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 set

A prewellordering on a set is a homogeneous binary relation on that satisfies the following conditions:[1]

  1. Reflexivity: for all
  2. Transitivity: if and then for all
  3. Total/Strongly connected: or for all
  4. 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 [1]

Examples

Hasse diagram of the prewellordering on the non-negative integers, shown up to 29. Cycles are indicated in red and denotes the floor function.
Hasse diagram of the prewellordering on the non-negative integers, shown up to 18. The associated equivalence relation is it identifies the numbers in each light red square.

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.[1]

Properties

If is a prewellordering on then the relation defined by

is an equivalence relation on and induces a wellordering on the quotient 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 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.

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 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

See also

References

  1. ^ a b c Moschovakis 2006, p. 106.