Continuous poset

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

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

If $a$ approximates $b$, we write $a\ll b$. The approximation relation $\ll$ is a transitive relation that is weaker than the original order, also antisymmetric if $P$ is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if $(P,\lesssim )$ satisfies the ascending chain condition.Template:Rp

For any $a\in P$, let

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

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

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

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

Properties

The interpolation property

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

Continuous dcpos

For any two elements $a,b\in P$ of a continuous dcpo $(P,\le )$, the following two conditions are equivalent.Template:Rp

• $a\ll b$ and $a\ne b$.
• For any directed set $D\subseteq P$ such that $b\le \sup D$, there is a $d\in D$ such that $a\ll d$ and $a\ne d$.

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any $a,b\in P$ such that $a\ll b$ and $a\ne b$, there is a $c\in P$ such that $a\ll c\ll b$ and $a\ne c$.Template:Rp

For a dcpo $(P,\le )$, the following conditions are equivalent.Template:Rp

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

In this case, the actual left adjoint is

$\Downarrow :P\to \mathrm{Ideal}(P)$

$\Downarrow ⊣\sup$

Continuous complete lattices

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

Let $L$ be a complete lattice. Then the following conditions are equivalent.

• $L$ is continuous.
• The supremum map $\sup :\mathrm{Ideal}(L)\to L$ from the complete lattice of ideals of $L$ to $L$ preserves arbitrary infima.
• For any family $𝒟$ of directed sets of $L$, $\underset{D\in 𝒟}{\inf }\sup D=\underset{f\in \prod 𝒟}{\sup }\underset{D\in 𝒟}{\inf }f(D)$.
• $L$ is isomorphic to the image of a Scott-continuous idempotent map $r:\{0,1{\}}^{\kappa }\to \{0,1{\}}^{\kappa }$ on the direct power of arbitrarily many two-point lattices $\{0,1\}$.Template:Rp

A continuous complete lattice is often called a continuous lattice.

Examples

Lattices of open sets

For a topological space $X$, the following conditions are equivalent.

• The complete Heyting algebra $\mathrm{Open}(X)$ of open sets of $X$ is a continuous complete Heyting algebra.
• The sobrification of $X$ is a locally compact space (in the sense that every point has a compact local base)
• $X$ is an exponentiable object in the category $\mathrm{Top}$ of topological spaces.Template:Rp That is, the functor $(-)\times X:\mathrm{Top}\to \mathrm{Top}$ has a right adjoint.

External links

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