标题： 无限求和的代数化 显示英文标题

标题： Algebraization of infinite summation

摘要： 一种用于研究无限求和的代数框架被提出，补充并增强了通常的拓扑工具。 该框架涵盖了文献中的许多例子。 它是通过许多不同的例子来发展的，特别强调将通常的群和环公理进行无限化。 比较这些例子表明，一些关键的代数性质在不同形式的无限求和行为中起着至关重要的作用。 特别关注结合性，这在正确地进行无限化时尤其困难。 在这一背景下，有一种称为Eilenberg-Mazer滑轮的重要技术被研究并大大推广。 一些特殊性质被单独指出作为潜在的公理。 分析了这些潜在公理之间的相互作用，并有许多结果探讨了如何在保留旧公理的同时强加新的公理。 在某些情况下，这些公理可以对给定的例子进行分类或归类。 令人惊讶的是，这样的公理化非常简洁，仅依赖于少数自然条件。 这些研究更精确地揭示了拓扑在无限求和形成中的作用。 特别关注部分求和和无条件求和的方法。 从相反的方向来看，证明仅从无限求和就可以创建一种精细的拓扑，介于原始拓扑及其序列核心反射之间。 这些思想的一个特别有趣的应用是构建新的代数商结构，这些结构保留处理无限求和的能力。 显示英文摘要

摘要： An algebraic framework in which to study infinite sums is proposed, complementing and augmenting the usual topological tools. The framework subsumes numerous examples in the literature. It is developed using many varied examples, with a particular emphasis on infinitizing the usual group and ring axioms. Comparing these examples reveals that a few key algebraic properties play a crucial role in the behaviors of different forms of infinite summation. Special attention is given to associativity, which is particularly difficult to properly infinitize. In that context, there is an important technique called the Eilenberg-Mazer swindle that is studied and greatly generalized. Some special properties are singled out as potential axioms. Interactions between these potential axioms are analyzed, and numerous results explore how to impose new axioms while retaining old ones. In some cases the axioms classify or categorize a given example. Surprisingly, such axiomatizations are very concise, relying on only a handful of natural conditions. These investigations reveal more precisely the part that topology plays in the formation of infinite sums. Special attention is given to the methods of partial summation and unconditional summation. In the opposite direction, it is proved that from the infinite sums alone one can create a refined topology, lying between the original topology and its sequential coreflection. Another especially interesting application of these ideas is the construction of new algebraic quotient structures that retain the ability to handle infinite summation.