• Images
• Videos
• Documents
• Books
• Articles

Definitions

Let a , b ∈ P {\displaystyle a,b\in P} be two elements of a preordered set ( P , ≲ ) {\displaystyle (P,\lesssim )} . Then we say that a {\displaystyle a} approximates b {\displaystyle b} , or that a {\displaystyle a} is way-below b {\displaystyle b} , if the following two equivalent conditions are satisfied.

• For any directed set D ⊆ P {\displaystyle D\subseteq P} such that b ≲ sup D {\displaystyle b\lesssim \sup D} , there is a d ∈ D {\displaystyle d\in D} such that a ≲ d {\displaystyle a\lesssim d} .
• For any ideal I ⊆ P {\displaystyle I\subseteq P} such that b ≲ sup I {\displaystyle b\lesssim \sup I} , a ∈ I {\displaystyle a\in I} .

If a {\displaystyle a} approximates b {\displaystyle b} , we write a ≪ b {\displaystyle a\ll b} . The approximation relation ≪ {\displaystyle \ll } is a transitive relation that is weaker than the original order, also antisymmetric if P {\displaystyle P} is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if ( P , ≲ ) {\displaystyle (P,\lesssim )} satisfies the ascending chain condition.¹: p.52, Examples I-1.3, (4) 

For any a ∈ P {\displaystyle a\in P} , let

⇑ ⁡ a = { b ∈ L ∣ a ≪ b } {\displaystyle \mathop {\Uparrow } a=\{b\in L\mid a\ll b\}} ⇓ ⁡ a = { b ∈ L ∣ b ≪ a } {\displaystyle \mathop {\Downarrow } a=\{b\in L\mid b\ll a\}}

Then ⇑ ⁡ a {\displaystyle \mathop {\Uparrow } a} is an upper set, and ⇓ ⁡ a {\displaystyle \mathop {\Downarrow } a} a lower set. If P {\displaystyle P} is an upper-semilattice, ⇓ ⁡ a {\displaystyle \mathop {\Downarrow } a} is a directed set (that is, b , c ≪ a {\displaystyle b,c\ll a} implies b ∨ c ≪ a {\displaystyle b\vee c\ll a} ), and therefore an ideal.

A preordered set ( P , ≲ ) {\displaystyle (P,\lesssim )} is called a continuous preordered set if for any a ∈ P {\displaystyle a\in P} , the subset ⇓ ⁡ a {\displaystyle \mathop {\Downarrow } a} is directed and a = sup ⇓ ⁡ a {\displaystyle a=\sup \mathop {\Downarrow } a} .

Properties

The interpolation property

For any two elements a , b ∈ P {\displaystyle a,b\in P} of a continuous preordered set ( P , ≲ ) {\displaystyle (P,\lesssim )} , a ≪ b {\displaystyle a\ll b} if and only if for any directed set D ⊆ P {\displaystyle D\subseteq P} such that b ≲ sup D {\displaystyle b\lesssim \sup D} , there is a d ∈ D {\displaystyle d\in D} such that a ≪ d {\displaystyle a\ll d} . From this follows the interpolation property of the continuous preordered set ( P , ≲ ) {\displaystyle (P,\lesssim )} : for any a , b ∈ P {\displaystyle a,b\in P} such that a ≪ b {\displaystyle a\ll b} there is a c ∈ P {\displaystyle c\in P} such that a ≪ c ≪ b {\displaystyle a\ll c\ll b} .

Continuous dcpos

For any two elements a , b ∈ P {\displaystyle a,b\in P} of a continuous dcpo ( P , ≤ ) {\displaystyle (P,\leq )} , the following two conditions are equivalent.²: p.61, Proposition I-1.19(i) 

• a ≪ b {\displaystyle a\ll b} and a ≠ b {\displaystyle a\neq b} .
• For any directed set D ⊆ P {\displaystyle D\subseteq P} such that b ≤ sup D {\displaystyle b\leq \sup D} , there is a d ∈ D {\displaystyle d\in D} such that a ≪ d {\displaystyle a\ll d} and a ≠ d {\displaystyle a\neq d} .

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any a , b ∈ P {\displaystyle a,b\in P} such that a ≪ b {\displaystyle a\ll b} and a ≠ b {\displaystyle a\neq b} , there is a c ∈ P {\displaystyle c\in P} such that a ≪ c ≪ b {\displaystyle a\ll c\ll b} and a ≠ c {\displaystyle a\neq c} .³: p.61, Proposition I-1.19(ii) 

For a dcpo ( P , ≤ ) {\displaystyle (P,\leq )} , the following conditions are equivalent.⁴: Theorem I-1.10 

• P {\displaystyle P} is continuous.
• The supremum map sup : Ideal ⁡ ( P ) → P {\displaystyle \sup \colon \operatorname {Ideal} (P)\to P} from the partially ordered set of ideals of P {\displaystyle P} to P {\displaystyle P} has a left adjoint.

In this case, the actual left adjoint is

⇓ : P → Ideal ⁡ ( P ) {\displaystyle {\Downarrow }\colon P\to \operatorname {Ideal} (P)} ⇓ ⊣ sup {\displaystyle {\mathord {\Downarrow }}\dashv \sup }

Continuous complete lattices

For any two elements a , b ∈ L {\displaystyle a,b\in L} of a complete lattice L {\displaystyle L} , a ≪ b {\displaystyle a\ll b} if and only if for any subset A ⊆ L {\displaystyle A\subseteq L} such that b ≤ sup A {\displaystyle b\leq \sup A} , there is a finite subset F ⊆ A {\displaystyle F\subseteq A} such that a ≤ sup F {\displaystyle a\leq \sup F} .

Let L {\displaystyle L} be a complete lattice. Then the following conditions are equivalent.

• L {\displaystyle L} is continuous.
• The supremum map sup : Ideal ⁡ ( L ) → L {\displaystyle \sup \colon \operatorname {Ideal} (L)\to L} from the complete lattice of ideals of L {\displaystyle L} to L {\displaystyle L} preserves arbitrary infima.
• For any family D {\displaystyle {\mathcal {D}}} of directed sets of L {\displaystyle L} , inf D ∈ D sup D = sup f ∈ ∏ D inf D ∈ D f ( D ) {\displaystyle \textstyle \inf _{D\in {\mathcal {D}}}\sup D=\sup _{f\in \prod {\mathcal {D}}}\inf _{D\in {\mathcal {D}}}f(D)} .
• L {\displaystyle L} is isomorphic to the image of a Scott-continuous idempotent map r : { 0 , 1 } κ → { 0 , 1 } κ {\displaystyle r\colon \{0,1\}^{\kappa }\to \{0,1\}^{\kappa }} on the direct power of arbitrarily many two-point lattices { 0 , 1 } {\displaystyle \{0,1\}} .⁵: p.56, Theorem 44 

A continuous complete lattice is often called a continuous lattice.

Examples

Lattices of open sets

For a topological space X {\displaystyle X} , the following conditions are equivalent.

• The complete Heyting algebra Open ⁡ ( X ) {\displaystyle \operatorname {Open} (X)} of open sets of X {\displaystyle X} is a continuous complete Heyting algebra.
• The sobrification of X {\displaystyle X} is a locally compact space (in the sense that every point has a compact local base)
• X {\displaystyle X} is an exponentiable object in the category Top {\displaystyle \operatorname {Top} } of topological spaces.⁶: p.196, Theorem II-4.12  That is, the functor ( − ) × X : Top → Top {\displaystyle (-)\times X\colon \operatorname {Top} \to \operatorname {Top} } has a right adjoint.

External links

• "Continuous lattice", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Core-compact space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Continuous poset at the nLab
• Continuous category at the nLab
• Exponential law for spaces at the nLab
• Continuous poset at PlanetMath.

References

1. Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001. 978-0-521-80338-0 ↩

2. Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001. 978-0-521-80338-0 ↩

3. Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001. 978-0-521-80338-0 ↩

4. Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001. 978-0-521-80338-0 ↩

5. Grätzer, George (2011). Lattice Theory: Foundation. Basel: Springer. doi:10.1007/978-3-0348-0018-1. ISBN 978-3-0348-0017-4. LCCN 2011921250. MR 2768581. Zbl 1233.06001. 978-3-0348-0017-4 ↩

6. Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. Vol. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0. MR 1975381. Zbl 1088.06001. 978-0-521-80338-0 ↩