Geodesic
On~filters in the lattice of quasivarieties of groups
Izvestiya. Mathematics , Tome 33 (1989) no. 1, pp. 201-207.

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

Suppose $\mathfrak M$ is a quasivariety of groups. Assume there exist finitely presented groups $A$ and $B$ such that $A\notin\mathfrak M$, $B\in\mathfrak M$, and $B$ is not contained in the quasivariety generated by $A$. It is proved that the principal filter generated in the lattice of quasivarieties of groups by has the cardinality of the continuum. In particular, the principal filters generated by a) the quasivariety generated by all proper varieties of groups, b) the quasivariety of all $RN$-groups and c) the quasivariety generated by all periodic groups, have the cardinality of the continuum. It is shown that the smallest quasivariety of groups containing all proper quasivarieties of groups in which nontrivial quasi-identities of the form $$ (\forall\,x_1)\dots(\forall\,x_n)(f(x_1,\dots,x_n)=1\to g(x_1,\dots,x_n)=1), $$ are true, where $f$ and $g$ are terms of group signature, does not coincide with the class of all groups. Bibliography: 12 titles. 
@article{IM2_1989_33_1_a8,
     author = {A. I. Budkin},
     title = {On~filters in the lattice of quasivarieties of groups},
     journal = {Izvestiya. Mathematics },
     pages = {201--207},
     publisher = {mathdoc},
     volume = {33},
     number = {1},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1989_33_1_a8/}
}
TY  - JOUR
AU  - A. I. Budkin
TI  - On~filters in the lattice of quasivarieties of groups
JO  - Izvestiya. Mathematics 
PY  - 1989
SP  - 201
EP  - 207
VL  - 33
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1989_33_1_a8/
LA  - en
ID  - IM2_1989_33_1_a8
ER  - 
%0 Journal Article
%A A. I. Budkin
%T On~filters in the lattice of quasivarieties of groups
%J Izvestiya. Mathematics 
%D 1989
%P 201-207
%V 33
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1989_33_1_a8/
%G en
%F IM2_1989_33_1_a8
A. I. Budkin. On~filters in the lattice of quasivarieties of groups. Izvestiya. Mathematics , Tome 33 (1989) no. 1, pp. 201-207. http://geodesic.mathdoc.fr/item/IM2_1989_33_1_a8/

[1] Smirnov D. M., “Varieties and quasivarieties of algebra”, Universal algebra (Esztergom, Hungary, 1977), Colloquia Math. Soc. Janes Bolyai, 29, North-Holland, Amsterdam, 1982, 745–751 | MR

[2] Belkin V. P., Gorbunov V. A., “Filtry reshetok kvazimnogoobrazii algebraicheskikh sistem”, Algebra i logika, 14 (1975), 373–392 | MR | Zbl

[3] Maltsev A. I., “Universalno-aksimatiziruemye podklassy lokalno-konechnykh klassov modelei”, Sib. matem. zhurn., 8 (1967), 1005–1014

[4] Budkin A. I., Gorbunov V. A., “K teorii kvazimnogoobrazii algebraicheskikh sistem”, Algebra i logika, 14 (1975), 123–142 | MR | Zbl

[5] Maltsev A. I., Algebraicheskie sistemy, Nauka, M., 1970 | MR

[6] Magnus V., Karras A., Soliter D., Kombinatornaya teoriya grupp, Nauka, M., 1974 | MR | Zbl

[7] Neumann P. M., “On the structure of standard wreath products of groups”, Math. Z., 84 (1964), 343–373 | DOI | MR | Zbl

[8] Higman G., Finitely presented infinite simple groups, Notes on Pure Mathematics, 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, 82 | MR

[9] Abels H., “An example of a finitely presented soluble group”, London. Math. Soc. Lect. Note Ser., 36, 1979, 205–211 | MR | Zbl

[10] Kargapolov M. I., Merzlyakov Yu. Ya., Osnovy teorii grupp, 3-e izd., Nauka, M., 1982 | MR | Zbl

[11] Rumyantsev A. K., “O kvazitozhdestvakh konechnykh grupp”, Algebra i logika, 19 (1980), 458–479 | MR | Zbl

[12] Boone W. W., Higman G., “On algebraic characterization of group with soluble word problem”, J. Austral. Math. Soc., 18 (1974), 41–53 | DOI | MR | Zbl