THE CLASS OF ALL S-PREGROUPS IS NOT FINITELY AXIOMATIZABLE

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.