dcpo—completion of posets zhao dongsheng national institute of education nanyang technological...
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Dcpo—Completion of posetsZhao Dongsheng
National Institute of EducationNanyang Technological University
Singapore
Fan TaiheDepartment of Mathematics
Zhejiang Science and Tech UniversityChina
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Outline
• Introduction
• D-completion of posets
• Properties
• Bounded complete dcpo completion
• Bounded sober spaces
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1.Introduction
The Scott open (closed ) set lattices:
For any poset P, let σ(P) (σop(P) ) be the
complete lattice of all Scott open ( closed ) sets of P .
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DomainsCompletely Distributive Lattices
σ
Continuous posets
Con Lattices
Stably Sup Cont Lattices
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Complete lattices
σop
Stably C-algebraiclattices
Complete semilattices
Weak Stably C-algebraiclattice
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dcpos σop
posets
L
M
M = L ?
σop
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2. Dcpo – completion of posets
DefinitionLet P be a poset. The D – completion of P is a dcpo S together with a Scott continuous mapping f: P → S such that for any Scott continuous mapping g: P → Q from P into a dcpo Q, there is a unique Scott continuous mapping g*: S → Q such that g=g*f.
P Sf
Qg
g*
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Every two D-completions of a poset are isomorphic !
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Theorem 1. For each poset P, the smallest subdcpo E(P) of σop (P) containing { ↓x: x ε P } is a D- completion of P. The universal Scott continuous mapping η : P→ E(P) sends x in P to ↓x.
Subdcpo--- a subset of a dcpo that is closed under taking supremum of directed sets.
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Let POSd be the category of posets and the Scott continuous mappings.
Corollary The subcategory DCPO of POSd consisting of dcpos is fully reflexive in POSd .
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2. Properties of D(P)
Theorem 2. For any poset P, σop (P) σop (E(P)).
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dcpos σop
posets
L
M
M = L
σop
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Theorem 3. A poset P is continuous if and only if E(P) is continuous.
Corollary If P is a continuous poset, then E(P)=Spec(σop (P)).
Spec(L): The set of co-prime elements of L
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Theorem 4 . A poset P is algebraic if and only if E(P) is algebraic.
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3. Local dcpo-completion
Definition A poset P is called a local dcpo, if every upper bounded directed subset has a supremum in P.
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DefinitionLet P be a poset. A local dcpo completion of P is a local dcpo S together with a Scott continuous mapping f: P → S such that for any Scott continuous mapping g: P → Q from P into a local dcpo Q, there is a unique Scott continuous mapping g*: S → Q such that g=g*f.
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Theorem 5For any poset P, the set BE(P) of all the members F of E(P) which has an upper bound in P and the mapping f: P → BE(P), that sends x to ↓x, is a local dcpo completion of P.
Corollary The subcategory LD of POSd consisting of local dcpos is reflexive in POSd .
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4. Bounded sober spaces
Definition A T0 space X is called bounded sober if every non empty co-prime closed subset of X that is upper bounded in the specialization order * is the closure of a point .
* If it is contained in some cl{x}.
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Example If P is a continuous local dcpo, then ΣP is bounded sober.
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For a T0 space X, let
B(X)={ F: F is a co-prime closed set and is upper bounded in the specialization order }.
For each open set U of X, let HU ={ Fε B(X): F∩ U≠ ᴓ }. Then {HU : U ε O(X)} is topology on B(X).
Let B(X) denote this topological space.
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Theorem 6 B(X) is the reflection of X in the subcategory BSober of bounded sobers spaces.
XB(X)
Top0BSober
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One question:
If P and Q are dcpos such that σ(P) is isomorphic to σ(Q), must P and Q be isomorphic?
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THANK YOU!