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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In this case, the actual left adjoint is

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

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

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

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

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

A continuous complete lattice is often called a continuous lattice.

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

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

• "Continuous lattice", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• "Core-compact space", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Continuous poset at the nLab
• Continuous category at the nLab
• Exponential law for spaces at the nLab
• Continuous poset at PlanetMath.

• Order theory

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• Short description is different from Wikidata