A NEW PROOF OF COMPLETENESS OF S4 WITH RESPECT TO THE REAL LINE
A NEW PROOF OF COMPLETENESS OF S4 WITH RESPECT TO THE REAL LINE
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S4关于实数线完备性的新证明
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
M. Gehrke
中科院分区：
文献类型：
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作者：
G. Bezhanishvili;M. Gehrke
McKinsey和Tarski在[7]中证明了每个有限良通闭包代数都嵌入到实直线R的幂集合的闭包代数中。Pucket[10]通过证明存在从R到任意有限连通拓扑空间的开映射，将这一结果推广到所有有限连通闭包代数。我们利用有限拓扑空间与有限拟序集之间的对应关系，大大简化了他的证明。作为结果，我们得到了Lewis的命题模式系统S4关于R的凸子集的可数并的布尔组合是完备的，这是对McKinsey和Tarski的原始结果的加强。我们还得到了Grzegorczyk的命题模系GRZ关于R的开子集的布尔组合是完备的。最后，通过证明没有包含无限升链的可数Alexandroff空间是R的开象，证明了McKinsey和Tarski的结果不能推广到可数连通闭包代数。
It was proved in McKinsey and Tarski [7] that every finite wellconnected closure algebra is embedded into the closure algebra of the power set of the real line R. Pucket [10] extended this result to all finite connected closure algebras by showing that there exists an open map fromR to any finite connected topological space. We simplify his proof considerably by using the correspondence between finite topological spaces and finite quasi-ordered sets. As a consequence, we obtain that the propositional modal system S4 of Lewis is complete with respect to Boolean combinations of countable unions of convex subsets of R, which is strengthening of McKinsey and Tarski’s original result. We also obtain that the propositional modal system Grz of Grzegorczyk is complete with respect to Boolean combinations of open subsets of R. Finally, we show that McKinsey and Tarski’s result can not be extended to countable connected closure algebras by proving that no countable Alexandroff space containing an infinite ascending chain is an open image of R.