Concerning basic notions of the measurement theory.

*(English)*Zbl 0577.08002The 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 \((B,\leq)\geq^ s{\mathcal B}\).

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) |

##### Keywords:

families of homomorphisms; strong homomorphisms; nominal scale; ordinal scale; strictly increasing functions; quantifier free formulae; first order language
PDF
BibTeX
XML
Cite

\textit{T. Bromek} et al., Czech. Math. J. 34(109), 570--587 (1984; Zbl 0577.08002)

Full Text:
EuDML

**OpenURL**

##### References:

[1] | Bartoszyňski R.: A metric structure derived from subjective judgements: scalling under perfect and imperfect discrimination. Econometrica 42, No 1 (1974). · Zbl 0335.62038 |

[2] | Chang C. C., Keisler H. J.: Model Theory. (1977). · Zbl 0697.03022 |

[3] | Roberts F. S.: Measurement Theory. (1979). · Zbl 0432.92001 |

[4] | Rudník K.: On regularity of scales. to appear in Bull. Acad. Polon. Sci. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.