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Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups

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Abstract

Two groups are constructed, these being elementarily equivalent, with one of them being nontrivially partially ordered but without being directionally ordered, while the other is directionally ordered. This proves the elementary nonclosure and nonaxiomatizability of the class of directionally ordered groups in the class of nontrivially partially ordered groups. It is shown that, in the decreasing chain of classes, i.e., all groups, nontrivially partially ordered groups, directionally ordered groups, and lattice ordered groups, each successive class is not elementarily closed, and hence, is not axiomatizable, in any earlier class.

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Literature cited

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Authors and Affiliations

  1. Ivanov Textile Institute, USSR

    A. A. Vinogradov

Authors
  1. A. A. Vinogradov
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Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 395–397, September, 1973.

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Vinogradov, A.A. Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups. Mathematical Notes of the Academy of Sciences of the USSR 14, 787–788 (1973). https://doi.org/10.1007/BF01147456

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  • DOI: https://doi.org/10.1007/BF01147456

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