Three levels of D - completions

Zhao Dongsheng

Ho Weng Kin, Wee Wen Shih

Mathematics and Mathematics EducationNational Institute of Education,

Nayang Technological University,Singapore

The 2nd International Conference on Quantitative Logic and Soft Computing

22-25, October, 2010 Jimei University, Xiamen , China

Outline:

• Subcategory and reflectivity

• Posets , dcpo’s and D – completion of posets

• Monotone convergence ( d - ) spaces, d-completion of T0 - spaces

• Convergence spaces, D-completion

1. Subcategory and reflectivity

Given a category C and a subcategory E of C, E is reflective in C if the inclusion functor from E into C has a left adjoint.

f

Y

G(X)ηX

X

f

C E

Examples

CRSP

KHSP

X βX

Stone–Čech compactification

TOP0

Sober

X SX

soberification

A space Y is sober if for any non-empty irreducible closed set F of Y, there is a unique element a in Y such that F = cl({a})

2. Directed complete posets and dcpo-completion

A non-empty subset D of a poset (P, ≤ ) is directed if any two elements in D has an upper bound in D.

A poset (P, ≤ ) is directed complete if every directed subset D of P has a supremum (join) sup D.

A directed complete poset is also called a dcpo.

A mapping f : P → Q between two posets is Scott continuous if it preserves joins of directed sets: f( sup D)=sup f(D)

for every directed D for which sup D exists.

Let POS ( DCPO) be the category of posets ( dcpo ) and Scott continuous mappings between posets.

Question: Is DCPO reflective in POS ?

Definition [11]The D-completion of a poset P is a dcpo D(P) , together with a Scott continuous mapping η : P → D(P) such that for any Scott continuous f: P → Q into a dcpo Q, there is a unique Scott continuous g: D(P) → Q such that f= g○η

A subset U of P is Scott open if the following conditions are satisfied: (i) U=↑U ;(ii) for any directed set D, sup D in U implies D∩U is non- empty

The set of all Scott closed (open ) sets of P is denoted by Г(P) ( σ(P)), which is a complete lattice under the inclusion relation.

A subset F of a poset P is called D-closed if for any directed subset, whenever sup D exists.

The D – closed sets defines a topology--- D - topology.

FDFD sup implies

* A subset F is Scott closed iff it is a down set and D-closed

Theorem [11]

For every poset P, the D - closure of ф={↓x: x is in P}in Г(P), or the smallest subdcpo containing ф, is the D-completion of P.

Theorem[11]

(1) P is continuous iff D(P) is continuous.

(2) P is algebraic iff P is algebraic.

(3) Г(P)≡ Г(D(P)).

*The directed completion o f continuous posets was considered in [10].

* The dcpo – completion of posets was also considered in [6] and [7] with different motivations.

* The local dcpo –completion was first considered in [8] and was revised in [11]

For any T0 space X, the specialization order on X is defined as : x y iff x is in cl({y}).

3. d-spaces and d-completion to topological spaces

Remark1. For any poset P, ΣP=(P, σ(P)) is a T0 space. The

specialization order is the original order on P.2. f: P → Q is Scott continuous iff it is continuous with

respect to the Scott topology. So the category POSd is a full subcategory of TOP0

3. If P is a dcpo, then (i) the specialization of ΣP is directed complete, and (ii) for any directed subset D, D converges to sup D, as a net.

Definition A topological space X satisfying (i) and (ii) is called a d - space, or monotone convergence space.

Remark:1. For any poset P, the Scott space ΣP is a d-space iff P is directed complete.

2. Every Sober space is a d-space.

3. If X is a d - space, then every open set U of X is a Scott open set of (X, ≤ ), where is the specialization order on X.

poset dcpoP

∑P

Q

∑D

TOP0

DTOP0

Question: Is DTOP reflexive in TOP0?

Theorem [9] [1] [7]

For each T0 space X there is a d –space D(X) and a continuous mapping η : X → D(X) such that for any continuous f: X → Y into a d-space Y, there is a unique Scott continuous g: D(X) → Y such that f= g∙η .

Theorem [7For any poset, the d-completion D( ∑P) of the space ∑P with the specialization order is the D-completion of P.

P

∑P D(∑P)

D(P)

specialization order

Posets

Space

So the D – completion of topological spaces is a generalization of D – completion of posets

4. Net convergence spaces and D-completion

Definition [ 5] A (net ) convergence space is a pair (X, →), where X is a non-empty set and → is a collection of pairs ( S, x ) with S a net in X and x an element in X ( we write S→x if (S, x) belongs to → and say that S converges to x ), such that the following conditions are satisfied:

1.( CONSTANT NET) For any x in X, {x}→x.

2. ( SUBNET) If S → x then S1 → x for any subnet S1 of S.

* A net S=(xi ) is a subnet of T=(yj ) if for any j0 there is i0 such that for any i0 ≤ i , xi belongs to the j0 - tail of T.

Continuous functions between convergence spaces are defined in the usual way. Let CONV be the category of convergence spaces and continuous functions between them.

* CONV is cartesian closed

ExampleEvery topological space (X, τ ) defines convergence space (X, ) in the ordinary sense. Such a convergence space is called a topological convergence space.

Also the assignment of (X, ) to (X, τ ) extends to a functor from TOP to CONV, setting TOP as a full subcategory of CONV.

Definition :A subset U of a convergence space ( X, → ) is open if S=(xi ) → x and x is in U, then xi is in U eventually.

The set of all open sets U of ( X, → ) form a topology , called the

induced topology and denoted by .

The specialization order ≤ of the topological space is called the specialization order of ( X, → ) .

),X(

A space ( X, → ) is up – nice if S=(xi ) converges to a and T=(yj ) is a net satisfying the condition below , then T converges to a: for any i0 there is j0 such that for any j0 ≤ j , yj belongs to ↑{xi : i0 ≤ i }.

A space ( X, → ) is down– nice if S=(xi ) converges to a and b ≤ a , then S converges to b .

A space is called nice, if it is both up-nice and down nice.

For any net S , let lim S={ x: S → x } be the set of all limit points of S.

Definition[5]A convergence space ( X, →) is called a d - space of 1). It is order – nice;

2). For any directed set D={ xi : i in I }, sup D exists and D → sup D ( take D as a net ). 3). For any net, lim S is closed under supremum of directed sets.

Example[5] For each poset P, define → by: S=(xi ) → x if there is a directed set E with sup E exist, x ≤ sup E , and for any e in E, e≤ xi eventually.

Then (i) (P, ≤ ) is order – nice; (ii) (P, ≤ ) is a d – space iff P is a dcpo; (iii) (P, ≤ ) is topological iff P is a continuous poset .

Theorem The category DCONV of d- convergence spaces ( as a subcategory of CONV) is Cartesian closed

Example A topological space (X, τ ) is a d- space iff the convergence space (X, ) is a d – space.

Remark (1) For any poset P, ( P, →d ) is an IL space if it is

topological, iff P is continuous.(2) Every topological convergence space is IL.

DefinitionA space ( X, → ) is called an IL space if it satisfies the Iterated Limit Axiom.

Definition[5] A D – completion of a space ( X, → ) is d – space Y together with a continuous map η : X →Y, such that for any continuous f: X → Z from X into a d – space Z, there is a unique continuous h: Y → Z such that f=h ○η.

Theorem [5]If X can be embedded into an IL d-space, then X has a d – completion.

Corollary Every T0 topological spaces has a d – completion which is also topological.

poset dcpoP

∑P

DTOP0

DCONV

TOP0

X

(X, )

CONV

Problem: Does every convergence space have a d – completion?

References[1] Erchov, Yu. L.: On d-spaces, Theoretical Computer Science, (1999), 224: 59-72.[2] Gierz, G. et al.: Continuous lattices and Domains, Encyclopedia of Mathematics and Its Applications, Vol.93.[3] R. Heckmann: A non- topological view of dcpos as convergence spaces. Theoretical Computer Science, 305(1193), 195 -186.[4] W. Ho, and D. Zhao, Lattices of Scott-closed sets, Comment. Math. Univer. Carolinae, 50(2009), 2:297-314.[5] W. Ho, D. Zhao and W. Wee, d – completion of net convergence space, Proceeding of this conference.[6] A. Jung, M. A. Moshier and S. Vickers, Presenting dcpos and dcpo algebras, (Preprint).[7] Keimel, K. and Lawson, J. D.: D-completions and the d-topology, Annals of Pure and Applied Logic 59(2009), 3:292-306. [8] M. W. Mislove, Local DCPOs, Local CPOs and Local completions, Electronic Notes in Theoretical Computer Science , 20(1999).[9] Wyler, O.: Dedekind complete posets and Scott topologies, In: Lecture Notes In Mathematics, Springer-Verlag, 1981.

[10] L. Xu, Continuity of posets via Scott topology and sobrification, Topology Appl., 153(2006), 11:1886-1894.[11] Zhao, D. and Fan, T., dcpo-completion of posets, Theoretical Computer Science, 411(2010), 2167-2173.

Thank you !