Scott Closed Set Lattices And 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

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 )

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.

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).

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.

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

Definition A mapping f : P→ Q is called Scott continuous if it preserves 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

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.

Domain Theory Theory of Computation General Topology Analysis and Algebra Category Theory and Logic

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.

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

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.

Definition A D-completion of a poset P is a dcpo 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 mapping h:A →B satisfying. PA B f h 2. Dcpo-completion of posets

Question: Dose every poset has a D- completion? What are the other connections of posets and their D- completions?

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. Every Scott closed set is a subdcpo All subdcpos form a co-topolgy

Theorem Let 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 this Is a D-completion of P

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 lattices Posets

Theorem A poset P is a continuous poset if and only if its D-completion is a continuous dcpo. posets dcpos Contin uous dcpos Continu ous posets P

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 ?

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.

Definition An element x of a lattice L is called co-prime if for any y,z in L, implies 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

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)). P L C(-) Spec(-)

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.

Two functors P Q C(P) C(Q) C(f) C CONPCDL CONPCDL L M Spec(L) Spec(M) g Spec

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.

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.

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 power structures -- powerdomains, in several ways

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.

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. An equation ( inequality ) is of the form ( ), where are terms in

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.

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. The free infaltionary semilattice of domain is called the Hoare Power domain

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

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?

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. Definition A complete lattice L is called C-continuous if for any a in L,

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

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,

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.

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

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.

Theorem For any complete semilattice L, 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.

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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Thank You! Zhao Dongsheng Mathematics and Mathematics Education National Institute of Education Nanyang Technological University Singapore