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Alonzo Church而命名的Alonzo，基于Church的类型论，即Church提出的简单类型论。小理论方法是一种将数学知识形式化为理论网络（称为理论图）的方法，该网络以理论为节点、理论态射为有向边。数学主题的展开在理论图中具有最合适抽象层次和最便利词汇的"小理论"中进行，随后通过理论图中的态射，将展开过程中产生的定义和定理按需传输到其他理论。本文旨在阐明如何运用小理论方法在Alonzo中对数学知识体系进行形式化。具体通过将幺半群理论——关于幺半群的数学知识体系——在Alonzo中进行形式化来实现这一目标。