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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The class of all $ S$-pregroups is not finitely axiomatizable


Author: Deko V. Dekov
Journal: Proc. Amer. Math. Soc. 115 (1992), 895-897
MSC: Primary 20A05
MathSciNet review: 1087461
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Abstract: In order to investigate the amalgamated free products of groups, in 1950 R. Baer (Free sums of groups and their generalizations. II, Amer. J. Math. 72 (1950), 625-646) introduced the concept of an $ S$-pregroup and gave an infinite set of elementary (i.e., of a first-order language) axioms for $ S$-pregroups. The term "$ S$-pregroup" was introduced by J. R. Stallings (Adian groups and pregroups, Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, New York, 1987, pp. 321-342), who suggested the problem of finding a finite set of elementary axioms for $ S$-pregroups (ibid, Question 5, The first part, p. 340). In the present paper we show that the class of all $ S$-pregroups is not finitely axiomatizable, i.e., it cannot be characterized by any finite set of elementary axioms.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1087461-8
PII: S 0002-9939(1992)1087461-8
Article copyright: © Copyright 1992 American Mathematical Society