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Theory of relations. Transl. from the French by P. Clote. With an appendix by Norbert Sauer. Revised ed. (English) Zbl 0965.03059

Studies in Logic and the Foundations of Mathematics. 145. Amsterdam: North-Holland. 451 p. (2000).
This book is a revision of the original edition (1986; Zbl 0593.04001). In addition it also treats the more recent developments in this area. In an introductory chapter a rather detailed review of axiomatic set theory is given, in which for most theorems also the proofs are worked out or at least sketched. Then the fundamental notions of order theory are introduced, and subsequenty a lot of concepts and theorems from the partition calculus of set theory, which later are used as tools.
The chapters 4-8 are mainly devoted to order relations. Among others the following concepts are discussed: cofinality (for linearly and partially ordered sets), well partial ordering, dimension theory for posets, embeddability between relations and chains, trees, scattered chains and scattered posets, well-quasi-ordering of scattered chains, barriers.
The chapters 9-13 deal mostly with more general relational systems. Among others the notions of age, \(\alpha\)-morphism, relative isomorphism, saturated relation, homogeneous relation, orbit, compatibility and chainability theorems are treated. An appendix on countable homogeneous systems by N. Sauer follows.
The book has an extensive bibliography (253 items) and includes many exercises. The text is concise but well comprehensible.
Reviewers remark: The theorem in 4.9.4 was first mentioned in a paper by the reviewer [Math. Nachr. 46, 183-188 (1970; Zbl 0175.01201)].
In the last line of p. 66 the comparison graph of the neighbor-relation, not that of \(<\), should be taken.
p. 54, line 16: replace “least” by “last”; p. 146 line 13 from below, replace \(a\) by \(c\); p. 147, in lines 20, 23, 26 replace the first \(\in\) by \(\not\in\).

MSC:

03E20 Other classical set theory (including functions, relations, and set algebra)
06A06 Partial orders, general
06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03E05 Other combinatorial set theory
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