Prewellordering

Transitive binary relations v t e
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 ${\displaystyle a,b}$ and ${\displaystyle S\neq \varnothing :}$ ${\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle aRa}$ ${\displaystyle {\text{not }}aRa}$ ${\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}$
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 ${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$ A term's definition may require additional properties that are not listed in this table.

In set theory, a prewellordering on a set ${\displaystyle X}$ is a preorder ${\displaystyle \leq }$ on ${\displaystyle X}$ (a transitive and reflexive relation on ${\displaystyle X}$) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation ${\displaystyle x$ defined by ${\displaystyle x\leq y{\text{ and }}y\nleq x}$ is a well-founded relation.

Prewellordering on a set

A prewellordering on a set ${\displaystyle X}$  is a homogeneous binary relation ${\displaystyle \,\leq \,}$  on ${\displaystyle X}$  that satisfies the following conditions:

1. Reflexivity: ${\displaystyle x\leq x}$  for all ${\displaystyle x\in X.}$ 
2. Transitivity: if ${\displaystyle x$  and ${\displaystyle y$  then ${\displaystyle x$  for all ${\displaystyle x,y,z\in X.}$ 
3. Total/Strongly connected: ${\displaystyle x\leq y}$  or ${\displaystyle y\leq x}$  for all ${\displaystyle x,y\in X.}$ 
4. for every non-empty subset ${\displaystyle S\subseteq X,}$  there exists some ${\displaystyle m\in S}$  such that ${\displaystyle m\leq s}$  for all ${\displaystyle s\in S.}$ 
   • This condition is equivalent to the induced strict preorder ${\displaystyle x$  defined by ${\displaystyle x\leq y}$  and ${\displaystyle y\nleq x}$  being a well-founded relation.

A homogeneous binary relation ${\displaystyle \,\leq \,}$  on ${\displaystyle X}$  is a prewellordering if and only if there exists a surjection ${\displaystyle \pi :X\to Y}$  into a well-ordered set ${\displaystyle (Y,\lesssim )}$  such that for all ${\displaystyle x,y\in X,}$  ${\textstyle x\leq y}$  if and only if ${\displaystyle \pi (x)\lesssim \pi (y).}$ 

Examples

 

Hasse diagram of the prewellordering ${\displaystyle \lfloor x/4\rfloor \leq \lfloor y/5\rfloor }$  on the non-negative integers, shown up to 29. Cycles are indicated in red and ${\displaystyle \lfloor \cdot \rfloor }$  denotes the floor function.

 

Hasse diagram of the prewellordering ${\displaystyle \lfloor x/4\rfloor \leq \lfloor y/4\rfloor }$  on the non-negative integers, shown up to 18. The associated equivalence relation is ${\displaystyle \lfloor x/4\rfloor =\lfloor y/4\rfloor ;}$  it identifies the numbers in each light red square.

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

Properties

If ${\displaystyle \leq }$  is a prewellordering on ${\displaystyle X,}$  then the relation ${\displaystyle \sim }$  defined by $${\displaystyle x\sim y{\text{ if and only if }}x\leq y\land y\leq x}$$  is an equivalence relation on ${\displaystyle X,}$  and ${\displaystyle \leq }$  induces a wellordering on the quotient ${\displaystyle X/{\sim }.}$  The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set ${\displaystyle X}$  is a map from ${\displaystyle X}$  into the ordinals. Every norm induces a prewellordering; if ${\displaystyle \phi :X\to Ord}$  is a norm, the associated prewellordering is given by $${\displaystyle x\leq y{\text{ if and only if }}\phi (x)\leq \phi (y)}$$  Conversely, every prewellordering is induced by a unique regular norm (a norm ${\displaystyle \phi :X\to Ord}$  is regular if, for any ${\displaystyle x\in X}$  and any ${\displaystyle \alpha <\phi (x),}$  there is ${\displaystyle y\in X}$  such that ${\displaystyle \phi (y)=\alpha }$ ).

Prewellordering property

If ${\displaystyle {\boldsymbol {\Gamma }}}$  is a pointclass of subsets of some collection ${\displaystyle {\mathcal {F}}}$  of Polish spaces, ${\displaystyle {\mathcal {F}}}$  closed under Cartesian product, and if ${\displaystyle \leq }$  is a prewellordering of some subset ${\displaystyle P}$  of some element ${\displaystyle X}$  of ${\displaystyle {\mathcal {F}},}$  then ${\displaystyle \leq }$  is said to be a ${\displaystyle {\boldsymbol {\Gamma }}}$ -prewellordering of ${\displaystyle P}$  if the relations ${\displaystyle <^{*}}$  and ${\displaystyle \leq ^{*}}$  are elements of ${\displaystyle {\boldsymbol {\Gamma }},}$  where for ${\displaystyle x,y\in X,}$ 

1. ${\displaystyle x<^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor (x\leq y\land y\not \leq x))}$ 
2. ${\displaystyle x\leq ^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor x\leq y)}$ 

${\displaystyle {\boldsymbol {\Gamma }}}$  is said to have the prewellordering property if every set in ${\displaystyle {\boldsymbol {\Gamma }}}$  admits a ${\displaystyle {\boldsymbol {\Gamma }}}$ -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

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

Consequences

Reduction

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

Separation

If ${\displaystyle {\boldsymbol {\Gamma }}}$  is an adequate pointclass whose dual pointclass has the prewellordering property, then ${\displaystyle {\boldsymbol {\Gamma }}}$  has the separation property: For any space ${\displaystyle X\in {\mathcal {F}}}$  and any sets ${\displaystyle A,B\subseteq X,}$  ${\displaystyle A}$  and ${\displaystyle B}$  disjoint sets both in ${\displaystyle {\boldsymbol {\Gamma }},}$  there is a set ${\displaystyle C\subseteq X}$  such that both ${\displaystyle C}$  and its complement ${\displaystyle X\setminus C}$  are in ${\displaystyle {\boldsymbol {\Gamma }},}$  with ${\displaystyle A\subseteq C}$  and ${\displaystyle B\cap C=\varnothing .}$ 

For example, ${\displaystyle {\boldsymbol {\Pi }}_{1}^{1}}$  has the prewellordering property, so ${\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}$  has the separation property. This means that if ${\displaystyle A}$  and ${\displaystyle B}$  are disjoint analytic subsets of some Polish space ${\displaystyle X,}$  then there is a Borel subset ${\displaystyle C}$  of ${\displaystyle X}$  such that ${\displaystyle C}$  includes ${\displaystyle A}$  and is disjoint from ${\displaystyle B.}$ 

See also

• Descriptive set theory – Subfield of mathematical logic
• Graded 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