On the variety generated by all nilpotent lattice-ordered groups.

*(English)*Zbl 1110.06019The variety of weakly abelian lattice-ordered groups was introduced in 1974 by J. Martinez. It is defined by the identity: \(x^{-1}(y\vee 1)x\vee (y\vee 1)^{2}=(y\vee 1)^{2}\). The present paper deals with the variety generated by all nilpotent lattice-ordered groups. Its main results are the following:

Theorem A. There is a centre-by-metabelian weakly Abelian ordered group that does not belong to the variety of lattice-ordered groups generated by all nilpotent lattice-ordered groups. (Note that this result answers two question of V. M. Kopytov.)

Theorem B. The quasivariety generated by all nilpotent lattice-ordered groups is the same as the variety generated by all nilpotent lattice-ordered groups. (Note that the proof of Theorem B also gives a set of defining identities for this variety.)

Theorem C. Every abelian-by-nilpotent weakly abelian lattice-ordered group belongs to the variety of lattice-ordered groups generated by all nilpotent lattice-ordered groups.

All results of the paper are new and carefully proved. The presentation is clear, with many examples, and so the paper contributes to the development of this important research domain.

Theorem A. There is a centre-by-metabelian weakly Abelian ordered group that does not belong to the variety of lattice-ordered groups generated by all nilpotent lattice-ordered groups. (Note that this result answers two question of V. M. Kopytov.)

Theorem B. The quasivariety generated by all nilpotent lattice-ordered groups is the same as the variety generated by all nilpotent lattice-ordered groups. (Note that the proof of Theorem B also gives a set of defining identities for this variety.)

Theorem C. Every abelian-by-nilpotent weakly abelian lattice-ordered group belongs to the variety of lattice-ordered groups generated by all nilpotent lattice-ordered groups.

All results of the paper are new and carefully proved. The presentation is clear, with many examples, and so the paper contributes to the development of this important research domain.

Reviewer: Marius Tarnauceanu (Iaşi)

##### Keywords:

nilpotent group; residually torsion-free-nilpotent; variety; quasi-variety; commutator calculus; lattice-ordered group; weakly abelian
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\textit{V. V. Bludov} and \textit{A. M. W. Glass}, Trans. Am. Math. Soc. 358, No. 12, 5179--5192 (2006; Zbl 1110.06019)

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##### References:

[1] | V. V. Bludov, On locally nilpotent groups [translation of Trudy Instituta Matematiki, Vol. 30, 26 – 47, Izdat. Ross. Akad. Nauk, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1996], Siberian Adv. Math. 8 (1998), no. 1, 49 – 79. · Zbl 0912.20027 |

[2] | V. V. Bludov, A. M. W. Glass, and Akbar H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003), no. 2, 225 – 239. · Zbl 1031.20036 · doi:10.4134/JKMS.2003.40.2.225 · doi.org |

[3] | V. V. Bludov, A. M. W. Glass, and A. H. Rhemtulla, On centrally orderable groups, J. Algebra 291 (2005), no. 1, 129 – 143. · Zbl 1083.06014 · doi:10.1016/j.jalgebra.2005.05.014 · doi.org |

[4] | A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. · Zbl 0933.06010 |

[5] | A. M. W. Glass, Weakly abelian lattice-ordered groups, Proc. Amer. Math. Soc. 129 (2001), no. 3, 677 – 684. · Zbl 0963.06017 |

[6] | S. A. Gurchenkov, About varieties of weakly abelian \?-groups, Math. Slovaca 42 (1992), no. 4, 437 – 441. · Zbl 0769.06009 |

[7] | P. Hall, Nilpotent Groups, Lectures given at the Canadian Mathematical Congress, University of Alberta, 1957. |

[8] | V. M. Kopytov, Lattice-ordered locally nilpotent groups, Algebra i Logika 14 (1975), no. 4, 407 – 413 (Russian). |

[9] | V. M. Kopytov and N. Ya. Medvedev, The theory of lattice-ordered groups, Mathematics and its Applications, vol. 307, Kluwer Academic Publishers Group, Dordrecht, 1994. · Zbl 0834.06015 |

[10] | V. D. Mazurov and E. I. Khukhro , Unsolved problems in group theory. The Kourovka notebook, Thirteenth augmented edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1995. · Zbl 0838.20001 |

[11] | Jorge Martinez, Varieties of lattice-ordered groups, Math. Z. 137 (1974), 265 – 284. · Zbl 0274.20034 · doi:10.1007/BF01214370 · doi.org |

[12] | Roberta Botto Mura and Akbar Rhemtulla, Orderable groups, Marcel Dekker, Inc., New York-Basel, 1977. Lecture Notes in Pure and Applied Mathematics, Vol. 27. · Zbl 0358.06038 |

[13] | Norman R. Reilly, Nilpotent, weakly abelian and Hamiltonian lattice ordered groups, Czechoslovak Math. J. 33(108) (1983), no. 3, 348 – 353. · Zbl 0553.06020 |

[14] | Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. · Zbl 0483.20001 |

[15] | Robert B. Warfield Jr., Nilpotent groups, Lecture Notes in Mathematics, Vol. 513, Springer-Verlag, Berlin-New York, 1976. · Zbl 0347.20018 |

[16] | The Black Swamp Problem Book is edited by W. Charles Holland (Bowling Green State University, Ohio 43403, U.S.A.) and is kept there by him (in the formerly Black Swamp region of Ohio). |

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