Algebras with the Same (Algebraic) Geometry

B. I. Plotkin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 176-207.

Résumé

Some basic notions of classical algebraic geometry can be defined on arbitrary varieties of algebras $\Theta$. For every algebra $H$ in $\Theta$, one can consider algebraic geometry in $\Theta$ over $H$. Correspondingly, algebras in $\Theta$ are considered with the emphasis on equations and geometry. We give examples of geometric properties of algebras in $\Theta$ and of geometric relations between them. The main problem considered in the paper is when different $H_1$ and $H_2$ have the same geometry.

Bibliographie

[1] Ananin A. Z., “Predstavimye mnogoobraziya algebr”, Algebra i logika, 28:2 (1989), 127–143 | MR

[2] Baumslag G., Myasnikov A., Remeslennikov V., “Algebraic geometry over groups. I: Algebraic sets and ideal theory”, J. Algebra, 219:1 (1999), 16–79 | DOI | MR | Zbl

[3] Baumslag G., Myasnikov A., Remeslennikov V., “Algebraic geometry over groups”, Algorithmic problems in groups and semigroups, Birkhäuser, Boston, 1999, 35–51 | MR

[4] Baumslag G., Myasnikov A., Roman'kov V., “Two theorems about equationally Noetherian groups”, J. Algebra, 194:2 (1997), 654–664 | DOI | MR | Zbl

[5] Belov A., Plotkin B., Abstract and natural Zariski topology, Preprint, Hebrew Univ., Jerusalem, 2003 | MR

[6] Berzins A., “Geometric equivalence of algebras”, Intern. J. Alg. and Comput., 11:4 (2001), 447–456 | DOI | MR | Zbl

[7] Berzins A., Plotkin B., Plotkin E., “Algebraic geometry in varieties of algebras with the given algebra of constants”, J. Math. Sci., 102:3 (2000), 4039–4070 | DOI | MR | Zbl

[8] Bludov V., Gusev D., “Geometricheskaya ekvivalentnost grupp”, Zap. Irkut. un-ta, 2002 | MR

[9] Cohn P. M., Free rings and their relation, Acad. Press, London, 1985 | MR | Zbl

[10] Dyer J., Formanek E., “The automorphism group of a free group is complete”, J. London Math. Soc. Ser. 2, 11:2 (1975), 181–190 | DOI | MR | Zbl

[11] Formanek E., “A question of B. Plotkin about semigroup of endomorphisms of a free group”, Proc. Amer. Math. Soc., 130 (2002), 935–937 | DOI | MR | Zbl

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

[13] Gorbunov V. A., Algebraicheskaya teoriya kvazimnogoobrazii, Dis. $\dots$ d-ra fiz.-mat. nauk, Novosibirsk, 1996; Gorbunov V. A., Algebraic theory of quasivarieties, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1998 | MR

[14] Gratzer G., Lakser H., “A note on implicational class generated by a class of structures”, Canad. Math. Bull., 16:4 (1974), 603–605 | MR

[15] Guba V. S., “Ekvivalentnost beskonechnykh sistem uravnenii v svobodnykh gruppakh i polugruppakh konechnym podsistemam”, Mat. zametki, 40:3 (1986), 321–324 | MR | Zbl

[16] Grigorchuk R. I., Kurchanov P. F., “Nekotorye voprosy teorii grupp, svyazannye s geometriei”, Algebra – 7, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 58, VINITI, M., 1990, 191–256 ; Grigorchuk R. I., Kurchanov P. F., “Some questions in group theory related to geometry”, Combinatorial group theory and applications to geometry, Springer, Berlin, 1998, 169–232 | MR | MR

[17] Kharlampovich O., Myasnikov A., “Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz”, J. Algebra, 200:2 (1998), 472–516 | DOI | MR | Zbl

[18] Kharlampovich O., Myasnikov A., “Irreducible affine varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups”, J. Algebra, 200:2 (1998), 517–570 | DOI | MR | Zbl

[19] Lyndon R. C., Schupp P. E., Combinatorial group theory, Springer, Berlin, Heidelberg, New York, 1977 | MR | Zbl

[20] Lichtman A., Passman D., “Finitely generated simple algebras: A question of B. I. Plotkin”, Proc. Amer. Math. Soc. (to appear)

[21] Malcev A. I., Algebraic systems, North-Holland, Amsterdam, 1973 | MR

[22] Maltsev A. I., “Neskolko zamechanii o kvazimnogoobraziyakh algebraicheskikh sistem”, Algebra i logika, 5:3 (1966), 3–9 | MR

[23] Mashevitzky G., Plotkin B., Plotkin E., “Automorphisms of categories of free algebras of varieties”, El. Res. Announc. AMS, 8 (2002), 1–10 | DOI | MR | Zbl

[24] Mashevitzky G., Plotkin B., Plotkin E., “Automorphisms of categories of free Lie algebras”, J. Algebra (to appear)

[25] MacLane S., Categories for the working mathematician, Springer, Berlin, 1971 | MR | Zbl

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

[27] Nikolova D., Plotkin B., “Some notes on universal algebraic geometry”, Algebra, Proc. Intern. Conf. on the Occasion of the 90th Birthday of A. G. Kurosh (Moscow, 1998), W. de Gruyter, Berlin, 1999, 237–261 | MR

[28] Plotkin B., “Algebraic logic, varieties of algebras and algebraic varieties”, Semigroups with applications, including semigroup rings, Proc. Intern. Conf. in honour of E. S. Lyapin (St. Petersburg, 1995), Severny Ochag, St. Petersburg, 1999, 189–271 | MR

[29] Plotkin B., “Varieties of algebras and algebraic varieties”, Israel J. Math., 96:2 (1996), 511–522 | DOI | MR | Zbl

[30] Plotkin B., “Varieties of algebras and algebraic varieties. Categories of algebraic varieties”, Siberian Adv. Math., 7:2 (1997), 64–97 | MR | Zbl

[31] Plotkin B. I., “Nekotorye ponyatiya algebraicheskoi geometrii v universalnoi algebre”, Algebra i analiz, 9:4 (1997), 224–248 | MR | Zbl

[32] Plotkin B., Seven lectures on the universal algebraic geometry, , 2002, 87 pp. http://arxiv.org/ abs/math.GM/0204245

[33] Plotkin B., “Infinitary quasi-identities and infinitary quasivarieties”, Proc. Latv. Acad. Sci. B., 56 (2002) | MR

[34] Plotkin B., Plotkin E., Tsurkov A., “Geometrical equivalence of groups”, Commun. Algebra, 27:8 (1999), 4015–4025 | DOI | MR | Zbl

[35] Rips E., Sela Z., “Cyclic splittings of the finitely presented groups and the canonical JSJ decomposition”, Ann. Math., 146:1 (1997), 53–109 | DOI | MR

[36] Sela Z., “Diophantine geometry over groups, I”, Publ. Math. IHES, 2001, no. 93, 31–105 | DOI | MR | Zbl

[37] Tsurkov A., Geometric equivalence of nilpotent groups, Preprint, Hebrew Univ., Jerusalem, 2003 | MR

[38] Tsalenko M. Sh., Shulgeifer E. G., Osnovy teorii kategorii, Nauka, M., 1974 | MR | Zbl

[39] Zhitomirskii G., “Autoequivalences of categories of free algebras of varieties”, Algebra Universalis (to appear)