UID:
almahu_9947367703902882
Format:
1 online resource (457 p.)
Edition:
Rev. ed.
ISBN:
1-281-02955-6
,
9786611029555
,
0-08-051911-3
Series Statement:
Studies in logic and the foundations of mathematics ; v. 145
Uniform Title:
Théorie des relations.
Content:
Relation theory originates with Hausdorff (Mengenlehre 1914) and Sierpinski (Nombres transfinis, 1928) with the study of order types, specially among chains = total orders = linear orders. One of its first important problems was partially solved by Dushnik, Miller 1940 who, starting from the chain of reals, obtained an infinite strictly decreasing sequence of chains (of continuum power) with respect to embeddability. In 1948 I conjectured that every strictly decreasing sequence of denumerable chains is finite. This was affirmatively proved by Laver (1968), in the more general case of denume
Note:
Description based upon print version of record.
,
Cover; Theory of Relations; Copyright Page; Contents; Introduction; Chapter 1. Review of axiomatic set theory, relation; 1.1 First group of axioms for ZF, finite set; 1.2 Second group of axioms, ordinal, integer; 1.3 Review of ordinal algebra; 1.4 Transitive closure, fundamental rank, cardinal; 1.5 Cardinality of the continuum; 1.6 Binary relation, poset, chain, aleph; 1.7 Relation, multirelation, restriction; 1.8 Axiom of dependent choice; 1.9 Exercises; Chapter 2. Coherence lemma, cofinality, tree, ideal; 2.1 Rational, real (chains Q and R); 2.2 Well-founded poset, maximal chain
,
2.3 Filter, ultrafilter axiom2.4 Coherence lemma, ordering axiom; 2.5 Set of initial intervals; 2.6 Ordinal sum and product of chains; Dedekind's statement; 2.7 Height, cofinal subset, cofinality; 2.8 Regular or singular aleph, accessibility; 2.9 Augmentation: relation, poset; 2.10 Partition in slices (Bonnet, Pouzet); 2.11 Tree; 2.12 Cofinality of a poset, cofinal height; 2.13 Net or directed poset, ideal; 2.14 Computation of posets (Chaunier, Lygerōs); 2.15 Exercises; Chapter 3. Ramsey theorem, partition, incidence matrix; 3.1 Ramsey's theorem, Ramsey number
,
3.2 Lexico ordered set: Galvin, Nash-Williams3.3 Partition theorems: Dushnik, Miller, Erdös, Rado; 3.4 Linear independence: Kantor, multicolor: Pouzet; 3.5 Combinatorial lemmas, color and inclusion; 3.6 Profile increase theorem (Pouzet); 3.7 Ramsey sequence for Galvin's theorem (Lopez); 3.8 Exercises; Chapter 4. Good, bad sequence, well partial ordering; 4.1 Less than relation, embedding between sequences; 4.2 Good, bad, minimal bad sequence; 4.3 Well partial ordering; 4.4 Initial intervals of a w.p.o. : Higman, Rado; 4.5 Extraction theorem, words: Higman; 4.6 Well-ordered restrictions
,
4.7 Ideal and finitely free poset: Bonnet4.8 Direct product of posets; 4.9 Dimension: Dushnik, Miller, Hiraguchi; 4.10 Bound; 4.11 Maximal augmented chain: De Jongh, Parikh; 4.12 Cofinality of a finitely free poset; 4.13 Cofinal restriction of a directed w.p.o (Pouzet); 4.14 Exercise; Chapter 5.Embeddability between relations and chains; 5.1 Embeddability, immediate extension (Hagendorf); 5.2 Embeddability between posets: Dilworth, Gleason; 5.3 Dense chain; embeddability conditions; 5.4 Well partial ordering of finite trees (Kruskal); 5.5 Decreasing sequences: Dushnik, Miller, Sierpinski
,
5.6 Immediate extension (chains): Hagendorf5.7 Dense chain for an infinite cardinal; 5.8 Suslin chain and Suslin tree; 5.9 Aronszajn tree, Specker chain; 5.10 Universal class (Tarski, Vaught); 5.11 Decreasing sequence of posets: K. Kunen, A. Miller; 5.12 Exercises; Chapter 6. Scattered chain, scattered poset; 6.1 Scattered chain; 6.2 Hausdorff decomposition, neighborhood; 6.3 Right or left indecomposable chain; 6.4 Covering by indecomposable chains or by doublets; 6.5 Scattered poset: Bonnet, Pouzet; 6.6 Simple convergence topology; 6.7 Topologically scattered poset
,
6.8 Indivisible relation or chain
,
English
Additional Edition:
ISBN 0-585-47454-0
Additional Edition:
ISBN 0-444-50542-3
Language:
English
Bookmarklink
