形式语言中的闭包与Kuratowski定理
Closures in Formal Languages and Kuratowski's Theorem
摘要：
Kuratowski的一个著名定理指出，在拓扑空间中，通过对给定集合重复应用闭包和补运算，最多可以产生14个不同的集合。我们在形式语言的背景下重新检验这个定理，其中闭包是Kleene闭包或正闭包。我们根据它们在补和闭包迭代下生成的代数结构对语言进行分类。我们证明了在正闭包的情况下有9个这样的代数，在Kleene闭包的情况下有12个。
英文摘要：
A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. We show that there are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure.
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