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A stronger form of the theorem on the existence of a rigid binary relation on any set. (English) Zbl 1096.03057

A relation \(R\subseteq A\times A\) on a set \(A\) is said to be rigid if there is no homomorphism \((A,R)\to (A,R)\) except the identity. P. Vopěnka, A. Pultr and Z. Hedrlín [Comment. Math. Univ. Carol. 6, 149–155 (1965; Zbl 0149.01402)] proved: On every set \(A\) there is a rigid binary relation. Here the author proves a stronger form of this theorem, namely: Let \(\kappa\) be an infinite cardinal and \(|A|\leq 2^\kappa\). Then there exists a relation \(R\subseteq A\times A\) with the property: For all \(x\in A\) there exists an \(x\) containing subset \(A(x)\) of \(A\) with cardinality \(\leq\kappa\) is such that for all \(f: A(x)\to A\) which are different from the identity mapping there holds: \(f\) is not a homomorphism of \(R\). In particular this implies that \(R\) is rigid. Further it is is proved that if a relation \(R\subseteq A\times A\) has the above property, then \(|A|\leq 2^\kappa\).

MSC:

03E05 Other combinatorial set theory
03E20 Other classical set theory (including functions, relations, and set algebra)
08A35 Automorphisms and endomorphisms of algebraic structures
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)

Citations:

Zbl 0149.01402
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