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Matematicheskie Zametki, 1973, Volume 14, Issue 3, Pages 395–397 (Mi mzm7269)  

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

A. A. Vinogradov

Ivanov Textile Institute
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.
Received: 24.01.1972
English version:
Mathematical Notes, 1973, Volume 14, Issue 3, Pages 787–788
DOI: https://doi.org/10.1007/BF01147456
Bibliographic databases:
UDC: 519.4
Language: Russian
Citation: A. A. Vinogradov, “Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups”, Mat. Zametki, 14:3 (1973), 395–397; Math. Notes, 14:3 (1973), 787–788
Citation in format AMSBIB
\Bibitem{Vin73}
\by A.~A.~Vinogradov
\paper Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 3
\pages 395--397
\mathnet{http://mi.mathnet.ru/mzm7269}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=327610}
\zmath{https://zbmath.org/?q=an:0294.06012}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 3
\pages 787--788
\crossref{https://doi.org/10.1007/BF01147456}
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