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59-561: For other uses, see Wadge hierarchy and Wadge determinacy . [REDACTED] Look up wadge in Wiktionary, the free dictionary. Wadge is a surname. Notable people with the surname include: Amy Wadge (born 1975), English singer and songwriter Eric Wadge , musician Richard Wadge (1864–1923), English football director See also [ edit ] Wedge (surname) [REDACTED] Surname list This page lists people with

118-491: A preorder or quasiorder if it is reflexive and transitive ; that is, if it satisfies: A set that is equipped with a preorder is called a preordered set (or proset ). Given a preorder ≲ {\displaystyle \,\lesssim \,} on S {\displaystyle S} one may define an equivalence relation ∼ {\displaystyle \,\sim \,} on S {\displaystyle S} such that

177-436: A ∼ b  if and only if  a ≲ b  and  b ≲ a . {\displaystyle a\sim b\quad {\text{ if and only if }}\quad a\lesssim b\;{\text{ and }}\;b\lesssim a.} The resulting relation ∼ {\displaystyle \,\sim \,} is reflexive since the preorder ≲ {\displaystyle \,\lesssim \,}

236-480: A ≠ b {\displaystyle a\lesssim b{\text{ and }}a\neq b} ) because if the preorder ≲ {\displaystyle \,\lesssim \,} is not antisymmetric then the resulting relation < {\displaystyle \,<\,} would not be transitive (consider how equivalent non-equal elements relate). This is the reason for using the symbol " ≲ {\displaystyle \lesssim } " instead of

295-399: A ≠ b {\displaystyle a\neq b} then a < b {\displaystyle a<b} or b < a . {\displaystyle b<a.} In computer science, one can find examples of the following preorders. Further examples: Example of a total preorder : Every binary relation R {\displaystyle R} on

354-487: A ≲ b  and not  a ∼ b ; {\displaystyle a\lesssim b{\text{ and not }}a\sim b;} and so the following holds a ≲ b  if and only if  a < b  or  a ∼ b . {\displaystyle a\lesssim b\quad {\text{ if and only if }}\quad a<b\;{\text{ or }}\;a\sim b.} The relation < {\displaystyle \,<\,}

413-459: A ≲ b , {\displaystyle a\lesssim b,} one may say that b covers a or that a precedes b , or that b reduces to a . Occasionally, the notation ← or → is also used. Let ≲ {\displaystyle \,\lesssim \,} be a binary relation on a set P , {\displaystyle P,} so that by definition, ≲ {\displaystyle \,\lesssim \,}

472-489: A ≲ b , {\displaystyle a\lesssim b,} the interval [ a , b ] {\displaystyle [a,b]} is the set of points x satisfying a ≲ x {\displaystyle a\lesssim x} and x ≲ b , {\displaystyle x\lesssim b,} also written a ≲ x ≲ b . {\displaystyle a\lesssim x\lesssim b.} It contains at least

531-429: A < x {\displaystyle a<x} and x < b , {\displaystyle x<b,} also written a < x < b . {\displaystyle a<x<b.} An open interval may be empty even if a < b . {\displaystyle a<b.} Also [ a , b ) {\displaystyle [a,b)} and (

590-411: A , {\displaystyle b\lesssim a,} then it is an equivalence relation . A preorder is total if a ≲ b {\displaystyle a\lesssim b} or b ≲ a {\displaystyle b\lesssim a} for all a , b ∈ P . {\displaystyle a,b\in P.} A preordered class is a class equipped with

649-536: A = b {\displaystyle a=b} ) and so in this case, the definition of < {\displaystyle \,<\,} can be restated as: a < b  if and only if  a ≲ b  and  a ≠ b ( assuming  ≲  is antisymmetric ) . {\displaystyle a<b\quad {\text{ if and only if }}\quad a\lesssim b\;{\text{ and }}\;a\neq b\quad \quad ({\text{assuming }}\lesssim {\text{

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708-430: A R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. In mathematics , especially in order theory , a preorder or quasiorder is a binary relation that is reflexive and transitive . The name preorder

767-401: A directed graph , with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph . A preorder that

826-401: A preorder on the subsets of Baire space. Degrees given by Lipschitz functions are called Lipschitz degrees , and degrees from Borel functions Borel–Wadge degrees . Preorder All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if

885-467: A continuous map f with the property that x is in A {\displaystyle A} if and only if f ( x ) is in B {\displaystyle B} . Martin and Monk proved in 1973 that AD implies the Wadge order for Baire space is well founded . Hence under AD, the Wadge classes modulo complements form a wellorder. The Wadge rank of a set A {\displaystyle A}

944-538: A linear order on the equivalence classes modulo complements. Wadge's lemma can be applied locally to any pointclass Γ, for example the Borel sets , Δ n sets, Σ n sets, or Π n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC , ZFC implies Wadge's lemma for Borel sets. Wadge's lemma is similar to the cone lemma from computability theory. The Wadge game

1003-438: A partition of a set S , {\displaystyle S,} it is possible to construct a preorder on S {\displaystyle S} itself. There is a one-to-one correspondence between preorders and pairs (partition, partial order). Example : Let S {\displaystyle S} be a formal theory , which is a set of sentences with certain properties (details of which can be found in

1062-485: A preorder called the left residual , where R T {\displaystyle R^{\textsf {T}}} denotes the converse relation of R , {\displaystyle R,} and R ¯ {\displaystyle {\overline {R}}} denotes the complement relation of R , {\displaystyle R,} while ∘ {\displaystyle \circ } denotes relation composition . If

1121-460: A preorder is also antisymmetric , that is, a ≲ b {\displaystyle a\lesssim b} and b ≲ a {\displaystyle b\lesssim a} implies a = b , {\displaystyle a=b,} then it is a partial order . On the other hand, if it is symmetric , that is, if a ≲ b {\displaystyle a\lesssim b} implies b ≲

1180-437: A preorder. Every set is a class and so every preordered set is a preordered class. Preorders play a pivotal role in several situations: Note that S ( n , k ) refers to Stirling numbers of the second kind . As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example: For

1239-426: A set S {\displaystyle S} can be extended to a preorder on S {\displaystyle S} by taking the transitive closure and reflexive closure , R + = . {\displaystyle R^{+=}.} The transitive closure indicates path connection in R : x R + y {\displaystyle R:xR^{+}y} if and only if there

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1298-463: A strict partial order defined by a < b {\displaystyle a<b} if and only if a ≲ b {\displaystyle a\lesssim b} and not b ≲ a {\displaystyle b\lesssim a} . Using the equivalence relation ∼ {\displaystyle \,\sim \,} introduced above, a < b {\displaystyle a<b} if and only if

1357-467: Is a countable intersection of open sets , then so is A {\displaystyle A} . The same works for all levels of the Borel hierarchy and the difference hierarchy . The Wadge hierarchy plays an important role in models of the axiom of determinacy . Further interest in Wadge degrees comes from computer science , where some papers have suggested Wadge degrees are relevant to algorithmic complexity . Wadge's lemma states that under

1416-415: Is a pointclass . Conversely, every pointclass is equal to some W {\displaystyle W} α . A pointclass is said to be self-dual if it is closed under complementation. It can be shown that W α is self-dual if and only if α is either 0, an even successor ordinal , or a limit ordinal of countable cofinality . Similar notions of reduction and degree arise by replacing

1475-421: Is a strict partial order and every strict partial order can be constructed this way. If the preorder ≲ {\displaystyle \,\lesssim \,} is antisymmetric (and thus a partial order) then the equivalence ∼ {\displaystyle \,\sim \,} is equality (that is, a ∼ b {\displaystyle a\sim b} if and only if

1534-408: Is a continuous function f {\displaystyle f} on ω with A = f − 1 [ B ] {\displaystyle A=f^{-1}[B]} . The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees , the degree of a set A {\displaystyle A}

1593-453: Is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle X} can equivalently be defined as an equivalence relation on X {\displaystyle X} , together with a partial order on the set of equivalence class. Like partial orders and equivalence relations, preorders (on a nonempty set) are never asymmetric . A preorder can be visualized as

1652-1050: Is a preorder on S {\displaystyle S} because A ⇐ A {\displaystyle A\Leftarrow A} always holds and whenever A ⇐ B {\displaystyle A\Leftarrow B} and B ⇐ C {\displaystyle B\Leftarrow C} both hold then so does A ⇐ C . {\displaystyle A\Leftarrow C.} Furthermore, for any A , B ∈ S , {\displaystyle A,B\in S,} A ∼ B {\displaystyle A\sim B} if and only if A ⇐ B  and  B ⇐ A {\displaystyle A\Leftarrow B{\text{ and }}B\Leftarrow A} ; that is, two sentences are equivalent with respect to ⇐ {\displaystyle \,\Leftarrow \,} if and only if they are logically equivalent . This particular equivalence relation A ∼ B {\displaystyle A\sim B}

1711-415: Is a simple infinite game used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game G ( A , B ) {\displaystyle G(A,B)} , player I and player II each in turn play integers, and

1770-552: Is an R {\displaystyle R} - path from x {\displaystyle x} to y . {\displaystyle y.} Left residual preorder induced by a binary relation Given a binary relation R , {\displaystyle R,} the complemented composition R ∖ R = R T ∘ R ¯ ¯ {\displaystyle R\backslash R={\overline {R^{\textsf {T}}\circ {\overline {R}}}}} forms

1829-433: Is antisymmetric}}).} But importantly, this new condition is not used as (nor is it equivalent to) the general definition of the relation < {\displaystyle \,<\,} (that is, < {\displaystyle \,<\,} is not defined as: a < b {\displaystyle a<b} if and only if a ≲ b  and 

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1888-1075: Is commonly denoted with its own special symbol A ⟺ B , {\displaystyle A\iff B,} and so this symbol ⟺ {\displaystyle \,\iff \,} may be used instead of ∼ . {\displaystyle \,\sim .} The equivalence class of a sentence A , {\displaystyle A,} denoted by [ A ] , {\displaystyle [A],} consists of all sentences B ∈ S {\displaystyle B\in S} that are logically equivalent to A {\displaystyle A} (that is, all B ∈ S {\displaystyle B\in S} such that A ⟺ B {\displaystyle A\iff B} ). The partial order on S / ∼ {\displaystyle S/\sim } induced by ⇐ , {\displaystyle \,\Leftarrow ,\,} which will also be denoted by

1947-630: Is consequently also a directed set. See Lindenbaum–Tarski algebra for a related example. If reflexivity is replaced with irreflexivity (while keeping transitivity) then we get the definition of a strict partial order on P {\displaystyle P} . For this reason, the term strict preorder is sometimes used for a strict partial order. That is, this is a binary relation < {\displaystyle \,<\,} on P {\displaystyle P} that satisfies: Any preorder ≲ {\displaystyle \,\lesssim \,} gives rise to

2006-477: Is denoted by [ A {\displaystyle A} ] W . The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy . Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A {\displaystyle A} ≤ W B {\displaystyle B} and B {\displaystyle B}

2065-456: Is different from Wikidata All set index articles Wadge hierarchy Suppose A {\displaystyle A} and B {\displaystyle B} are subsets of Baire space ω . Then A {\displaystyle A} is Wadge reducible to B {\displaystyle B} or A {\displaystyle A} ≤ W B {\displaystyle B} if there

2124-428: Is in an R {\displaystyle R} -cycle with y {\displaystyle y} . In any case, on S / ∼ {\displaystyle S/\sim } it is possible to define [ x ] ≤ [ y ] {\displaystyle [x]\leq [y]} if and only if x ≲ y . {\displaystyle x\lesssim y.} That this

2183-640: Is meant to suggest that preorders are almost partial orders , but not quite, as they are not necessarily antisymmetric . A natural example of a preorder is the divides relation "x divides y" between integers, polynomials , or elements of a commutative ring . For example, the divides relation is reflexive as every integer divides itself. But the divides relation is not antisymmetric, because 1 {\displaystyle 1} divides − 1 {\displaystyle -1} and − 1 {\displaystyle -1} divides 1 {\displaystyle 1} . It

2242-470: Is reflexive; transitive by applying the transitivity of ≲ {\displaystyle \,\lesssim \,} twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ∼ , {\displaystyle S/\sim ,} which is the set of all equivalence classes of ∼ . {\displaystyle \,\sim .} If

2301-404: Is some subset of P × P {\displaystyle P\times P} and the notation a ≲ b {\displaystyle a\lesssim b} is used in place of ( a , b ) ∈ ≲ . {\displaystyle (a,b)\in \,\lesssim .} Then ≲ {\displaystyle \,\lesssim \,} is called

2360-424: Is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components. As a binary relation, a preorder may be denoted ≲ {\displaystyle \,\lesssim \,} or ≤ {\displaystyle \,\leq \,} . In words, when

2419-472: Is the order type of the set of Wadge degrees modulo complements strictly below [ A {\displaystyle A} ] W . The length of the Wadge hierarchy has been shown to be Θ . Wadge also proved that the length of the Wadge hierarchy restricted to the Borel sets is φ ω 1 (1) (or φ ω 1 (2) depending on the notation), where φ γ is the γ Veblen function to the base ω 1 (instead of

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2478-411: Is to this preorder that "greatest" and "lowest" refer in the phrases " greatest common divisor " and " lowest common multiple " (except that, for integers, the greatest common divisor is also the greatest for the natural order of the integers). Preorders are closely related to equivalence relations and (non-strict) partial orders. Both of these are special cases of a preorder: an antisymmetric preorder

2537-436: Is well-defined, meaning that its defining condition does not depend on which representatives of [ x ] {\displaystyle [x]} and [ y ] {\displaystyle [y]} are chosen, follows from the definition of ∼ . {\displaystyle \,\sim .\,} It is readily verified that this yields a partially ordered set. Conversely, from any partial order on

2596-478: The axiom of determinacy ( AD ), for any two subsets A , B {\displaystyle A,B} of Baire space, A {\displaystyle A} ≤ W B {\displaystyle B} or B {\displaystyle B} ≤ W ω \ A {\displaystyle A} . The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle for Γ or SLO(Γ). Any semilinear order defines

2655-407: The surname Wadge . If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Wadge&oldid=1234245069 " Category : Surnames Hidden categories: Articles with short description Short description

2714-415: The "less than or equal to" symbol " ≤ {\displaystyle \leq } ", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that a ≤ b {\displaystyle a\leq b} implies a < b  or  a = b . {\displaystyle a<b{\text{ or }}a=b.} Using

2773-891: The article on the subject ). For instance, S {\displaystyle S} could be a first-order theory (like Zermelo–Fraenkel set theory ) or a simpler zeroth-order theory . One of the many properties of S {\displaystyle S} is that it is closed under logical consequences so that, for instance, if a sentence A ∈ S {\displaystyle A\in S} logically implies some sentence B , {\displaystyle B,} which will be written as A ⇒ B {\displaystyle A\Rightarrow B} and also as B ⇐ A , {\displaystyle B\Leftarrow A,} then necessarily B ∈ S {\displaystyle B\in S} (by modus ponens ). The relation ⇐ {\displaystyle \,\Leftarrow \,}

2832-435: The construction above, multiple non-strict preorders can produce the same strict preorder < , {\displaystyle \,<,\,} so without more information about how < {\displaystyle \,<\,} was constructed (such knowledge of the equivalence relation ∼ {\displaystyle \,\sim \,} for instance), it might not be possible to reconstruct

2891-459: The continuous functions by any class of functions F that contains the identity function and is closed under composition . Write A {\displaystyle A} ≤ F B {\displaystyle B} if A = f − 1 [ B ] {\displaystyle A=f^{-1}[B]} for some function f {\displaystyle f} in F . Any such class of functions again determines

2950-574: The equivalence classes. All that has been said of ⇐ {\displaystyle \,\Leftarrow \,} so far can also be said of its converse relation ⇒ . {\displaystyle \,\Rightarrow .\,} The preordered set ( S , ⇐ ) {\displaystyle (S,\Leftarrow )} is a directed set because if A , B ∈ S {\displaystyle A,B\in S} and if C := A ∧ B {\displaystyle C:=A\wedge B} denotes

3009-581: The original non-strict preorder from < . {\displaystyle \,<.\,} Possible (non-strict) preorders that induce the given strict preorder < {\displaystyle \,<\,} include the following: If a ≤ b {\displaystyle a\leq b} then a ≲ b . {\displaystyle a\lesssim b.} The converse holds (that is, ≲ = ≤ {\displaystyle \,\lesssim \;\;=\;\;\leq \,} ) if and only if whenever

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3068-475: The other hand player II has a winning strategy then you have a reduction of A {\displaystyle A} to B {\displaystyle B} . For example, suppose that player II has a winning strategy. Map every sequence x to the sequence y that player II plays in G ( A , B ) {\displaystyle G(A,B)} if player I plays the sequence x , and player II follows his or her winning strategy. This defines

3127-498: The outcome is different. Sometimes this is also called the Lipschitz game , and the variant where player II has the option to pass finitely many times is called the Wadge game. Suppose that the game is determined . If player I has a winning strategy, then this defines a continuous (even Lipschitz ) map reducing B {\displaystyle B} to the complement of A {\displaystyle A} , and if on

3186-460: The outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B , respectively. Player II wins if the outcome is the same for both players, i.e. x {\displaystyle x} is in A {\displaystyle A} if and only if y {\displaystyle y} is in B {\displaystyle B} . Player I wins if

3245-417: The points a and b . One may choose to extend the definition to all pairs ( a , b ) {\displaystyle (a,b)} The extra intervals are all empty. Using the corresponding strict relation " < {\displaystyle <} ", one can also define the interval ( a , b ) {\displaystyle (a,b)} as the set of points x satisfying

3304-464: The preorder is denoted by R + = , {\displaystyle R^{+=},} then S / ∼ {\displaystyle S/\sim } is the set of R {\displaystyle R} - cycle equivalence classes: x ∈ [ y ] {\displaystyle x\in [y]} if and only if x = y {\displaystyle x=y} or x {\displaystyle x}

3363-563: The same symbol ⇐ , {\displaystyle \,\Leftarrow ,\,} is characterized by [ A ] ⇐ [ B ] {\displaystyle [A]\Leftarrow [B]} if and only if A ⇐ B , {\displaystyle A\Leftarrow B,} where the right hand side condition is independent of the choice of representatives A ∈ [ A ] {\displaystyle A\in [A]} and B ∈ [ B ] {\displaystyle B\in [B]} of

3422-513: The sentence formed by logical conjunction ∧ , {\displaystyle \,\wedge ,\,} then A ⇐ C {\displaystyle A\Leftarrow C} and B ⇐ C {\displaystyle B\Leftarrow C} where C ∈ S . {\displaystyle C\in S.} The partially ordered set ( S / ∼ , ⇐ ) {\displaystyle \left(S/\sim ,\Leftarrow \right)}

3481-433: The usual ω). As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy . If we associate with each set A {\displaystyle A} the collection of all sets strictly below A {\displaystyle A} on the Wadge hierarchy, this forms a pointclass. Equivalently, for each ordinal α  ≤ θ the collection W α of sets that show up before stage α

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