无环性和还原性

摘要：

文献提供了涉及同态（如G 0二分法）或减少（如表征集潜在的Wadge类的博雷尔集，它持有的一个子集的产品）的二分法。然而，后一个结果背后的部分动机是得到整个产品的减少，就像在分析等价关系研究中考虑的经典概念Borel reductions。这在一般情况下是不可能的。我们表明，在一些无环性（和拓扑）的假设下，这是广泛可能的。特别地，我们证明了对于任何非自对偶Borel类Γ，存在一个具体的有限的=<c-反链基给Borel关系类，它的闭包具有非循环对称性，并且它不可能在Γ中.沿着类似的思路，我们提供了一个充分条件=<c-约化G 0。我们还证明了一个类似的结果，给出了一个最小集，而不是一个反链，如果我们允许矩形减少。

英文摘要：

The literature provides dichotomies involving homomorphisms (like the G 0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of analytic equivalence relations. This is not possible in general. We show that, under some acyclicity (and also topological) assumptions, this is widely possible. In particular, we prove that, for any non-self dual Borel class Γ, there is a concrete finite =< c-antichain basis for the class of Borel relations, whose closure has acyclic symmetrization, and which are not potentially in Γ. Along similar lines, we provide a sufficient condition for =< c-reducing G 0. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.

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