Orders and lattices. Basics, approaches and applications (Q2637247)

In this book the author gives a precise and elementary introduction to the theory of partial orders and lattice theory. The book is based essentially on the first six chapters of the second edition of his book [Orders, lattices and relations with applications. Wiesbaden: Vieweg+Teubner (2008; Zbl 1146.06001)]. In the first chapter the author introduces basic notions. In chapter two he describes lattices from the algebraic point of view and also in form of special partial orders. He investigates the duality principle for lattices. In the next chapter he characterizes some important classes of lattices such as modular, distributive and Boolean lattices. He also investigates complete lattices. In chapter four he studies fixed-point theorems. As an application he gives a proof of the Schröder-Bernstein theorem. He considers Galois connections. Chapter five is devoted to completions of partial orders and lattices. He describes completions by cuts and by ideals and compares both methods. Chapter six deals with well-orders and the axiom of choice. He shows some consequences of the axiom of choice and gives statements which are equivalent to this axiom. He proves the Stone representation theorem and also the compactness theorem for propositional logic. In the last chapter he studies some applications of lattices and partial orders in computer science. So he deals with denotational semantics and distributed systems.