Continuous poset

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

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

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

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

${\displaystyle \mathop {\Uparrow } a=\{b\in L\mid a\ll b\}}$

${\displaystyle \mathop {\Downarrow } a=\{b\in L\mid b\ll a\}}$

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

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

Properties

The interpolation property

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

Continuous dcpos

For any two elements ${\displaystyle a,b\in P}$ of a continuous dcpo ${\displaystyle (P,\leq )}$, the following two conditions are equivalent.^{[1]: p.61, Proposition I-1.19(i)}

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

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any ${\displaystyle a,b\in P}$ such that ${\displaystyle a\ll b}$ and ${\displaystyle a\neq b}$, there is a ${\displaystyle c\in P}$ such that ${\displaystyle a\ll c\ll b}$ and ${\displaystyle a\neq c}$.^{[1]: p.61, Proposition I-1.19(ii)}

For a dcpo ${\displaystyle (P,\leq )}$, the following conditions are equivalent.^{[1]: Theorem I-1.10}

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

In this case, the actual left adjoint is

${\displaystyle {\Downarrow }\colon P\to \operatorname {Ideal} (P)}$

${\displaystyle {\mathord {\Downarrow }}\dashv \sup }$

Continuous complete lattices

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

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

• ${\displaystyle L}$ is continuous.
• The supremum map ${\displaystyle \sup \colon \operatorname {Ideal} (L)\to L}$ from the complete lattice of ideals of ${\displaystyle L}$ to ${\displaystyle L}$ preserves arbitrary infima.
• For any family ${\displaystyle {\mathcal {D}}}$ of directed sets of ${\displaystyle L}$, ${\displaystyle \textstyle \inf _{D\in {\mathcal {D}}}\sup D=\sup _{f\in \prod {\mathcal {D}}}\inf _{D\in {\mathcal {D}}}f(D)}$.
• ${\displaystyle L}$ is isomorphic to the image of a Scott-continuous idempotent map ${\displaystyle r\colon \{0,1\}^{\kappa }\to \{0,1\}^{\kappa }}$ on the direct power of arbitrarily many two-point lattices ${\displaystyle \{0,1\}}$.^{[2]: p.56, Theorem 44}

A continuous complete lattice is often called a continuous lattice.

Examples

Lattices of open sets

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

• The complete Heyting algebra ${\displaystyle \operatorname {Open} (X)}$ of open sets of ${\displaystyle X}$ is a continuous complete Heyting algebra.
• The sobrification of ${\displaystyle X}$ is a locally compact space (in the sense that every point has a compact local base)
• ${\displaystyle X}$ is an exponentiable object in the category ${\displaystyle \operatorname {Top} }$ of topological spaces.^{[1]: p.196, Theorem II-4.12} That is, the functor ${\displaystyle (-)\times X\colon \operatorname {Top} \to \operatorname {Top} }$ has a right adjoint.