Product of sets as complexes

Question

What does it mean to take the product of two sets of complex numbers as complexes?

Reading this paper: "The Determinant of the Sum of Two Normal Matrices with Prescribed Eigenvalues" by N. Bebiano and J. F. Queiro

Here's the paper:

http://www.sciencedirect.com/science/article/pii/0024379585902319/pdf?md5=4ad22968e71962565f02aeaad8f27c8b&pid=1-s2.0-0024379585902319-main.pdf

It refers to products of sets as complexes. Here's a quote

LEMMA 1. If $S$ and $T$ are subsets of a real algebra, then $(\operatorname{conv} S)\cdot (\operatorname{conv} T)$ is a subset of $\operatorname{conv}(S\cdot T)$. [Here $X \cdot Y$ means the product (as complexes) of the sets $X$ and $Y$.]

$\operatorname{conv}$ means convex hull.

I don't understand what kind of product of sets this is. I suspect it is related to topology.

Answer 1

From p. 22 of B. L. van der Waerden, Modern Algebra, Volume I, Revised English Edition, Frederick Ungar Publishing Co., New York, Copyright 1949, 1953:

In group theory a complex is defined as an arbitrary set of elements of a group $\frak G.$

By the product $\frak{gh}$ of two complexes $\frak g$ and $\frak h$ we understand the set of all products $gh$ where $g$ is taken from $\frak g,$ and $h$ from $\frak h.$ If in the product $\frak{gh}$ one of the complexes, say $\frak g,$ consists of only one element $g,$ we may simply write $g\frak h$ instead of $\frak{gh}.$