A stronger form of the theorem on the existence of a rigid binary relation on any set

• Autores: Apoloniusz Tyszka
• Localización: Aequationes mathematicae, ISSN 0001-9054, Vol. 66, Nº. 3, 2003, págs. 257-260
• Idioma: inglés
• DOI: 10.1007/s00010-003-2676-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • On every set A there is a rigid binary relation, i.e. such a relation $ R \subseteq A \times A $ that there is no homomorphism $ \langle A, R \rangle \rightarrow \langle A, R \rangle $ except the identity (Vopnka et al. [1965]). We prove that for each infinite cardinal number $ \kappa $ if card $ A \leq 2^\kappa $, then there exists a relation $ R \subseteq A \times A $ with the property $ \forall x \in A \exists^{x \in A(x) \subseteq A}_{{\rm card} A(x) \leq \kappa} \, \forall^{f:A(x) \rightarrow A}_{f \neq {\rm id}_{A(x)\rightarrow A}} $ f is not a homomorphism of R which implies that R is rigid. If a relation $ R \subseteq A \times A $ has the above property, then $ {\rm card} A \leq 2^\kappa $.