关于Medvedev格的结构
On the structure of the Medvedev lattice
摘要：
我们调查的梅德韦杰夫格的结构作为一个偏序。我们证明了格中的每个区间要么是有限的，在这种情况下，它同构于有限布尔代数，要么包含大小为2^2^ \aleph_0的反链，即格本身的大小。我们还证明了格上有这种大小的链是相容的，事实上，这些大链出现在每一个有大反链的区间上。我们还研究了格和代数的嵌入。我们表明，大布尔代数可以嵌入到梅德韦杰夫格上半格，但布尔代数可以嵌入格，只有当它是可数的。最后，我们讨论这些结果持有密切相关的Muchnik格。
英文摘要：
We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size 2^2^\aleph_0, the size of the lattice itself. We also prove that it is consistent that the lattice has chains of this size, and in fact that these big chains occur in every interval that has a big antichain. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
