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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: demirci@akdeniz.edu.tr 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 REFERENCES Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of logic in computer science. vol. 3. , pp. 1–168. The Clarendon Press, Oxford University Press, New York (1994) Adámek, J., Koubek, V., Nelson E., Reiterman, J.: Arbitrarily large continuous algebras on one generator. Trans. Amer. Math. Soc. 291, 681-699 (1985) Banaschewski, B., Bruns, G.: The fundamental duality of partially ordered sets. Order 5, 61-74 (1988) Bandelt, H. -J., Erné, M.: The category of Z-continuous posets. J. Pure Appl. 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Montreal, 1992, pp. 113-192, Kluwer Academic Publishers, Amsterdam, Netherlands (1993) Erné, M: Z-continuous posets and their topological manifestation, Appl. Categ. Structures 7, 31-70 (1999) 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 Erné, M: General Stone duality. Topology and its Applications 137, 125-158 (2004) Erné, M.: Choiceless, pointless, but not useless: dualities for preframes. Appl. Categ. Structures 15, 541-572 (2007) Gierz, G., Hofmann, K.H., Keimel K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003) Markowsky, G.: Categories of chain-complete posets. Theor. Comput. Sci. 4, 125-135 (1977) Nelson, E.: Z-continuous algebras. In : Banaschewski, B., Hoffmann, R.E. (eds.) Continuous Lattices. Proc. Conf., Bremen 1979. Lect. Notes Math. 871, pp. 315-334. Springer-Verlag, Berlin (1981) 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 Pasztor, A.: The epis of Pos(Z). Comment. Math. Univ. Carolin. 23, 285-299 (1982) Powers, R.C., Riedel, T.: Z-Semicontinuous posets. Order 20, 365-371 (2003) Venugopalan, G.: Z-continuous posets. Houston J. Math. 12, 275-294 (1986) Wright, J. B., Wagner, E. G., Thatcher, J. W.: A uniform approach to inductive posets and inductive closure. Theoret. Comput. Sci. 7, 57-77 (1978) Zhao, D.: On projective Z-frames. Can. Math. Bull. 40, 39-46 (1997) Mustafa Demirci Akdeniz University, Faculty of Sciences,DepartmentOn ofAugmented Mathematics, Posets 07058-Antalya, And (Z1 , ZTurkey,e-mail: demirci@akdeniz 1 )-Complete Posets
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