Sets and Ordered Structures

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.....

As explained in the Preface, the concept of order is a tool used in mathematics both in its more general form and under certain supplementary conditions. This chapter first deals with partially ordered sets at the general level, then with totally ordered sets and well-ordered sets, thus beginning the specialization line which leads to ordinals (to be studied in the next chapter)....

The theory of ordinal numbers is a natural and very powerful generalization of the order-theoretical properties of natural numbers. In particular it furnishes transfinite induction, a method for constructing rather complicated mathematical concepts and for proving properties valid beyond the natural numbers. Ordinal numbers can also serve as a basis for introducing cardinal numbers. The latter evaluate “how many elements” a set possesses, being thus a kind of “quantitative” generalization of natural numbers, widely used in mathematics.

After the essentials of the theory of transfinite numbers, investigated in the previous chapter, we now present a few topics from another specialization of the concept of order: lattice theory. Our aim is to provide lattice-theoretical tools broadly used in various fields of mathematics and in particular some latticetheoretical background necessary to universal algebra....

The title of this chapter summarizes three types of representation theorems dealt with: representations of the elements of certain lattices as meets/joins of elements from prescribed subsets, isomorphic representation of several types of posets (semilattices, lattices) as posets (semilattices, lattices) of sets with inclusion as partial order, and finally a more sophisticated development of the latter representations in the case of distributive and Boolean lattices: the duality between these categories and certain categories of topological spaces. These types of problems are treated in §§ 2, 3 and 5, respectively. The first section is devoted to ideals and filters both as a preparation to the subsequent sections and in view of the numerous other applications. The topological prerequisites necessary to §5 are collected in §4.

We have selected in five chapters what we believe to be the core of sets and order in mathematics. Of course,we are aware that this was an ambitious plan and that our selection could not be infallible. Yet we did our best and we conclude the eBook with several examples which illustrate the applications of lattices to topology (§§1-3), to universal algebra (§4), to a new field of applied mathematics called formal concept analysis (§5) and to logic (§6). Besides the references to the literature given in the sections of this chapter, we recommend a nice paper by Birkhoff [1970], with the suggestive title “What can lattices do for you?”.