Continuous poset#Continuous complete lattices

Let $a,b\backslash in\; P$ be two elements of a preordered set $(P,\backslash lesssim)$. 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 $D\backslash subseteq\; P$ such that $b\backslash lesssim\backslash sup\; D$, there is a $d\backslash in\; D$ such that $a\backslash lesssim\; d$.
• For any ideal $I\backslash subseteq\; P$ such that $b\backslash lesssim\backslash sup\; I$, $a\backslash in\; I$.

If $a$ approximates $b$, we write $a\backslash ll\; b$. The approximation relation $\backslash ll$ 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 $(P,\backslash lesssim)$ satisfies the ascending chain condition.{{cite book|author-link6=Dana Scott|date=2003|doi=10.1017/CBO9780511542725|first1=Gerhard|first2=Karl|first3=Klaus|first4=Jimmie|first5=Michael|first6=Dana S.|isbn=978-0-521-80338-0|language=en|last1=Gierz|last2=Hofmann|last3=Keimel|last4=Lawson|last5=Mislove|last6=Scott|location=Cambridge|mr=1975381|publisher=Cambridge University Press|series=Encyclopedia of Mathematics and Its Applications|title=Continuous lattices and domains|volume=93|zbl=1088.06001}}{{rp|p.52, Examples I-1.3, (4)}}

For any $a\backslash in\; P$, let

:$\backslash mathop\backslash Uparrow\; a=\backslash \{b\backslash in\; L\backslash mid\; a\backslash ll\; b\backslash \}$

:$\backslash mathop\backslash Downarrow\; a=\backslash \{b\backslash in\; L\backslash mid\; b\backslash ll\; a\backslash \}$

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

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

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

For any two elements $a,b\backslash in\; P$ of a continuous dcpo $(P,\backslash le)$, the following two conditions are equivalent.{{rp|p.61, Proposition I-1.19(i)}}

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

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any $a,b\backslash in\; P$ such that $a\backslash ll\; b$ and $a\backslash ne\; b$, there is a $c\backslash in\; P$ such that $a\backslash ll\; c\backslash ll\; b$ and $a\backslash ne\; c$.{{rp|p.61, Proposition I-1.19(ii)}}

For a dcpo $(P,\backslash le)$, the following conditions are equivalent.{{rp|Theorem I-1.10}}

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

In this case, the actual left adjoint is

:$\{\backslash Downarrow\}\; \backslash colon\; P\backslash to\backslash operatorname\{Ideal\}(P)$

:$\backslash mathord\backslash Downarrow\backslash dashv\backslash sup$

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

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

• $L$ is continuous.
• The supremum map $\backslash sup\; \backslash colon\; \backslash operatorname\{Ideal\}(L)\backslash to\; L$ from the complete lattice of ideals of $L$ to $L$ preserves arbitrary infima.
• For any family $\backslash mathcal\; D$ of directed sets of $L$, $\backslash textstyle\backslash inf\_\{D\backslash in\backslash mathcal\; D\}\backslash sup\; D=\backslash sup\_\{f\backslash in\backslash prod\backslash mathcal\; D\}\backslash inf\_\{D\backslash in\backslash mathcal\; D\}f(D)$.
• $L$ is isomorphic to the image of a Scott-continuous idempotent map $r\; \backslash colon\; \backslash \{0,1\backslash \}^\backslash kappa\backslash to\backslash \{0,1\backslash \}^\backslash kappa$ on the direct power of arbitrarily many two-point lattices $\backslash \{0,1\backslash \}$.{{cite book|date=2011|doi=10.1007/978-3-0348-0018-1|first=George|isbn=978-3-0348-0017-4|language=en|last=Grätzer|authorlink = George Grätzer|lccn=2011921250|location=Basel|mr=2768581|publisher=Springer|title=Lattice Theory: Foundation|zbl=1233.06001}}{{rp|p.56, Theorem 44}}

A continuous complete lattice is often called a continuous lattice.

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

• The complete Heyting algebra $\backslash operatorname\{Open\}(X)$ 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 $\backslash operatorname\{Top\}$ of topological spaces.{{rp|p.196, Theorem II-4.12}} That is, the functor $(-)\backslash times\; X\backslash colon\backslash operatorname\{Top\}\backslash to\backslash operatorname\{Top\}$ has a right adjoint.

{{reflist}}

• {{eom|title=Continuous lattice}}
• {{eom|title=Core-compact space}}
• {{nlab|id=continuous_poset|title=Continuous poset}}
• {{nlab|id=continuous_category|title=Continuous category}}
• {{nlab|id=exponential_law_for_spaces|title=Exponential law for spaces}}
• {{PlanetMath|urlname=ContinuousPoset|title=Continuous poset}}

Category:Order theory