Continuous poset explained
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
be two elements of a preordered set
. Then we say that
approximates
, or that
is
way-below
, if the following two equivalent conditions are satisfied.
such that
, there is a
such that
.
such that
,
.If
approximates
, we write
. The approximation relation
is a
transitive relation that is weaker than the original order, also
antisymmetric if
is a
partially ordered set, but not necessarily a
preorder. It is a preorder if and only if
satisfies the
ascending chain condition.
[1] For any
, let
\Uparrowa=\{b\inL\mida\llb\}
\Downarrowa=\{b\inL\midb\lla\}
Then
is an
upper set, and
a
lower set. If
is an
upper-semilattice,
is a
directed set (that is,
implies
), and therefore an
ideal.
is called a
continuous preordered set if for any
, the subset
is
directed and
.
Properties
The interpolation property
For any two elements
of a continuous preordered set
,
if and only if for any
directed set
such that
, there is a
such that
. From this follows the interpolation property of the continuous preordered set
: for any
such that
there is a
such that
.
Continuous dcpos
For any two elements
of a continuous
dcpo
, the following two conditions are equivalent.
and
.
such that
, there is a
such that
and
.Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any
such that
and
, there is a
such that
and
.
, the following conditions are equivalent.
is continuous.
\sup\colon\operatorname{Ideal}(P)\toP
from the
partially ordered set of
ideals of
to
has a
left adjoint.In this case, the actual left adjoint is
{\Downarrow}\colonP\to\operatorname{Ideal}(P)
Continuous complete lattices
For any two elements
of a
complete lattice
,
if and only if for any subset
such that
, there is a finite subset
such that
.
Let
be a
complete lattice. Then the following conditions are equivalent.
is continuous.
\sup\colon\operatorname{Ideal}(L)\toL
from the
complete lattice of
ideals of
to
preserves arbitrary
infima.
of
directed sets of
,
styleinfD\inlD\supD=\supf\in\prodlDinfD\inlDf(D)
.
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
.
[2] A continuous complete lattice is often called a continuous lattice.
Examples
Lattices of open sets
, the following conditions are equivalent.
of open sets of
is a continuous
complete Heyting algebra.
is a
locally compact space (in the sense that every point has a
compact local base)
is an exponentiable object in the category
of
topological spaces. That is, the
functor (-) x X\colon\operatorname{Top}\to\operatorname{Top}
has a
right adjoint.
Notes and References
- Book: Dana Scott. 2003. 10.1017/CBO9780511542725. Gerhard. Karl. Klaus. Jimmie. Michael. Dana S.. 978-0-521-80338-0. en. Gierz. Hofmann. Keimel. Lawson. Mislove. Scott. Cambridge. 1975381. Cambridge University Press. Encyclopedia of Mathematics and Its Applications. Continuous lattices and domains. 93. 1088.06001.
- Book: Grätzer, George. George Grätzer
. 2011. 10.1007/978-3-0348-0018-1. 978-3-0348-0017-4. en. George Grätzer. 2011921250. Basel. 2768581. Springer. Lattice Theory: Foundation. 1233.06001.
