Infinite sums in totally ordered abelian groups. (English) Zbl 1459.06012

Oman, Greg; Randall, Caitlin; Robinson, Logan

Involve 12, No. 2, 281-300 (2019).

Summary: The notion of convergence is absolutely fundamental in the study of calculus. In particular, it enables one to define the sum of certain infinite sets of real numbers as the limit of a sequence of partial sums, thus obtaining so-called convergent series. Convergent series, of course, play an integral role in real analysis (and, more generally, functional analysis) and the theory of differential equations. An interesting textbook problem is to show that there is no canonical way to “sum” uncountably many positive real numbers to obtain a finite (i.e., real) value. Plenty of solutions to this problem, which make strong use of the completeness property of the real line, can be found both online and in textbooks. In this note, we show that there is a more general reason for the nonfiniteness of uncountable sums. In particular, we present a canonical definition of “convergent series”, valid in any totally ordered abelian group, which extends the usual definition encountered in elementary analysis. We prove that there are convergent real series of positive numbers indexed by an arbitrary countable well-ordered set and, moreover, that any convergent series in a totally ordered abelian group indexed by an arbitrary well-ordered set has but countably many nonzero terms.

MSC:

06F20	Ordered abelian groups, Riesz groups, ordered linear spaces
40J99	Summability in abstract structures

Keywords:

infinite series; totally ordered abelian group; transfinite induction; transfinite recursion; well-ordered set

References:

[1]	; Botto Mura, Orderable groups. Lecture Notes in Pure and Applied Mathematics, 27 (1977) · Zbl 0358.06038
[2]	10.1090/gsm/126
[3]	; Tao, Analysis, I. Texts and Readings in Mathematics, 37 (2014)