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.

翻译：代数语言理论的起点在于：有限单词的正则语言恰好是那些能被有限幺半群识别的语言。这一有限性条件导出了一个拓扑空间，其点（称为profinite单词）编码了单词相对于有限幺半群的极限行为。在本工作中，我们从单词和幺半群转向树与克隆（clone）——确定性自底向上树自动机背后的代数结构。利用范畴论中的余密度单子（codensity monad）概念，我们为克隆引入了一种profinite完备化构造。我们证明，克隆上的这一构造同时推广了集合上的超滤子单子（ultrafilter monad）与幺半群的profinite完备化。当应用于分级字母表上的自由克隆时，克隆的profinite完备化产生了一种profinite树的概念，从而为有限树的正则语言提供了一种拓扑学进路。我们证明这些profinite树与profinite λ演算中一个已被明确识别的片段相吻合。