Geodesic
Universal equivalence of general and special linear groups over fields
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 3, pp. 73-106.

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

In this paper, we study universal equivalence of general and special linear groups over fields. We give the following criterion for this relation to hold: two groups $\mathbf G_n(K)$ and $\mathbf G_m(L)$ ($\mathbf G=\mathrm{GL}, \mathrm{SL}$, $K$ and $L$ are infinite fields) are universally equivalent if and only if $n=m$ and the fields $K$ and $L$ are universally equivalent. 
@article{FPM_2016_21_3_a4,
     author = {E. I. Bunina and G. A. Kaleeva},
     title = {Universal equivalence of general and special linear groups over fields},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {73--106},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a4/}
}
TY  - JOUR
AU  - E. I. Bunina
AU  - G. A. Kaleeva
TI  - Universal equivalence of general and special linear groups over fields
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2016
SP  - 73
EP  - 106
VL  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a4/
LA  - ru
ID  - FPM_2016_21_3_a4
ER  - 
%0 Journal Article
%A E. I. Bunina
%A G. A. Kaleeva
%T Universal equivalence of general and special linear groups over fields
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2016
%P 73-106
%V 21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a4/
%G ru
%F FPM_2016_21_3_a4
E. I. Bunina; G. A. Kaleeva. Universal equivalence of general and special linear groups over fields. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 3, pp. 73-106. http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a4/

[1] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad koltsami i telami”, UMN, 53:2 (1998), 137–138 | DOI | MR | Zbl

[2] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad polyami”, Fundament. i prikl. matem., 4:4 (1998), 1265–1278 | MR | Zbl

[3] Bunina E. I., “Elementarnaya ekvivalentnost grupp Shevalle”, UMN, 56:1 (2001), 157–158 | DOI | MR | Zbl

[4] Bunina E. I., “Avtomorfizmy grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s neobratimoi dvoikoi”, Fundament. i prikl. matem., 15:7 (2009), 47–80

[5] Bunina E. I., Primenenie metoda lokalizatsii dlya opisaniya avtomorfizmov lineinykh grupp nad kommutativnymi koltsami, , 2010 http://halgebra.math.msu.su/wiki/lib/exe/fetch.php/specialcourses:speckurs10.pdf

[6] Bunina E. I., “Elementarnaya ekvivalentnost grupp Shevalle nad lokalnymi koltsami”, Matem. sb., 201:3 (2010), 3–20 | DOI | MR | Zbl

[7] Bunina E. I., Mikhalev A. V., Pinus A. G., Elementarnaya i blizkie k nei logicheskie ekvivalentnosti klassicheskikh i universalnykh algebr, MTsNMO, M., 2015

[8] Gurevich Yu. Sh., Kokorin A. I., “Universalnaya ekvivalentnost uporyadochennykh abelevykh grupp”, Algebra i logika, 2:1 (1963), 37–39 | Zbl

[9] Ershov Yu. L., Palyutin E. A., Matematicheskaya logika, Nauka, M., 1987 | MR

[10] Maltsev A. I., “Ob elementarnykh svoistvakh lineinykh grupp”, Problemy matematiki i mekhaniki, Novosibirsk, 1961, 110–132 | Zbl

[11] O'Mira O., “Lektsii o lineinykh gruppakh”, Avtomorfizmy klassicheskikh grupp, Mir, M., 1976, 57–167 | MR

[12] Taimanov A. D., “Kharakteristiki aksiomatiziruemykh klassov modelei”, Algebra i logika, 1:4 (1962), 5–31 | Zbl

[13] Timoshenko E. I., “Ob universalno ekvivalentnykh razreshimykh gruppakh”, Algebra i logika, 39:2 (2000), 227–240 | MR | Zbl

[14] Khisamiev N. G., “Universalnaya teoriya strukturno uporyadochennykh abelevykh grupp”, Algebra i logika, 5:3 (1966), 71–76 | Zbl

[15] Beidar C. I., Mikhalev A. V., “On Mal'cev's theorem on elementary equivalence of linear groups”, Proc. of the Int. Conf. on Algebra, Dedicated to the Memory of A. I. Malcev (Novosibirsk, 1989), Contemp. Math., 131, eds. L. A. Bokut', A. I. Mal'cev, A. I. Kostrikin, Amer. Math. Soc., 1992, 29–35 | DOI | MR

[16] Eklof P. C., “Some model theory for Abelian groups”, J. Symb. Logic, 37 (1972), 335–342 | DOI | MR | Zbl

[17] Mishchenko A. A., Timoshenko E. I., “Universal equivalence of partially commutative nilpotent groups”, Sib. Math. J., 52:5 (2011), 884–891 | DOI | MR | Zbl

[18] Poroshenko E. N., Timoshenko E. I., “Universal equivalence of partially commutative metabelian Lie algebras”, J. Algebra, 384 (2013), 143–168 | DOI | MR | Zbl