基本説明
Treats in detail lexicographic products and their connections with universally ordered sets, and also provides thorough investigations on the structure of power sets.
Full Description
This book is written particularly for mathematics students and, of course, for mathematicians interested in set theory. Only some fun- mental parts of naive set theory are presupposed, - not more than is treated in a textbook on set theory, even if this restricts us only to the most basic facts of this field. We have summarized all of this in Chapter 0 without longer discusssions and explanations, because there are s- eral textbooks which can be consulted by the reader, e.g. HrbacekIJech [88], KneeboneIRotman [99], ShenIVereshchagin [159]. Besides this only elementary facts of analysis are used. The theory of ordered sets can be divided into two parts, depending on whether the sets under consideration are finite or infinite. The first part is grounded mainly in combinatorics and graph theory and does not make essential use of set-theorical concepts, whereas the second part presupposes a knowledge of the fundamental notions of set theory, in particular of the system of ordinal and cardinal numbers. In this book we mainly deal with general infinite ordered sets. In this field the textbook literature is still very small. Therefore this book supplements the existing literature.
Contents
Fundamental notions of set theory.- Fundamental notions.- General relations between posets and their chains and antichains.- Linearly ordered sets.- Products of orders.- Universally ordered sets.- Applications of the splitting method.- The dimension of posets.- Well-founded posets, pwo-sets and trees.- On the order structure of power sets.- Comparison of order types.- Comparability graphs.
