In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.

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.

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.

  • 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}.

For a dcpo ( P , ≤ ) {\displaystyle (P,\leq )}, the following conditions are equivalent.

  • 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\}}.

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.

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