Topological aspects of the Medvedev lattice
Topological aspects of the Medvedev lattice
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梅德韦杰夫晶格的拓扑方面
DOI：
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发表时间：
2011
影响因子：
0.3
通讯作者：
A. Sorbi
中科院分区：
文献类型：
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作者：
A. Lewis;R. Shore;A. Sorbi
我们研究具有不同拓扑性质的Medvedev质量度问题，如稠密性、封闭性或离散性。我们研究了由这些度生成的子格；由稠密度及其补集生成的素理想，一个素滤子；由非零闭度生成的滤子和由非零离散度生成的滤子。我们给出了这些子格、这些滤子和这个理想之间包含保持的关系的完整图景。我们证明了闭Medvedev度的子格不是Brouwer代数。我们研究了在图灵等价下闭合的稠密质量度问题，证明了稠密度构成了Medvedev格的自同构基。所得结果对Baire空间上的Medvedev格和Cantor空间上的Medvedev格都成立。
We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.