profinite trees 通过 Lawvere 理论和 lambda-算子
Profinite trees, through Lawvere theories and the lambda-calculus
AI总结：
本文通过 Lawvere 理论和 lambda-算子引入了克隆的 profinite 完成，从而为有限树的正则语言提供了一种拓扑方法，并证明了 profinite 树与 profinite lambda-算子的片段一致。
摘要：
代数语言理论的起点是有限词的正则语言恰好是那些由有限单形识别的。这个有限性条件导致了一个拓扑空间，其点被称为 profinite 词，编码了词相对于有限单形的极限行为。在本工作中，我们从词和单形转向树和克隆，即确定性自底向上的树自动机的代数结构。利用范畴中的码密度单形概念，我们引入了克隆的 profinite 完成。我们证明了这个构造在克隆上同时推广了集合上的超滤单形和单形的 profinite 完成。当应用于带排名字母表的自由克隆时，克隆的 profinite 完成提供了一种 profinite 树的概念，为有限树的正则语言提供了一种拓扑方法。我们证明这些 profinite 树与 profinite lambda-算子中已确定的片段一致。
英文摘要：
The starting point of algebraic language theory is that regular languages of finite words are exactly those recognized by finite monoids. This finiteness condition gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. In this work, we move from words and monoids to trees and clones, the algebraic structures underlying deterministic bottom-up tree automata. Using the categorical notion of codensity monad, we introduce a profinite completion for clones. We prove that this construction on clones simultaneously generalizes the ultrafilter monad on sets and the profinite completion of monoids. When applied to free clones on a ranked alphabet, the profinite completion of clones yields a notion of profinite tree, providing a topological approach to regular languages of finite trees. We prove that these profinite trees coincide with a well-identified fragment of the profinite lambda-calculus.
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。
