Geodesic
Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 4, pp. 75-100.

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

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometrical, universal geometrical, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, $\mathrm{q}_\omega$-compact, $\mathrm{u}_\omega$-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class $\mathbf K$, which do not? (2) With respect to which equivalences a given class $\mathbf K$ is invariant, with respect to which it is not? 
@article{FPM_2019_22_4_a5,
     author = {E. Yu. Daniyarova and A. G. Myasnikov and V. N. Remeslennikov},
     title = {Algebraic geometry over algebraic structures. {VIII.} {Geometric} equivalences and special classes of algebraic structures},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {75--100},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2019_22_4_a5/}
}
TY  - JOUR
AU  - E. Yu. Daniyarova
AU  - A. G. Myasnikov
AU  - V. N. Remeslennikov
TI  - Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2019
SP  - 75
EP  - 100
VL  - 22
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2019_22_4_a5/
LA  - ru
ID  - FPM_2019_22_4_a5
ER  - 
%0 Journal Article
%A E. Yu. Daniyarova
%A A. G. Myasnikov
%A V. N. Remeslennikov
%T Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2019
%P 75-100
%V 22
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2019_22_4_a5/
%G ru
%F FPM_2019_22_4_a5
E. Yu. Daniyarova; A. G. Myasnikov; V. N. Remeslennikov. Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 4, pp. 75-100. http://geodesic.mathdoc.fr/item/FPM_2019_22_4_a5/

[1] Gorbunov V. A., Algebraicheskaya teoriya kvazimnogoobrazii, Nauch. kniga, Novosibirsk, 1999

[2] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Algebraicheskaya geometriya nad algebraicheskimi sistemami. IV. Ekvatsionalnye oblasti i kooblasti”, Algebra i logika, 49:6 (2010), 715–756 | MR

[3] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Universalnaya algebraicheskaya geometriya”, Dokl. RAN, 439:6 (2011), 730–732 | MR | Zbl

[4] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Algebraicheskaya geometriya nad algebraicheskimi sistemami. II. Osnovaniya”, Fundament. i prikl. matem., 17:1 (2011/2012), 65–106 | MR

[5] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Algebraicheskaya geometriya nad algebraicheskimi sistemami. V. Sluchai proizvolnoi signatury”, Algebra i logika, 51:1 (2012), 41–60 | MR | Zbl

[6] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Razmernost v universalnoi algebraicheskoi geometrii”, Dokl. RAN, 457:3 (2014), 265–267 | MR | Zbl

[7] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Algebraicheskaya geometriya nad algebraicheskimi sistemami. VI. Geometricheskaya ekvivalentnost”, Algebra i logika, 56:4 (2017), 421–442 | MR | Zbl

[8] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Universalnaya geometricheskaya ekvivalentnost algebraicheskikh sistem odnoi signatury”, Sib. matem. zhurn., 58:5 (2017), 1035–1050 | MR | Zbl

[9] Plotkin B. I., “Problemy algebry, inspirirovannye universalnoi algebraicheskoi geometriei”, Fundament. i prikl. matem., 10:3 (2004), 181–197 | MR | Zbl

[10] Shevlyakov A. N., “Ob elementarnoi i geometricheskoi ekvivalentnosti ekvatsionalnykh kooblastei”, Fundament. i prikl. matem. (to appear) | MR

[11] Berzins A., “Geometrical equivalence of algebras”, Internat. J. Algebra Comput., 11:4 (2001), 447–456 | MR | Zbl

[12] Daniyarova E., Myasnikov A., Remeslennikov V., “Unification theorems in algebraic geometry”, Algebra Discrete Math., 1 (2008), 80–112 | MR

[13] Daniyarova E., Myasnikov A., Remeslennikov V., “Algebraic geometry over algebraic structures. III. Equationally Noetherian property and compactness”, Southeast Asian Bull. Math., 35:1 (2011), 35–68 | MR | Zbl

[14] Göbel R., Shelah S., “Radicals and Plotkin's problem concerning geometrically equivalent groups”, Proc. Amer. Math. Soc., 130:3 (2002), 673–674 | MR | Zbl

[15] Grätzer G., Lakser H., “A note on implicational class generated by class of structures”, Can. Math. Bull., 16:4 (1973), 603–605 | MR

[16] Myasnikov A., Remeslennikov V., “Algebraic geometry over groups. II. Logical foundations”, J. Algebra, 234:1 (2000), 225–276 | MR | Zbl

[17] Plotkin B. I., Seven lectures on the universal algebraic geometry, 2002, arXiv: math/0204245 [math.GM] | MR

[18] Plotkin B., “Algebras with the same (algebraic) geometry”, Proc. Steklov Inst. Math., 242 (2003), 165–196 | MR | Zbl

[19] Plotkin B. I., “Geometrical equivalence, geometrical similarity, and geometrical compatibility of algebras”, J. Math. Sci., 140:5 (2007), 716–728 | MR

[20] Plotkin B., Tsurkov A., “Action type geometrical equivalence of representations of groups”, Algebra Discrete Math., 4 (2005), 48–79 | MR | Zbl

[21] Plotkin B., Zhitomirski G., “Some logical invariants of algebras and logical relations between algebras”, Algebra i analiz, 19:5 (2007), 214–245 | MR

[22] Shevlyakov A. N., “Commutative idempotent semigroups at the service of universal algebraic geometry”, Southeast Asian Bull. Math., 35:1 (2011), 111–136 | MR | Zbl