Slides.

Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
On Augmented Posets And (Z1 , Z1 )-Complete
Posets
Mustafa Demirci
Akdeniz University, Faculty of Sciences,
Department of Mathematics,
07058-Antalya, Turkey,
e-mail: [email protected]
July 11, 2011
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
1
Banaschewski and Bruns’s approach (BB-approach)
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
1
Banaschewski and Bruns’s approach (BB-approach)
2
Subset selection-based approach (Z-approach)
Q-spaces and their category Q-SPC
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
1
Banaschewski and Bruns’s approach (BB-approach)
2
Subset selection-based approach (Z-approach)
Q-spaces and their category Q-SPC
3
Relation between BB-approach and Z-approach
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Banaschewski and Bruns’s approach (BB-approach)
Primary concept in their approach is augmented poset that is a
triple U = (|U| , JU, MU), consisting of a poset |U|, a subset JU
of P (U) in which each member has the join in |U| and a subset
MU of P (U) in which each member has the meet in |U|.
Augmented posets together with structure preserving maps
constitute a category P. A structure preserving map h : U → V
here means a monotone map h : |U| → |V | with the properties
W
W
(a) For all S ∈ JU, h (S) ∈ JV and h ( S) = h (S),
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Banaschewski and Bruns’s approach (BB-approach)
Primary concept in their approach is augmented poset that is a
triple U = (|U| , JU, MU), consisting of a poset |U|, a subset JU
of P (U) in which each member has the join in |U| and a subset
MU of P (U) in which each member has the meet in |U|.
Augmented posets together with structure preserving maps
constitute a category P. A structure preserving map h : U → V
here means a monotone map h : |U| → |V | with the properties
W
W
(a) For all S ∈ JU, h (S) ∈ JV and h ( S) = h (S),
V
V
(b) For all R ∈ MU, h (R) ∈ MV and h ( R) = h (R).
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
The category of spaces, denoted by S, is another central concept
in BB-approach. The objects of S (the so-called spaces) and its
morphisms generalize the notions of topological spaces and
continuous functions. A space is defined to be a quadruple
W = (|W | , D(W ), Σ(W ), ∆(W )) fulfilling the properties
(S1) |W | is a set,
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
The category of spaces, denoted by S, is another central concept
in BB-approach. The objects of S (the so-called spaces) and its
morphisms generalize the notions of topological spaces and
continuous functions. A space is defined to be a quadruple
W = (|W | , D(W ), Σ(W ), ∆(W )) fulfilling the properties
(S1) |W | is a set,
(S2) D(W ) is a subset of P (|W |),
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
The category of spaces, denoted by S, is another central concept
in BB-approach. The objects of S (the so-called spaces) and its
morphisms generalize the notions of topological spaces and
continuous functions. A space is defined to be a quadruple
W = (|W | , D(W ), Σ(W ), ∆(W )) fulfilling the properties
(S1) |W | is a set,
(S2) D(W ) is a subset of P (|W |),
(S3) Σ(W ) consists of all U ⊆ D(W ) for which ∪U ∈ D(W ),
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
The category of spaces, denoted by S, is another central concept
in BB-approach. The objects of S (the so-called spaces) and its
morphisms generalize the notions of topological spaces and
continuous functions. A space is defined to be a quadruple
W = (|W | , D(W ), Σ(W ), ∆(W )) fulfilling the properties
(S1) |W | is a set,
(S2) D(W ) is a subset of P (|W |),
(S3) Σ(W ) consists of all U ⊆ D(W ) for which ∪U ∈ D(W ),
(S4) ∆(W ) consists of all B ⊆ D(W ) with ∩B ∈ D(W ).
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A morphism f : W1 → W2 in S is a function f : |W1 | → |W2 |
satisfying the next properties
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A morphism f : W1 → W2 in S is a function f : |W1 | → |W2 |
satisfying the next properties
(S5) (f ← )→ (D(W2 )) ⊆ D(W1 ),
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A morphism f : W1 → W2 in S is a function f : |W1 | → |W2 |
satisfying the next properties
(S5) (f ← )→ (D(W2 )) ⊆ D(W1 ),
(S6) (f ← )→ (U) ∈ Σ(W1 ) for all U ∈ Σ(W2 ),
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A morphism f : W1 → W2 in S is a function f : |W1 | → |W2 |
satisfying the next properties
(S5) (f ← )→ (D(W2 )) ⊆ D(W1 ),
(S6) (f ← )→ (U) ∈ Σ(W1 ) for all U ∈ Σ(W2 ),
(S7) (f ← )→ (B) ∈ ∆(W1 ) and for all B ∈ ∆(W2 ).
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A morphism f : W1 → W2 in S is a function f : |W1 | → |W2 |
satisfying the next properties
(S5) (f ← )→ (D(W2 )) ⊆ D(W1 ),
(S6) (f ← )→ (U) ∈ Σ(W1 ) for all U ∈ Σ(W2 ),
(S7) (f ← )→ (B) ∈ ∆(W1 ) and for all B ∈ ∆(W2 ).
Theorem
[3] P is dually adjoint to S, i.e. there are functors Ψ :Pop →S and
T :S→Pop such that T a Ψ :Pop →S.
Corollary
[3] The full subcategory of P of all spatial objects ( SpaP) and the
full subcategory of S of all sober objects ( SobS) are dually
equivalent.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Subset selection-based approach (Z-approach)
Z-approach uses the notion of subset selection: A subset selection
Z is a rule assigning to each poset P a subset Z (P) of its power
set P (P).
Subset selection Z
V
V⊥
Pn
F
CN
D
BN
elements of Z (P)
no subset of P
the empty set ∅
nonempty subsets of P with cardinality less
than or equal to n
finite subsets of P
countable subsets of P
directed subsets of P
bounded subsets of P
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
C
C⊥
P
A
W
Q-spaces and their category Q-SPC
nonempty linearly ordered subsets (chains) of P
linearly ordered subsets (including ∅) of P
subsets of P
downsets of P
well-ordered chains of P
A subset selection Z is called a subset system [2, 6, 14] iff for each
order-preserving function f : P → Q, the implication
M ∈ Z (P) ⇒ f (M) ∈ Z (Q) holds.
During this talk, Z will be assumed as a subset selection unless
further assumptions are made.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Definition
[8, 18] A poset P with the property that each M ∈ Z (P) has the
join (meet) in P is called a Z-join(meet)-complete poset.
For simplicity, we call a Z1 -join-complete and Z2 -meet-complete
poset a (Z1 , Z2 )-complete poset. There are two useful subset
selections Z1sup and Z2inf derived from given two original subset
selections Z1 and Z2 by the formulas
n
o
_
M ∈ Z1 (P) |
M exists in P ,
n
o
^
Z2inf (P) =
N ∈ Z2 (P) |
N exists in P ,
Z1sup (P) =
With the help of these derivedsubset selections, every poset can
be introduced as a Z1sup , Z2inf -complete poset.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Definition
A monotone map f : P → Q is (Z1 , Z2 )-continuous iff the
following two conditions are satisfied:
W
W
(i) For each M ∈ Z1sup (P), f ( M) = f (M),
V
V
(ii) For each N ∈ Z2inf (P), f ( N) = f (N).
Under the assumption “Z3 and Z4 are subset systems”,
(Z1 , Z2 )-complete posets and (Z3 , Z4 )-continuous maps constitute
a category Q-CPos, where Q stands for (Z1 , Z2 , Z3 , Z4 ). Q-CPos
provides a practically useful categorical framework for many
order-theoretic structures.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Example
(Examples of Q-CPos)
(F, V, V, V)-CPos=Category of join-semilattices with ⊥ and
monotone maps (SEMI [12]),
(D, V, D, V)-CPos=Category of directed-complete posets and
Scott-continuous functions (DCPO [1]),
(P, P, C, V)-CPos=Category of complete lattices and maps
preserving joins of chains (LC [13]),
(P, P, V, V)-CPos=Category of complete lattices and monotone
maps (LI [13]).
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Q-spaces and their category Q-SPC
Z-approach suggests another generalization of topological space
called Q-space: A Q-space is, by definition, a pair (X , τ )
consisting of a set X and a subset τ (so-called a Q-system on X )
of P (X ) such that the inclusion map iτ : (τ, ⊆) ,→ (P (X ) , ⊆) is a
Q-CPos-morphism.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Example
(Examples of Q-systems)
(V, V, V, V)-system=System [7, 8], ,
(P, F, P, F)-system=Topology [7, 8],
(F, P, F, P)-system=Topological closure system [7, 8],
(V, P, V, P)-system=Closure system [7, 8],
(D, P, D, P)-system=Algebraic closure system [7, 8],
(D, F, D, F)-system=Pretopology [11],
(D, F, P, F)-system τ =Pretopology τ such that for
S each V ⊆ τ , if
V is not directed but has the join in (τ, ⊆) , then V ∈ τ ,
(P, P, C⊥ , V)-system τ =(τ, ⊆) is a complete lattice such that joins
of chains are exactly unions of chains.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Q-spaces and their category Q-SPC
Continuous functions turn into Q-space-continuous functions in
Z-approach:
A map f : (X , τ ) → (Y , ν) between Q-spaces (X , τ ) and (Y , ν) is
Q-space-continuous if the usual requirement of continuity (i.e.
(f ← )→ (ν) ⊆ τ ) is satisfied.
Q-spaces and Q-space-continuous maps form a category Q-SPC
extending the familiar category of topological spaces (Top) to the
Z-approach.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Relation between BB-approach and Z-approach
We describe this relation via the functors GQ : Q-CPos→P and
HQ : Q-SPC→S, defined by
P, Z3sup (P) , Z4inf (P) , GQ (f ) = f ,
HQ (X , τ ) =
X , τ, Z3sup (τ ) , Z4inf (τ ) and HQ (g ) = g
GQ (P) =
It is easy to check that GQ and HQ are full embeddings, and so
BB-approach is more general than Z-approach. Using these full
embeddings, we may formulate spatiality and sobriety in
Z-approach as follows:
Definition
(i) A (Z1 , Z2 )-complete poset P is Q-spatial iff GQ (P) is spatial,
(ii) A Q-space (X , τ ) is Q-sober iff HQ (X , τ ) is sober.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Theorem
(Main result) Assume that Z1 , Z2 are iso-invariant subset
selections and Z3 , Z4 are subset systems. Let Q-CPoss and
Q-SPCs denote the full subcategory of Q-CPos of all Q-spatial
objects and the full subcategory of Q-SPC of all Q-sober objects.
(i) Q-CPoss and Q-SPC are dually adjoint to each other.
(ii) Q-CPoss is dually equivalent to Q-SPCs .
(iii) If Z1 and Z2 are surjectivity-preserving subset systems, then
(Z1 , Z2 , Z1 , Z2 )-CPos is dually adjoint to (Z1 , Z2 , Z1 , Z2 )-SPC.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
A subset selection Z is iso-invariant [6] iff for each
order-isomorphism f : P → Q, the implication
M ∈ Z (P) ⇒ f (M) ∈ Z (Q) holds
A subset system Z is surjectivity-preserving iff for each surjective
monotone map f : P → Q and for each M ∈ Z (Q), there exists
at least one N ∈ Z (P) such that f (N) = M.
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
Thank you for attending....
Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
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Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
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Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
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Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets
Outline
Banaschewski and Bruns’s approach (BB-approach)
Subset selection-based approach (Z-approach)
Relation between BB-approach and Z-approach
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Carolin. 23, 285-299 (1982)
Powers, R.C., Riedel, T.: Z-Semicontinuous posets. Order 20,
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Venugopalan, G.: Z-continuous posets. Houston J. Math. 12,
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Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn
ofAugmented
Mathematics,
Posets
07058-Antalya,
And (Z1 , ZTurkey,e-mail:
demirci@akdeniz
1 )-Complete Posets