an:05120557
Zbl 1110.06019
Bludov, V. V.; Glass, A. M. W.
On the variety generated by all nilpotent lattice-ordered groups
EN
Trans. Am. Math. Soc. 358, No. 12, 5179-5192 (2006).
00186356
2006
j
06F15 20F18 20F12
nilpotent group; residually torsion-free-nilpotent; variety; quasi-variety; commutator calculus; lattice-ordered group; weakly abelian
The 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.
Marius Tarnauceanu (Ia??i)