|Transitive binary relations|
|✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. indicates that the column's property is always true the row's term (at the very left), while |
All definitions tacitly require the homogeneous relation be transitive: for all if and then
In set theory, a prewellordering on a set is a preorder on (a transitive and reflexive relation on ) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation defined by is a well-founded relation.
Prewellordering on a setEdit
A prewellordering on a set is a homogeneous binary relation on that satisfies the following conditions:
- Reflexivity: for all
- Transitivity: if and then for all
- Total/Strongly connected: or for all
- for every non-empty subset there exists some such that for all
- This condition is equivalent to the induced strict preorder defined by and being a well-founded relation.
A homogeneous binary relation on is a prewellordering if and only if there exists a surjection into a well-ordered set such that for all if and only if 
Given a set the binary relation on the set of all finite subsets of defined by if and only if (where denotes the set's cardinality) is a prewellordering.
If is a prewellordering on then the relation defined by
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
If is a pointclass of subsets of some collection of Polish spaces, closed under Cartesian product, 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.
and both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every and have the prewellordering property.
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
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 are in with and
For example, has the prewellordering property, so has the separation property. This means that if and are disjoint analytic subsets of some Polish space then there is a Borel subset of such that includes and is disjoint from
- Descriptive set theory – Subfield of mathematical logic
- Graded poset – partially ordered set equipped with a rank function, sometimes called a ranked poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the natural numbers
- Scale property – kind of object in descriptive set theory
- ^ a b c Moschovakis 2006, p. 106.
- Moschovakis, Yiannis N. (1980). Descriptive Set Theory. Amsterdam: North Holland. ISBN 978-0-08-096319-8. OCLC 499778252.
- Moschovakis, Yiannis N. (2006). Notes on set theory. New York: Springer. ISBN 978-0-387-31609-3. OCLC 209913560.