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\(\kappa\)-homogeneous relations and tournaments. (English) Zbl 0678.04001

An r-ary relation R on an infinite set \(\Omega\) is called \(\kappa\)- homogeneous (elementary \(\kappa\)-homogeneous) if whenever A,B\(\subseteq \Omega\) are two subsets with cardinality \(\kappa\), then the structure \((A,R\cap A^ r)\) and \((B,R\cap B^ r)\) are isomorphic (elementary equivalent). The author characterizes all infinitary binary relation structures (\(\Omega\),R) which are \(\kappa\)-homogeneous (or elementary \(\kappa\)-homogeneous) for some fixed finite or infinite cardinal \(\kappa\).
Also, for a certain class of infinite tournaments he generalizes some properties of infinite linearly ordered sets.
Reviewer: S.S.Starchenko

MSC:

03E20 Other classical set theory (including functions, relations, and set algebra)
06A99 Ordered sets
05C20 Directed graphs (digraphs), tournaments
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