On identities in groups of fractions of cancellative semigroups

Ivanov, S.; Storozhev, A.

doi:10.1090/S0002-9939-05-07903-7

On identities in groups of fractions of cancellative semigroups
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by S. V. Ivanov and A. M. Storozhev PDF

Proc. Amer. Math. Soc. 133 (2005), 1873-1879 Request permission

Abstract:

To solve two problems of Bergman stated in 1981, we construct a group $G$ such that $G$ contains a free noncyclic subgroup (hence, $G$ satisfies no group identity) and $G$, as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity.

References

George M. Bergman, Hyperidentities of groups and semigroups, Aequationes Math. 23 (1981), no. 1, 50–65. MR 667217, DOI 10.1007/BF02188011
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR 0218472
S.V. Ivanov and A.M. Storozhev, Non-hopfian relatively free groups, submitted, preprint available at http://arXiv.org.
Jan Krempa and Olga Macedońska, On identities of cancellative semigroups, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 125–133. MR 1175878, DOI 10.1080/00927872.2013.833212
P. Longobardi, M. Maj, and A. H. Rhemtulla, Groups with no free subsemigroups, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1419–1427. MR 1277124, DOI 10.1090/S0002-9947-1995-1277124-5
A.I. Mal’tsev, On the immersion of an algebraic ring into a field, Math. Ann. 113(1937), 686–691.
A. I. Mal′cev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki 4 (1953), 107–111 (Russian). MR 0075959
B. H. Neumann and Tekla Taylor, Subsemigroups of nilpotent groups, Proc. Roy. Soc. London Ser. A 274 (1963), 1–4. MR 159884, DOI 10.1098/rspa.1963.0110
A. Yu. Ol′shanskiĭ, Varieties in which all finite groups are abelian, Mat. Sb. (N.S.) 126(168) (1985), no. 1, 59–82, 143 (Russian). MR 773429
A. Yu. Ol′shanskiĭ, Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR 1191619, DOI 10.1007/978-94-011-3618-1
A. Yu. Ol′shanskii and A. Storozhev, A group variety defined by a semigroup law, J. Austral. Math. Soc. Ser. A 60 (1996), no. 2, 255–259. MR 1375590, DOI 10.1017/S1446788700037642
Joseph Max Rosenblatt, Invariant measures and growth conditions, Trans. Amer. Math. Soc. 193 (1974), 33–53. MR 342955, DOI 10.1090/S0002-9947-1974-0342955-9
L. N. Shevrin and E. V. Sukhanov, Structural aspects of the theory of varieties of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 6 (1989), 3–39 (Russian); English transl., Soviet Math. (Iz. VUZ) 33 (1989), no. 6, 1–34. MR 1017775
Andrei Storozhev, On abelian subgroups of relatively free groups, Comm. Algebra 22 (1994), no. 7, 2677–2701. MR 1271632, DOI 10.1080/00927879408824986