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.^[1]^{: p.52, Examples I-1.3, (4)}

For any $a\in P$ , let

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

\mathop {\Downarrow } a=\{b\in L\mid b\ll 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 $(P,\leq )$ , the following two conditions are equivalent.^[1]^{: p.61, Proposition I-1.19(i)}

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

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 $a\neq b$ , there is a $c\in P$ such that $a\ll c\ll b$ and $a\neq c$ .^[1]^{: p.61, Proposition I-1.19(ii)}

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

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

In this case, the actual left adjoint is

{\Downarrow }\colon P\to \operatorname {Ideal} (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 $b\leq \sup A$ , there is a finite subset $F\subseteq A$ such that $a\leq \sup F$ .

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

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

A continuous complete lattice is often called a continuous lattice.

Examples

Lattices of open sets

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

The complete Heyting algebra $\operatorname {Open} (X)$ 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 $\operatorname {Top}$ of topological spaces.^[1]^{: p.196, Theorem II-4.12} That is, the functor $(-)\times X\colon \operatorname {Top} \to \operatorname {Top}$ has a right adjoint.

References

^ ^a ^b ^c ^d ^e 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.
^ 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.