# Theory of Relations

The first part of this book concerns the present state of the theory of chains (= total or linear orderings), in connection with some refinements of Ramsey's theorem, due to Galvin and Nash-Williams. This leads to the fundamental Laver's embeddability theorem for scattered chains, using Nash-Williams' better quasi-orderings, barriers and forerunning.The second part (chapters 9 to 12) extends to general relations the main notions and results from order-type theory. An important connection appears with permutation theory (Cameron, Pouzet, Livingstone and Wagner) and with logics (existence criterion of Pouzet-Vaught for saturated relations). The notion of bound of a relation (due to the author) leads to important calculus of thresholds by Frasnay, Hodges, Lachlan and Shelah. The redaction systematically goes back to set-theoretic axioms and precise definitions (such as Tarski's definition for finite sets), so that for each statement it is mentioned either that ZF axioms suffice, or what other axioms are needed (choice, continuum, dependent choice, ultrafilter axiom, etc.).

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Published: January 1986

Imprint: North-holland

ISBN: 978-0-444-87865-6

## Contents

• Review of Axiomatic Set Theory. Relation, Partial Ordering, Chain, Isomorphism, Cofinality. Ramsey Theorems, Partitions, Combinatorial Principles. Good and Bad Sequence, Finitely Free Partial Ordering, Well Partial Ordering, Ideal, Tree, Dimension. Embeddability Between Partial or Total Orderings. Scattered Chain, Neighborhood, Indecomposability. Use of Scattered Chains for the Study of Finitely Free and Well Partial Orderings. Barrier, Barrier Sequence, Forerunning, Embeddability Theorem for Scattered Chains, Better Partial Ordering. Isomorphism and Embeddability Between Relations, Local Isomorphism, Free Interpretability, Constant Relation, Chainable and Monomorphic Relation. Age, Rich Relation, Inexhaustible Relation, Saturated Relation, Existence Criterion for a Rich Relation of a Given Age. Homogeneous Relation, Relational System, Connection with Permutation Groups, Orbit. Bound of a Relation; Well Relation, Reassembling Theorem. Bibliography. Index.