Alonzo is a practice-oriented classical higher-order logic that extends first-order logic and that admits undefined expressions. Named in honor of Alonzo Church, Alonzo is based on Church's type theory, Church's formulation of simple type theory. The little theories method is a method for formalizing mathematical knowledge as a network of theories called a theory graph consisting of theories as nodes and theory morphisms as directed edges. The development of a mathematical topic is done in the "little theory" in the theory graph that has the most convenient level of abstraction and the most convenient vocabulary, and then the definitions and theorems produced in the development are transported, as needed, to other theories via the theory morphisms in the theory graph. The purpose of this paper is to illustrate how a body of mathematical knowledge can be formalized in Alonzo using the little theories method. This is done by formalizing monoid theory -- the body of mathematical knowledge about monoids -- in Alonzo.

翻译：Alonzo 是一种面向实践的经典高阶逻辑，它扩展了一阶逻辑并允许未定义表达式。为纪念阿隆佐·邱奇而命名的 Alonzo 基于邱奇类型理论，即邱奇对简单类型理论的表述。小理论方法是一种将数学知识形式化为理论网络的方法，该网络称为理论图，其中理论作为节点，理论态射作为有向边。数学主题的发展是在理论图中具有最便捷抽象层次和最便捷词汇的“小理论”中完成的，然后根据需要，通过理论图中的理论态射将发展中产生的定义和定理迁移到其他理论中。本文旨在说明如何利用小理论方法在 Alonzo 中形式化一个数学知识体系。这通过 Alonzo 中幺半群理论（关于幺半群的数学知识体系）的形式化来实现。