In
set theory, a prewellordering on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
on
(a
transitive and
strongly connected relation on
) that is
wellfounded in the sense that the relation
is wellfounded. If
is a prewellordering on
then the relation
defined by
is an
equivalence relation on
and
induces a
wellordering on the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
The
order-type of this induced wellordering is an
ordinal, referred to as the length of the prewellordering.
A norm on a set
is a map from
into the ordinals. Every norm induces a prewellordering; if
is a norm, the associated prewellordering is given by
Conversely, every prewellordering is induced by a unique regular norm (a norm
is regular if, for any
and any
there is
such that
).
Prewellordering property
If
is a
pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
of subsets of some collection
of
Polish spaces,
closed under
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
, and if
is a prewellordering of some subset
of some element
of
then
is said to be a
-prewellordering of
if the relations
and
are elements of
where for
#
#
is said to have the prewellordering property if every set in
admits a
-prewellordering.
The prewellordering property is related to the stronger
scale property; in practice, many pointclasses having the prewellordering property also have the scale property, which allows drawing stronger conclusions.
Examples
and
both have the prewellordering property; this is provable in
ZFC alone. Assuming sufficient
large cardinals, for every
and
have the prewellordering property.
Consequences
Reduction
If
is an
adequate pointclass with the prewellordering property, then it also has the reduction property: For any space
and any sets
and
both in
the union
may be partitioned into sets
both in
such that
and
Separation
If
is an
adequate pointclass whose
dual pointclass has the prewellordering property, then
has the separation property: For any space
and any sets
and
''disjoint'' sets both in
there is a set
such that both
and its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
are in
with
and
For example,
has the prewellordering property, so
has the separation property. This means that if
and
are disjoint
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
subsets of some Polish space
then there is a
Borel subset
of
such that
includes
and is disjoint from
See also
*
* – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the integers
*
References
*
{{Order theory
Binary relations
Descriptive set theory
Order theory
Wellfoundedness