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Prewellordering information


Transitive binary relations
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Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Green tickY ✗ ✗ ✗ ✗ ✗ Green tickY ✗ ✗
Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Green tickY ✗ ✗
Partial order ✗ Green tickY ✗ ✗ ✗ ✗ Green tickY ✗ ✗
Total preorder ✗ ✗ Green tickY ✗ ✗ ✗ Green tickY ✗ ✗
Total order ✗ Green tickY Green tickY ✗ ✗ ✗ Green tickY ✗ ✗
Prewellordering ✗ ✗ Green tickY Green tickY ✗ ✗ Green tickY ✗ ✗
Well-quasi-ordering ✗ ✗ ✗ Green tickY ✗ ✗ Green tickY ✗ ✗
Well-ordering ✗ Green tickY Green tickY Green tickY ✗ ✗ Green tickY ✗ ✗
Lattice ✗ Green tickY ✗ ✗ Green tickY Green tickY Green tickY ✗ ✗
Join-semilattice ✗ Green tickY ✗ ✗ Green tickY ✗ Green tickY ✗ ✗
Meet-semilattice ✗ Green tickY ✗ ✗ ✗ Green tickY Green tickY ✗ ✗
Strict partial order ✗ Green tickY ✗ ✗ ✗ ✗ ✗ Green tickY Green tickY
Strict weak order ✗ Green tickY ✗ ✗ ✗ ✗ ✗ Green tickY Green tickY
Strict total order ✗ Green tickY Green tickY ✗ ✗ ✗ ✗ Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY indicates that the column's property is always true 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 Green tickY in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

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.

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Prewellordering

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-prewellordering. The prewellordering property is related to the stronger scale property; in practice, many pointclasses having the prewellordering property...

Word Count : 1246

Norm

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group Norm map, a map from a pointset into the ordinals inducing a prewellordering Norm group, a group in class field theory that is the image of the...

Word Count : 490

Inductive set

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determinacy, the class of inductive sets has the scale property and thus the prewellordering property. The term can have a number of different meanings: Russell's...

Word Count : 344

List of set theory topics

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Infinity-Borel set Lightface analytic game Perfect set property Polish space Prewellordering Projective set Property of Baire Uniformization (set theory) Universally...

Word Count : 448

Transitive relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 2073

Weak ordering

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 4360

Transitive closure

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 2306

Reflexive relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 1390

Total order

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 3094

Partially ordered set

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 5396

Separation property

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to simplify the process of building a portfolio of financial assets Prewellordering in mathematics, a component of set theory Separation axiom in mathematics...

Word Count : 70

Equivalence relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 4422

List of order theory topics

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theory) Tree (descriptive set theory) Suslin's problem Absorption law Prewellordering Stone duality Stone's representation theorem for Boolean algebras Specialization...

Word Count : 396

Outline of logic

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equivalence relation Partial function Partially ordered set Preorder Prewellordering Propositional function Quasitransitive relation Reflexive relation...

Word Count : 2084

Antisymmetric relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 589

Preorder

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is antisymmetric and total Directed set Category of preordered sets Prewellordering Well-quasi-ordering For "proset", see e.g. Eklund, Patrik; Gähler,...

Word Count : 3351

Determinacy

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then Π12n+1 and Σ12n+2 have the prewellordering property (and that Σ12n+1 and Π12n+2 do not have the prewellordering property, but rather have the separation...

Word Count : 4090

Binary relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 8932

Symmetric relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 385

List of mathematical logic topics

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Infinity-Borel set Lightface analytic game Perfect set property Polish space Prewellordering Projective set Property of Baire Uniformization (set theory) Universally...

Word Count : 1012

Graded poset

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power set has the Sperner property Graded (mathematics) Prewellordering – a prewellordering with a norm is analogous to a graded poset, replacing a map...

Word Count : 1934

Semilattice

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 2397

Descriptive set theory

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theory and second-order arithmetic. This box: view talk edit Pointclass Prewellordering Scale property Kechris, Alexander S. (1994). Classical Descriptive...

Word Count : 1595

Asymmetric relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 833

Join and meet

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 2262

Connected relation

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✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering...

Word Count : 1391
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