Orbits of Maximal Vector Spaces
Orbits of Maximal Vector Spaces
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最大向量空间的轨道
DOI：
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发表时间：
2016
期刊：
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通讯作者：
V. Harizanov
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作者：
R. Dimitrov;V. Harizanov
令 V∞ 为有理数域上的标准可计算无限维向量空间。 V∞ 的可计算向量子空间的latticeL$$ mathfrak{L} $$(V∞) 及其商格模有限维L$$ mathfrak{L} $$*(V∞) 已被广泛研究。与此同时，许多重要问题仍然悬而未决。 1998 年，R. Downey 和 J. Remmel 提出了在 L$$ mathfrak{L} $$*(V∞) 中寻找有意义的轨道的问题 [4，问题 5.8]。这个问题既重要又困难，它的答案取决于格子L$$ mathfrak{L} $$*(V∞)结构理论的重大进展，以及对其自同构的更好理解。这里我们给出了具有可扩展基的拟最大（因此最大）向量空间位于 L$$ mathfrak{L} $$*(V∞) 相同轨道上的充要条件。更具体地说，我们考虑两个向量空间 V1 和 V2，它们由 V∞ 的两个可能不同的可计算基的拟极大子集跨越。我们给出了由 V1 和 V2inL$$ mathfrak{L} $$*(V∞) 确定的主滤波器同构的充要条件。我们还指定 L$$ mathfrak{L} $$*(V∞) 的自同构 Φ 存在的充分必要条件，使得 Φ 将 V1 的等价类映射到 V2 的等价类。我们的结果使用 m 度的相关向量集来表示。这项研究与 R. Soare 在 [13] 中对可计算可枚举集的格 ε 及其商格模有限集 ε* 中的拟极大集轨道的研究并行。然而，我们的结论和证明机制与索亚雷的有很大不同。特别地，我们确定由 L$$ mathfrak{L} $$*(V∞) 中的拟极大向量空间确定的主滤波器的结构通常比由 ε* 中的拟极大集确定的主滤波器的结构复杂得多。我们还指出，与 ε* 不同，L$$ mathfrak{L} $$*(V∞) 中具有同构主滤波器仅仅是两个拟极大向量空间的等价类位于 L$$ mathfrak{L} $$*(V∞) 的同一轨道上的必要条件。
Let V∞be a standard computable infinite-dimensional vector space over the field of rationals. The latticeL$$ mathfrak{L} $$(V∞) of computably enumerable vector subspaces of V∞and its quotient lattice modulo finite dimension,L$$ mathfrak{L} $$*(V∞), have been studied extensively. At the same time, many important questions still remain open. In 1998, R. Downey and J. Remmel posed the question of finding meaningful orbits inL$$ mathfrak{L} $$*(V∞) [4, Question 5.8]. This question is important and difficult and its answer depends on significant progress in the structure theory for the latticeL$$ mathfrak{L} $$*(V∞), and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit ofL$$ mathfrak{L} $$*(V∞). More specifically, we consider two vector spaces, V1and V2, which are spanned by two quasimaximal subsets of, possibly different, computable bases of V∞. We give a necessary and sufficient condition for the principal filters determined by V1and V2inL$$ mathfrak{L} $$*(V∞) to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism Φ ofL$$ mathfrak{L} $$*(V∞) such that Φ maps the equivalence class of V1to the equivalence class of V2. Our results are expressed using m-degrees of relevant sets of vectors. This study parallels the study of orbits of quasimaximal sets in the lattice ε of computably enumerable sets, as well as in its quotient lattice modulo finite sets, ε*, carried out by R. Soare in [13]. However, our conclusions and proof machinery are quite different from Soare’s. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space inL$$ mathfrak{L} $$*(V∞) is generally much more complicated than the one of a principal filter determined by a quasimaximal set in ε*. We also state that, unlike in ε*, having isomorphic principal filters inL$$ mathfrak{L} $$*(V∞) is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit ofL$$ mathfrak{L} $$*(V∞).