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Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. (Here, by abuse of terminology, a quasiorder \le is said to be well-founded if the corresponding strict order x\le y\land y leq x is a well-founded relation.) However the class of well-founded quasiorders is not closed under certain operations—that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well- founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded. An example of this is the power set operation.
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Suppose A and B are subsets of Baire space ωω. Then A is Wadge reducible to B or A ≤W B if there is a continuous function f on ωω with 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 is denoted by [A]W.
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In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder. An antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric.
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Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive; i.e., for all a, b and c in P, we have that: : a ≤ a (reflexivity) : if a ≤ b and b ≤ c then a ≤ c (transitivity) A set that is equipped with a preorder is called a preordered set (or proset).For "proset", see e.g. . If a preorder is also antisymmetric, that is, and implies , then it is a partial order.
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