The aim of this introductory chapter is twofold. First, to refresh the reader’s memory about the basics before entering into the study of more serious topics. A fairly large portion of the material can be found in standard textbooks on algebra. So, the reader may skip some or all of the sections and turn to the appropriate places when needed. Secondly, we have to fix the fundamental terminology to be used throughout this volume. In the 1950s, as in every rapidly developing field, the terminology in abelian groups often varied from author to author. Slowly standardization took place, and we will use here the widely accepted, the most familiar or the more appropriate names for the concepts, with a couple of exceptions where new terminology is warranted.
The topics covered in this chapter include maps, diagrams, sets, categories. Most of the proofs will be omitted as they are standard and available in textbooks.
For the comfort of the reader, especially, of those who look for references, we avoid the use of abbreviations in the text as far as reasonable.