部分遗传不可解空间的模态逻辑
Modal Logics of Some Hereditarily Irresolvable Spaces
AI总结：
该研究引入遗传k-不可解空间概念，为K4$\u2102_n$等模态逻辑建立拓扑语义，刻画了不同空间类下的逻辑扩展，并给出K4M的拓扑语义解释。
摘要：
若一个拓扑空间的任意子空间都无法划分为k个稠密子集，则称其为“遗传k-不可解（hereditarily k-irresolvable）”空间。我们利用这一概念为一系列模态逻辑提供拓扑语义，该序列的第n个逻辑K4$\u2102_n$的特征是在周长不超过n的传递Kripke框架中有效。我们证明，在将模态算子$\u25c7$解释为导集（即极限点集）运算的情况下，K4$\u2102_n$的有效性可由所有满足遗传n+1-不可解性且具有T$_D$分离性质的空间刻画。\n我们还确定了当涉及的空间类限制为弱散点空间、拥挤空间或开不可解空间时，K4$\u2102_n$的扩展逻辑，其中开不可解指每个非空开子空间都是2-不可解的。最后，我们为K4M（其中M是McKinsey公理）提供了拓扑语义。
英文摘要：
A topological space is \emph{hereditarily $k$-irresolvable} if none of its subspaces can be partitioned into $k$ dense subsets, We use this notion to provide a topological semantics for a sequence of modal logics whose $n$-th member K4$\mathbb{C}_n$ is characterised by validity in transitive Kripke frames of circumference at most $n$. We show that under the interpretation of the modality $\Diamond$ as the derived set (of limit points) operation, K4$\mathbb{C}_n$ is characterised by validity in all spaces that are hereditarily $n+1$-irresolvable and have the T$_D$ separation property. We also identify the extensions of K4$\mathbb{C}_n$ that result when the class of spaces involved is restricted to those that are weakly scattered, or crowded, or openly irresolvable, the latter meaning that every non-empty open subspace is 2-irresolvable. Finally we give a topological semantics for K4M, where M is the McKinsey axiom.
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