# Prewellordering

Transitive binary relations vte :
| | Symmetric | Antisymmetric | Connected | Well-founded | Has joins | Has meets | Reflexive | Irreflexive | Asymmetric |
| | | | Total, Semiconnex | | | | | Anti- reflexive | |
| Equivalence relation | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Preorder (Quasiorder) | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Partial order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Total preorder | ✗ | ✗ | Y | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Total order | ✗ | Y | Y | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Prewellordering | ✗ | ✗ | Y | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Well-quasi-ordering | ✗ | ✗ | ✗ | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Well-ordering | ✗ | Y | Y | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Lattice | ✗ | Y | ✗ | ✗ | Y | Y | Y | ✗ | ✗ |
| Join-semilattice | ✗ | Y | ✗ | ✗ | Y | ✗ | Y | ✗ | ✗ |
| Meet-semilattice | ✗ | Y | ✗ | ✗ | ✗ | Y | Y | ✗ | ✗ |
| Strict partial order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | Y |
| Strict weak order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | Y |
| Strict total order | ✗ | Y | Y | ✗ | ✗ | ✗ | ✗ | Y | Y |
| | Symmetric | Antisymmetric | Connected | Well-founded | Has joins | Has meets | Reflexive | Irreflexive | Asymmetric |
| Definitions, for all $a,b$ and $S\neq \varnothing :$ | ${\begin{aligned}&aRb\\\Rightarrow&bRa\end{aligned}}$ | ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a=&b\end{aligned}}$ | ${\begin{aligned}a\neq&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ | ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ | ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ | ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ | $aRa$ | ${\text{not }}aRa$ | ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$ |
Y indicates that the column's property is always true for the row's term (at the very left), while ✗ 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 Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table. :
In set theory, a prewellordering on a set $X$ is a preorder $\leq$ on $X$ (a transitive and reflexive relation on $X$) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation $x<y$ defined by $x\leq y{\text{ and }}y\nleq x$ is a well-founded relation.

## Prewellordering on a set

A prewellordering on a set $X$ is a homogeneous binary relation $\,\leq \,$ on $X$ that satisfies the following conditions:
* Reflexivity: $x\leq x$ for all $x\in X.$
* Transitivity: if $x<y$ and $y<z$ then $x<z$ for all $x,y,z\in X.$
* Total/Strongly connected: $x\leq y$ or $y\leq x$ for all $x,y\in X.$
* for every non-empty subset $S\subseteq X,$ there exists some $m\in S$ such that $m\leq s$ for all $s\in S.$
** This condition is equivalent to the induced strict preorder $x<y$ defined by $x\leq y$ and $y\nleq x$ being a well-founded relation.
A homogeneous binary relation $\,\leq \,$ on $X$ is a prewellordering if and only if there exists a surjection $\pi :X\to Y$ into a well-ordered set $(Y,\lesssim )$ such that for all $x,y\in X,$ x≤y{\textstyle x\leq y} if and only if $\pi (x)\lesssim \pi (y).$

### Examples

Given a set $A,$ the binary relation on the set $X:=\operatorname {Finite} (A)$ of all finite subsets of $A$ defined by $S\leq T$ if and only if $|S|\leq |T|$ (where $|\cdot |$ denotes the set's cardinality) is a prewellordering.

### Properties

If $\leq$ is a prewellordering on $X,$ then the relation $\sim$ defined by $x\sim y{\text{ if and only if }}x\leq y\land y\leq x$ is an equivalence relation on $X,$ and $\leq$ induces a wellordering on the quotient $X/{\sim }.$ The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.
A norm on a set $X$ is a map from $X$ into the ordinals. Every norm induces a prewellordering; if $\phi :X\to Ord$ is a norm, the associated prewellordering is given by $x\leq y{\text{ if and only if }}\phi (x)\leq \phi (y)$ Conversely, every prewellordering is induced by a unique regular norm (a norm $\phi :X\to Ord$ is regular if, for any $x\in X$ and any $\alpha <\phi (x),$ there is $y\in X$ such that $\phi (y)=\alpha$).

## Prewellordering property

If ${\boldsymbol {\Gamma }}$ is a pointclass of subsets of some collection ${\mathcal {F}}$ of Polish spaces, ${\mathcal {F}}$ closed under Cartesian product, and if $\leq$ is a prewellordering of some subset $P$ of some element $X$ of ${\mathcal {F}},$ then $\leq$ is said to be a ${\boldsymbol {\Gamma }}$-prewellordering of $P$ if the relations $<^{*}$ and $\leq ^{*}$ are elements of ${\boldsymbol {\Gamma }},$ where for $x,y\in X,$
* $x<^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor (x\leq y\land y\not \leq x))$
* $x\leq ^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor x\leq y)$
${\boldsymbol {\Gamma }}$ is said to have the prewellordering property if every set in ${\boldsymbol {\Gamma }}$ admits a ${\boldsymbol {\Gamma }}$-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

${\boldsymbol {\Pi }}_{1}^{1}$ and ${\boldsymbol {\Sigma }}_{2}^{1}$ both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every $n\in \omega ,$ ${\boldsymbol {\Pi }}_{2n+1}^{1}$ and ${\boldsymbol {\Sigma }}_{2n+2}^{1}$ have the prewellordering property.

### Consequences
#### Reduction

If ${\boldsymbol {\Gamma }}$ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space $X\in {\mathcal {F}}$ and any sets $A,B\subseteq X,$ $A$ and $B$ both in ${\boldsymbol {\Gamma }},$ the union $A\cup B$ may be partitioned into sets $A^{*},B^{*},$ both in ${\boldsymbol {\Gamma }},$ such that $A^{*}\subseteq A$ and $B^{*}\subseteq B.$

#### Separation

If ${\boldsymbol {\Gamma }}$ is an adequate pointclass whose dual pointclass has the prewellordering property, then ${\boldsymbol {\Gamma }}$ has the separation property: For any space $X\in {\mathcal {F}}$ and any sets $A,B\subseteq X,$ $A$ and $B$ disjoint sets both in ${\boldsymbol {\Gamma }},$ there is a set $C\subseteq X$ such that both $C$ and its complement $X\setminus C$ are in ${\boldsymbol {\Gamma }},$ with $A\subseteq C$ and $B\cap C=\varnothing .$
For example, ${\boldsymbol {\Pi }}_{1}^{1}$ has the prewellordering property, so ${\boldsymbol {\Sigma }}_{1}^{1}$ has the separation property. This means that if $A$ and $B$ are disjoint analytic subsets of some Polish space $X,$ then there is a Borel subset $C$ of $X$ such that $C$ includes $A$ and is disjoint from $B.$