As stated in the Preface, the aim of this eBook is to study several ordertheoretical
tools which are currently used in various fields of mathematics, particularly
in set theory, which, in its turn, provides a tool for the whole mathematics.
Thus we begin quite naturally with the construction of the essentials
of the set-theoretical framework, with special emphasis on order-theoretical aspects.
The first chapter could be more precisely titled “Set theory before the introduction
of the concept of order” and comprises three sections. The first one
sketches an axiomatic construction of set theory. This is motivated by our belief
that although most mathematicians work within the framework of naive set
theory, no mathematician could ignore the existence of foundations. However,
as the axiomatic line will not be followed in the sequel, the presentation is quite
informal. The next section gathers in a systematic presentation the most frequently
used properties of correspondences, relations and functions; thus e.g.
each of the concepts of injection, surjection and bijection is characterized by
6-8 equivalent conditions. The last section sketches a few categorical prerequisites
which will enable us to use the language of categories whenever it will be
convenient in the subsequent sections.....
