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 [math]\displaystyle{ a,b\in P }[/math] be two elements of a preordered set [math]\displaystyle{ (P,\lesssim) }[/math]. Then we say that [math]\displaystyle{ a }[/math] approximates [math]\displaystyle{ b }[/math], or that [math]\displaystyle{ a }[/math] is way-below [math]\displaystyle{ b }[/math], if the following two equivalent conditions are satisfied.
- For any directed set [math]\displaystyle{ D\subseteq P }[/math] such that [math]\displaystyle{ b\lesssim\sup D }[/math], there is a [math]\displaystyle{ d\in D }[/math] such that [math]\displaystyle{ a\lesssim d }[/math].
- For any ideal [math]\displaystyle{ I\subseteq P }[/math] such that [math]\displaystyle{ b\lesssim\sup I }[/math], [math]\displaystyle{ a\in I }[/math].
If [math]\displaystyle{ a }[/math] approximates [math]\displaystyle{ b }[/math], we write [math]\displaystyle{ a\ll b }[/math]. The approximation relation [math]\displaystyle{ \ll }[/math] is a transitive relation that is weaker than the original order, also antisymmetric if [math]\displaystyle{ P }[/math] is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if [math]\displaystyle{ (P,\lesssim) }[/math] satisfies the ascending chain condition.[1]:p.52, Examples I-1.3, (4)
For any [math]\displaystyle{ a\in P }[/math], let
- [math]\displaystyle{ \mathop\Uparrow a=\{b\in L\mid a\ll b\} }[/math]
- [math]\displaystyle{ \mathop\Downarrow a=\{b\in L\mid b\ll a\} }[/math]
Then [math]\displaystyle{ \mathop\Uparrow a }[/math] is an upper set, and [math]\displaystyle{ \mathop\Downarrow a }[/math] a lower set. If [math]\displaystyle{ P }[/math] is an upper-semilattice, [math]\displaystyle{ \mathop\Downarrow a }[/math] is a directed set (that is, [math]\displaystyle{ b,c\ll a }[/math] implies [math]\displaystyle{ b\vee c\ll a }[/math]), and therefore an ideal.
A preordered set [math]\displaystyle{ (P,\lesssim) }[/math] is called a continuous preordered set if for any [math]\displaystyle{ a\in P }[/math], the subset [math]\displaystyle{ \mathop\Downarrow a }[/math] is directed and [math]\displaystyle{ a=\sup\mathop\Downarrow a }[/math].
Properties
The interpolation property
For any two elements [math]\displaystyle{ a,b\in P }[/math] of a continuous preordered set [math]\displaystyle{ (P,\lesssim) }[/math], [math]\displaystyle{ a\ll b }[/math] if and only if for any directed set [math]\displaystyle{ D\subseteq P }[/math] such that [math]\displaystyle{ b\lesssim\sup D }[/math], there is a [math]\displaystyle{ d\in D }[/math] such that [math]\displaystyle{ a\ll d }[/math]. From this follows the interpolation property of the continuous preordered set [math]\displaystyle{ (P,\lesssim) }[/math]: for any [math]\displaystyle{ a,b\in P }[/math] such that [math]\displaystyle{ a\ll b }[/math] there is a [math]\displaystyle{ c\in P }[/math] such that [math]\displaystyle{ a\ll c\ll b }[/math].
Continuous dcpos
For any two elements [math]\displaystyle{ a,b\in P }[/math] of a continuous dcpo [math]\displaystyle{ (P,\le) }[/math], the following two conditions are equivalent.[1]:p.61, Proposition I-1.19(i)
- [math]\displaystyle{ a\ll b }[/math] and [math]\displaystyle{ a\ne b }[/math].
- For any directed set [math]\displaystyle{ D\subseteq P }[/math] such that [math]\displaystyle{ b\le\sup D }[/math], there is a [math]\displaystyle{ d\in D }[/math] such that [math]\displaystyle{ a\ll d }[/math] and [math]\displaystyle{ a\ne d }[/math].
Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any [math]\displaystyle{ a,b\in P }[/math] such that [math]\displaystyle{ a\ll b }[/math] and [math]\displaystyle{ a\ne b }[/math], there is a [math]\displaystyle{ c\in P }[/math] such that [math]\displaystyle{ a\ll c\ll b }[/math] and [math]\displaystyle{ a\ne c }[/math].[1]:p.61, Proposition I-1.19(ii)
For a dcpo [math]\displaystyle{ (P,\le) }[/math], the following conditions are equivalent.[1]:Theorem I-1.10
- [math]\displaystyle{ P }[/math] is continuous.
- The supremum map [math]\displaystyle{ \sup \colon \operatorname{Ideal}(P)\to P }[/math] from the partially ordered set of ideals of [math]\displaystyle{ P }[/math] to [math]\displaystyle{ P }[/math] has a left adjoint.
In this case, the actual left adjoint is
- [math]\displaystyle{ {\Downarrow} \colon P\to\operatorname{Ideal}(P) }[/math]
- [math]\displaystyle{ \mathord\Downarrow\dashv\sup }[/math]
Continuous complete lattices
For any two elements [math]\displaystyle{ a,b\in L }[/math] of a complete lattice [math]\displaystyle{ L }[/math], [math]\displaystyle{ a\ll b }[/math] if and only if for any subset [math]\displaystyle{ A\subseteq L }[/math] such that [math]\displaystyle{ b\le\sup A }[/math], there is a finite subset [math]\displaystyle{ F\subseteq A }[/math] such that [math]\displaystyle{ a\le\sup F }[/math].
Let [math]\displaystyle{ L }[/math] be a complete lattice. Then the following conditions are equivalent.
- [math]\displaystyle{ L }[/math] is continuous.
- The supremum map [math]\displaystyle{ \sup \colon \operatorname{Ideal}(L)\to L }[/math] from the complete lattice of ideals of [math]\displaystyle{ L }[/math] to [math]\displaystyle{ L }[/math] preserves arbitrary infima.
- For any family [math]\displaystyle{ \mathcal D }[/math] of directed sets of [math]\displaystyle{ L }[/math], [math]\displaystyle{ \textstyle\inf_{D\in\mathcal D}\sup D=\sup_{f\in\prod\mathcal D}\inf_{D\in\mathcal D}f(D) }[/math].
- [math]\displaystyle{ L }[/math] is isomorphic to the image of a Scott-continuous idempotent map [math]\displaystyle{ r \colon \{0,1\}^\kappa\to\{0,1\}^\kappa }[/math] on the direct power of arbitrarily many two-point lattices [math]\displaystyle{ \{0,1\} }[/math].[2]:p.56, Theorem 44
A continuous complete lattice is often called a continuous lattice.
Examples
Lattices of open sets
For a topological space [math]\displaystyle{ X }[/math], the following conditions are equivalent.
- The complete Heyting algebra [math]\displaystyle{ \operatorname{Open}(X) }[/math] of open sets of [math]\displaystyle{ X }[/math] is a continuous complete Heyting algebra.
- The sobrification of [math]\displaystyle{ X }[/math] is a locally compact space (in the sense that every point has a compact local base)
- [math]\displaystyle{ X }[/math] is an exponentiable object in the category [math]\displaystyle{ \operatorname{Top} }[/math] of topological spaces.[1]:p.196, Theorem II-4.12 That is, the functor [math]\displaystyle{ (-)\times X\colon\operatorname{Top}\to\operatorname{Top} }[/math] has a right adjoint.
References
- ↑ 1.0 1.1 1.2 1.3 1.4 Gierz, Gerhard; Hofmann, Karl; Keimel, Klaus; Lawson, Jimmie; Mislove, Michael; Scott, Dana S. (2003) (in en). Continuous lattices and domains. Encyclopedia of Mathematics and Its Applications. 93. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511542725. ISBN 978-0-521-80338-0.
- ↑ Grätzer, George (2011) (in en). Lattice Theory: Foundation. Basel: Springer. doi:10.1007/978-3-0348-0018-1. ISBN 978-3-0348-0017-4.
External links
- Hazewinkel, Michiel, ed. (2001), "Continuous lattice", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Hazewinkel, Michiel, ed. (2001), "Core-compact space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Continuous poset in nLab
- Continuous category in nLab
- Exponential law for spaces in nLab
- Continuous poset at PlanetMath.org.
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