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Concerning basic notions of the measurement theory. (English) Zbl 0577.08002
The authors study families of homomorphisms between pairs of relational structures. All homomorphisms considered are strong homomorphisms. Let H be a family of mappings from a set A to a set B. H is called regular if for every f,g\(\in H\) there is a mapping h from f(A) to B such that \(hf=g\). The set of all mappings h from f(A) to B such that hf\(\in H\) will be denoted by H(f). If \(H(f)=H(g)\) for all f,g\(\in H\), then H is called homogeneous. Various results are presented. We will only phrase here three of these results.
Theorem 1: Let \({\mathcal A},{\mathcal B}\) be relational structures of a given type and \(H=Hom({\mathcal A},{\mathcal B})\). Then the following statements are equivalent: (i) every member of H is surjective, (ii) for some \(f\in H\), \(H(f)=Aut({\mathcal B})\), (iii) for all \(f\in H\), \(H(f)=Aut({\mathcal B})\). In the following the universe B of the structure \({\mathcal B}\) is a subset of the set of real numbers \({\mathbb{R}}\). A nominal scale is any \(f\in H\) such that H(f) is all the injections from f(A) to B. An ordinal scale is any \(f\in H\) such that H(f) is all the strictly increasing functions from f(A) to B. Let \({\mathcal B}=(B,T)\), \({\mathcal B}'=(B,T')\) be relational structures. \({\mathcal B}\geq^ s{\mathcal B}'\) if T’ is definable by means of the family s of quantifier free formulae in the first order language of \({\mathcal B}\); \({\mathcal B}\equiv^{s,t}{\mathcal B}'\) if \({\mathcal B}\geq^ s{\mathcal B}'\) and \({\mathcal B}'\geq^ t{\mathcal B}\). Theorem 2: Let \({\mathcal A},{\mathcal B}\) be relational structures of a given type and let \(f\in Hom({\mathcal A},{\mathcal B})\). If \({\mathcal B}\equiv^{s,t}(B,=)\) and \({\mathcal B}/f(A)\equiv^{s,t}(f(A),=)\), then f is a nominal scale. If \({\mathcal B}\equiv^{s,t}(B,\leq)\) and \({\mathcal B}/f(A)\equiv^{s,t}(f(A),\leq)\), then f is an ordinal scale. Theorem 3: Let \({\mathcal A},{\mathcal B}\) be relational structures of a given type. If there is a surjective nominal scale \(f\in Hom({\mathcal A},{\mathcal B})\), then \((B,=)\geq^ s{\mathcal B}\). If there is a surjective ordinal scale \(f\in Hom({\mathcal A},{\mathcal B})\), then \((B,\leq)\geq^ s{\mathcal B}\).
Reviewer: A.A.Iskander

MSC:
08A02 Relational systems, laws of composition
03C60 Model-theoretic algebra
92F05 Other natural sciences (mathematical treatment)
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References:
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