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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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