Immunity and Non-Cupping for Closed Sets
Immunity and Non-Cupping for Closed Sets
复制标题
封闭组的免疫和非拔罐
DOI：
10.32513/tbilisi/1528768843
复制
发表时间：
2009
影响因子：
0.5
通讯作者：
Guohua Wu
中科院分区：
文献类型：
--
作者：
D. Cenzer;Takayuki Kihara;Rebecca Weber;Guohua Wu
我们将免疫的概念扩展到闭集，特别是在两个方面：免疫意味着相应的树没有无限可计算的子集，树免疫意味着它没有无限可计算的子树。我们将这些概念彼此分开，并将其特殊化，并显示可计算不可分的c. e.的分离类。集合是免疫的，完美瘦类是树免疫的。我们定义了即时免疫的概念，并构造了一个积极的措施，迅速免疫类。我们证明了无免疫的P_01类P杯到对角不可计算集的Medvedev完全类DNC，其中P_01类的Medvedv度中的P杯到Q，如果存在类R使得乘积P <$R <$M Q.我们刻画了（树）免疫与Medvedev相遇和连接之间的相互作用，显示了Medvedev格中素理想的（树）免疫度。证明了每个随机闭集都是免疫的且不小，每个小的特殊类都是免疫的。
We extend the notion of immunity to closed sets and to Π1 classes in particular in two ways: immunity meaning the corresponding tree has no infinite computable subset, and tree-immunity meaning it has no infinite computable subtree. We separate these notions from each other and that of being special, and show separating classes for computably inseparable c.e. sets are immune and perfect thin classes are tree-immune. We define the notion of prompt immunity and construct a positive-measure promptly immune Π1 class. We show that no immune-free Π 0 1 class P cups to the Medvedev complete class DNC of diagonally noncomputable sets, where P cups to Q in the Medvedv degrees of Π1 classes if there is a class R such that the product P ⊗ R ≡M Q. We characterize the interaction between (tree-)immunity and Medvedev meet and join, showing the (tree-)immune degrees form prime ideals in the Medvedev lattice. We show that every random closed set is immune and not small, and every small special class is immune.