In the classical theory of regular languages the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called "Substitution Principle". In this paper we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil's Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.

翻译：在正则语言的经典理论中，通过投射有限幺半群进行识别的概念是一个重要工具。超越正则性，具有内部幺半群的布尔空间（BiMs）最近被提出作为推广。另一方面，定义正则语言的逻辑片段可以通过所谓的"替换原则"进行归纳研究。本文显式地阐明了该原则的逻辑基础，并利用Stone对偶性将其推广到任意语言。随后我们展示了如何利用该原则获取由广泛的一阶逻辑片段定义的语言类别的拓扑代数识别器。这自然引出了BiMs的半直积概念，它推广了投射有限幺半群的经典构造。我们的主要结果是将Almeida和Weil关于半直积的分解定理从投射有限设定推广到BiMs设定。这是将正则语言理论的投射有限方法推广到复杂性理论框架的关键步骤。