed sets of X with the Vietoris
topology ),
• The lattice Sub(H) of all closed subspaces of a
Hilbert space H.
In domain theory, one can construct the power
structures -- powerdomains, in several ways
Definition A directed complete partially ordered  algebra, or a dcpo-algebra, is a dcpo that is also a
 -algebra for which all the operations are Scott
continuous ( from the appropriate products equipped
with the Scott topology).
A homomorphism is a function between dcpo-algebras of
the same signature that is Scott continuous and a
homomorphism for each of the operations.

P  P  P 

P
Given any set X and any signature  , there is a free
 -algebra over X, T (X ), consisting of terms

that can be built up recursively from X by formally
applying the various operations in  .
Every mapping f:X A from X to a  -algebra extends
uniquely to an algebra homomorphism
 
from T (X ) into A.
1
2
An equation ( inequality ) is of the form  1   2
(  1   2 ), where 1 ,  2 are terms in T (X )
Let  be the signature consists of a single binary
operation, denoted by 
Let E be the inequality
(i) x  x  y
and equations
(ii) x  y  y  x
(iii) x  ( y  z )  ( x  y)  z
(iv) x  x  x
A dcpo  - algebra satisfying inequality and equations in
E is called an inflationary semilattice.
Theorem
Let P be a dcpo. Then the free inflationary semillatice over
P consists of all nonempty Scott closed sets of P with
binary union as the operation, inclusion relation as the
order and the embedding of P given by
 () : P C0 ( P)
which sends x in P to  x  { y  P : y  x
.}
The free infaltionary semilattice of domain is called the
Hoare Power domain
If P is a domain( continuous dcpo ), its Hoare power
domain is the dcpo consists of all nonempty Scott closed
sets of P, and
hence is also a continuous dcpo.
Other power domains:
•Smyth power domain
• Plotkin power domain
5. Scott closed set lattices of complete semilattices
Question:
(1)What are the general order properties of C(P)?
(2)What are the lattices C(P) of complete lattice P,
complete semilattices?
(3)What are the lattices C(P) of dcpo P?
Definition
Let L be a complete lattice and x, y be elements of L.
Define x  y , if for every nonempty Scott-closed set E
of P, the relation E  y
always implies that x  E .
Definition A complete lattice L is called C-continuous if
for any a in L,
a  sup{x  L : x  a}
Theorem
For any dcpo P, C(P) is C-continuous
Definition
An element x of a complete lattice is called C-algebraic if
xx
The set of all C-algebraic elements of L is denoted by
KC (L)
Definition
(1) A complete lattice L is called C-prealgebraic if for any
element a of L,
a  sup [( a)  KC (L)]
(2) L is called C-algebraic if it is C-prealgebraic and for
each a in L,
 [( a)  KC (L)  C(P)]
Definition
A complete lattice L is called C-stable if
(i)
1L  1L , and
(ii) for any element x of L and a Scott closed set D of L
such that x  y
for all y in D, then
x  inf D
A complete lattice satisfying only condition (ii) is called a
weakly C-stable lattice.
A complete semilattice is a dcpo P in which every upper
bounded subset has a supermum in P
Example
(1) The poset of all partial functions from N to N with the
order of extension.
(2) The poset of all nonempty closed sets of R under the
order 
(3) Let End(X) be the set of all mappings f: X
f g
if
f ( x)  x then f ( x)  g ( x)
X . Define
Theorem Let M be a complete lattice.
(i)
M is order isomorphic to C(L) for a complete lattice L
iff M is a C-stable and C-algebraic lattice.
(ii) M is isomorphic to C(L) for a complete semilattice L
iff M is a weakly C-Stable and C-algebraic lattice .
Theorem For any complete semilattice L,
KC (L)  { x : x  L }
Corollary Let L and M be two complete semilattices such
that C(L) is order isomorphic to C(M), then L is order
isomorphic to M.
6. Some problems and remarks for further research
1. Study the D-completion of the continuous poset
C(X, R*) of continuous functions on a compact space X.
2. Is it true that for any two dcpos P and Q, if C(P) is
order isomorphic to C(Q) then P is order isomorphic to
Q?
3. Charcterize the dcpo P such C(P)  C(Q)
implies P  Q for all dcpo Q. [ Conjecture: P is
continuous]
4. Is the product of two Scott closed set lattices a Scott
closed set lattice?
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6.
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Thank You!
Zhao Dongsheng
Mathematics and Mathematics Education
National Institute of Education
Nanyang Technological University
Singapore
E-mail:
dongsheng.zhao@nie.edu.sg