Algebraization of infinite summation

Abstract: 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.