Geodesic
Quasivarieties of nilpotent groups of axiomatic rank~$4$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2131-2141.

Voir la notice de l'article provenant de la source Math-Net.Ru

We say that the axiomatic rank of a quasivariety $K$ is equal to $n$ if $K$ can be defined by a system of quasi-identities in $n$ variables and cannot be defined by any set of quasi-identities in fewer variables. If there is no such $n$, then $K$ has an infinite axiomatic rank. We prove that the set of quasivarieties of nilpotent torsion-free groups of class at most $2$ of axiomatic rank $4$ is continual.
Keywords: nilpotent group, quasivariety, variety, axiomatic rank. 
@article{SEMR_2020_17_a42,
     author = {A. I. Budkin},
     title = {Quasivarieties of nilpotent groups of axiomatic rank~$4$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {2131--2141},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a42/}
}
TY  - JOUR
AU  - A. I. Budkin
TI  - Quasivarieties of nilpotent groups of axiomatic rank~$4$
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 2131
EP  - 2141
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a42/
LA  - en
ID  - SEMR_2020_17_a42
ER  - 
%0 Journal Article
%A A. I. Budkin
%T Quasivarieties of nilpotent groups of axiomatic rank~$4$
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 2131-2141
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a42/
%G en
%F SEMR_2020_17_a42
A. I. Budkin. Quasivarieties of nilpotent groups of axiomatic rank~$4$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2131-2141. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a42/

[1] A. Yu. Ol'shanskii, “Conditional identities in finite groups”, Sib. Math. J., 15 (1975), 1000–1003 | DOI | Zbl

[2] A.K. Rumyantsev, “Quasiidentities of finite groups”, Algebra Logic, 19 (1981), 297–311 | DOI | Zbl

[3] A.I. Budkin, “Quasi-identities in a free group”, Algebra Logic, 15 (1977), 25–33 | DOI | MR | Zbl

[4] A.I. Budkin, “A lattice of quasivarieties of nilpotent groups”, Algebra Logic, 33:1 (1994), 14–21 | DOI | MR | Zbl

[5] A.I. Budkin, “Axiomatic rank of a quasivariety with a free solvable group”, Mat. Sb., N. Ser., 112:4 (1980), 647–655 | MR | Zbl

[6] A.I. Budkin, “Quasiidentities of nilpotent groups and of groups with one defining relation”, Algebra Logic, 18:2 (1979), 127–136 | DOI | MR | Zbl

[7] V.A. Gorbunov, “The structure of the lattices of quasivarieties”, Algebra Univers., 32:4 (1994), 493–530 | DOI | MR | Zbl

[8] S.A. Shakhova, “On the cardinality of the lattice of quasivarieties of nilpotent groups”, Algebra Logic, 38:3 (1999), 202–206 | DOI | MR | Zbl

[9] S.A. Shakhova, “On a lattice of quasivarieties of nilpotent groups of class 2”, Sib. Adv. Math., 7:3 (1997), 98–125 | MR | Zbl

[10] A.I. Budkin, “Quasivarieties with torsion-free nilpotent groups”, Algebra Logic, 40:6 (2001), 353–364 | DOI | MR | Zbl

[11] E.S. Polovnikova, “On the axiomatic rank quasivarieties”, Sib. Math. J., 40:1 (1999), 143–152 | DOI | MR | Zbl

[12] A.I. Malcev, Algebraic Systems, Springer-Verlag, Berlin etc, 1973 | MR | Zbl

[13] M.I. Kargapolov, Yu.I. Merzlyakov, Fundamentals of the Theory of Groups, Springer-Verlag, New York etc, 1979 | MR | Zbl

[14] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York etc, 1990 | MR | Zbl