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 $a,b\in P$ 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.

For any $a\in P$, let
:$\mathop\Uparrow a=\$
:$\mathop\Downarrow a=\$
Then $\mathop\Uparrow a$ is an upper set, and $\mathop\Downarrow a$ a lower set. If $P$ is an upper-semilattice, $\mathop\Downarrow a$ 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 $\mathop\Downarrow a$ is directed and $a=\sup\mathop\Downarrow a$.

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.
* $a\ll b$ 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 $a\ne c$.

For a dcpo $(P,\le)$, the following conditions are equivalent.
* $P$ is continuous.
* The supremum map $\sup \colon \operatorname(P)\to P$ from the partially ordered set of ideals of $P$ to $P$ has a left adjoint.
In this case, the actual left adjoint is
:$\colon P\to\operatorname(P)$
:$\mathord\Downarrow\dashv\sup$

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 $\sup \colon \operatorname(L)\to L$ from the complete lattice of ideals of $L$ to $L$ preserves arbitrary infima.
* For any family $\mathcal D$ of directed sets of $L$, $\textstyle\inf_\sup D=\sup_\inf_f(D)$.
* $L$ is isomorphic to the image of a Scott-continuous idempotent map $r \colon \^\kappa\to\^\kappa$ on the direct power of arbitrarily many two-point lattices $\$.

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 $\operatorname(X)$ 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 $\operatorname$ of topological spaces. That is, the functor $(-)\times X\colon\operatorname\to\operatorname$ has a right adjoint.

References

External links

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* {{PlanetMath, urlname=ContinuousPoset, title=Continuous poset

Order theory