The class of all 𝑆-pregroups is not finitely axiomatizable

Dekov, Deko V.

doi:10.1090/S0002-9939-1992-1087461-8

The class of all $S$-pregroups is not finitely axiomatizable
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Proc. Amer. Math. Soc. 115 (1992), 895-897 Request permission

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.

References

Reinhold Baer, Free sums of groups and their generalizations. II, Amer. J. Math. 72 (1950), 625–646. MR 38974, DOI 10.2307/2372280
Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132
Gérard Lallement, Semigroups and combinatorial applications, Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 530552
Frank Rimlinger, Pregroups and Bass-Serre theory, Mem. Amer. Math. Soc. 65 (1987), no. 361, viii+73. MR 874086, DOI 10.1090/memo/0361
John R. Stallings, Adian groups and pregroups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 321–342. MR 919831, DOI 10.1007/978-1-4613-9586-7_{5}