scott closed set lattices and applications 1. some preliminaries 2. dcpo-completion 3. equivalence...
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Scott Closed Set LatticesAnd 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
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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 )
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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.
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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).
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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.
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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
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DefinitionA mapping f : P→ Q is called Scott continuous if itpreserves 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
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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.
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Domain Theory
Theory of Computation General Topology
Analysis and Algebra
Category Theory and Logic
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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.
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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
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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.
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DefinitionA D-completion of a poset P is adcpo A together with a Scott continuous mapping ,
such that for any Scott continuous mapping f:P → B into a dcpo B there exists a unique Scott continuous mappingh:A →B satisfying .
APP :
Phf
PP
A
B
fh
2. Dcpo-completion of posets
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Question: Dose every poset has a D- completion?
What are the other connections of posets and their D-completions?
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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 be the intersection of all subdcpo containing X.
)(Xcld
• Every Scott closed set is a subdcpo• All subdcpos form a co-topolgy
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TheoremLet 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
by
Then 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 thisIs a D-completion of P
)(: PEPP
.,)( PxallforxxP
P
P
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Proposition
If E(P) is a D-completion of poset P, then
C(P) is isomorphic to C(E(P)).
P
E(P)
C(P)
Scott closed set
latticesPosets
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Theorem
A poset P is a continuous poset if and only if its
D-completion is a continuous dcpo.
posets
dcpos
Continuous dcpos
Continuous posets
P
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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
CONP CDL
P
Q
L
M
?
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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.
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Definition
An element x of a lattice L is called co-prime if for any y,z
in L, implies zyx .zxoryx
•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
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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)).
PL
C(-)
Spec(-)
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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
is a morphism in CONP.
)()(:| )( MSpecLSpecg LSpec
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Two functors
P
Q
C(P)
C(Q)
C(f)
C
CONP CDL
CONP CDL
L
M
Spec(L)
Spec(M)
g)(| LSpecgSpec
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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.
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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 .
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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 powerstructures -- powerdomains, in several ways
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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.
PPPP
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Given any set X and any signature , there is a free
-algebra over 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 into A.
)(XT
)(XT
An equation ( inequality ) is of the form ( ), where are terms in
21
21 21 21, )(XT
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Let be the signature consists of a single binary
operation, denoted by
Let E be the inequality
(i) and equations
(ii)
(iii)
(iv)
A dcpo - algebra satisfying inequality and equations in E is called an inflationary semilattice.
yxx
xyyx
zyxzyx )()(
xxx
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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
which sends x in P to .
)(:)( 0 PCP
}:{ xyPyx
The free infaltionary semilattice of domain is called the Hoare Power domain
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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
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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?
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Definition
Let L be a complete lattice and x, y be elements of L.
Define , if for every nonempty Scott-closed set E
of P, the relation
always implies that .
yx yE Ex
Definition A complete lattice L is called C-continuous if for any a in L,
}:sup{ axLxa
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Theorem
For any dcpo P, C(P) is C-continuous
Definition
An element x of a complete lattice is called C-algebraic if
The set of all C-algebraic elements of L is denoted by
xx
)(LKC
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Definition
(1) A complete lattice L is called C-prealgebraic if for any element a of L,
(2) L is called C-algebraic if it is C-prealgebraic and for each a in L,
)]()[(sup LKaa C
)]()()[( PCLKa C
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Definition
A complete lattice L is called C-stable if
(i) , and
(ii) for any element x of L and a Scott closed set D of L such that
for all y in D, then
A complete lattice satisfying only condition (ii) is called a weakly C-stable lattice.
LL 11
yx
Dx inf
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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 X . Define
if
gf
)()(then)( xgxfxxf
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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 .
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Theorem For any complete semilattice L,
}:{)( LxxLKC
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.
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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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References1. S. Abramsky and A. Jung, Domain Theory, Volume 3 of Handbook for
Logic in Computer Science, Clarendo Press 1994.
2. B. Ern|’e and D. Zhao, Z-join spectra of Z-supercompactly generated lattices, Applied categorical Structures, 9(2001), 41-63
3. 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.
4. W. Ho, and D. Zhao, On the characterization of Scott-closed set lattices, (2007)(Preprint)
5. R. E. Hoffmann, Continuous posets-prime spectra of
completely distributive lattices, and Hausdorff compactification,
Lecture Note in Mathematics, 871(1981), 159-208
6. J. D. Lawson, The duality of continuous posets, Houston
Journal of Mathematics, 5(1979), 357-394.
7. M. W. Mislove, Local DCPOs, Local CPOs and Local
completions, Electronic Notes in Theoretical Computer Science,
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Thank You!
Zhao Dongsheng
Mathematics and Mathematics Education
National Institute of Education
Nanyang Technological University
Singapore
E-mail: [email protected]