Relations: Concrete, Abstract, And Applied - An Introduction

The book is intended as an invitation to the topic of relations on a rather general basis. It fills the gap between the basic knowledge offered in countless introductory papers and books (usually comprising orders and equivalences) and the highly specialized monographs on mainly relation algebras, many-valued (fuzzy) relations, or graphs. This is done not only by presenting theoretical results but also by giving hints to some of the many interesting application areas (also including their respective theoretical basics). This book is a new — and the first of its kind — compilation of known results on binary relations. It offers relational concepts in both reasonable depth and broadness, and also provides insight into the vast diversity of theoretical results as well as application possibilities beyond the commonly known examples. This book is unique by the spectrum of the topics it handles. As indicated in its title these are: Concrete aspects are covered in detail in Part 2 for two-valued, and in Part 3 for many-valued relations. Various kinds of relations and their properties are presented in the familiar form of first order formulas, but mainly algebraically in the point-free version as used in the so-called calculus of relations. Abstract points of view are the content of Part 4: The connections of relations to semigroups, as well as the concept of relation algebras, and how relations are handled within category theory are presented in some detail. Applied relations in Part 5 give a broad spectrum of classical and newer topics where relations of various kinds play a dominant role. This way, readers should get an idea how diverse the possibilities are where relations can be used, and how mighty their modeling abilities turn out to be.