Abstract
We describe the general approach to a nonstandard geometry with the emphasis on associative algebras.
Similar content being viewed by others
REFERENCES
B. Plotkin, “Varieties of algebras and algebraic varieties. Categories of algebraic varieties,” Sib. Adv. Math., 7, No. 2, 64–97 (1997).
B. Plotkin, “Varieties of algebras and algebraic varieties,” Isr. J. Math., 96, No. 2, 511–522 (1996).
B. Plotkin, Seven Lectures in Universal Algebraic Geometry, Preprint, Hebrew University, Jerusalem (2000).
B. Plotkin, Algebras with the Same Algebraic Geometry, Preprint, Hebrew University, Jerusalem (2002).
S. MacLane, Categories for the Working Mathematicians, Springer, New York (1971).
A. Berzins, B. Plotkin, and E. Plotkin, “Algebraic geometry in varieties of algebras with the given algebra of constants,” J. Math. Sci., 102, No. 3, 4039–4070 (2000).
B. Plotkin, E. Plotkin, and A. Tsurkov, “Geometrical equivalence in groups,” Commun. Algebra, 27, 4015–4025 (1999).
A. I. Malcev, “Some remarks on quasi-varieties of algebraic structures,” Algebra Logic, 5, No. 3, 3–9 (1966).
A. I. Malcev, Algebraic Structures [in Russian], Nauka, Moscow (1987).
V. A. Gorbunov, Algebraic Theory of Quasi-Varieties, Doctor Thesis, Novosibirsk (1996).
A. Myasnikov and V. Remeslennikov, “Algebraic geometry over groups II. Logical foundations,” J. Algebra, 234, 225–276 (2000).
G. Baumslag, A. Myasnikov, and V. N. Remeslennikov, “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, 219, 16–79 (1999).
G. Baumslag, A. Myasnikov, and V. Romankov, “Two theorems about equationally Noetherian groups,” J. Algebra, 194, 654–664 (1997).
G. Gratzer and H. Lakser, “A note on implicational class generated by a class of structures,” Can. Math. Bull., 16, No. 4, 603–605 (1974).
R. Gobel and S. Shelah, “Radicals and Plotkin's problem concerning geometrically equivalent groups,” Proc. Amer. Math. Soc., 130, No. 3, 673–674 (2002).
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, New York (1977).
V. Guba, “Equivalence of infinite systems of equations in free groups and semigroups to finite systems,” Mat. Zametki, 40, No. 3, 321–324 (1986).
A. Mashevitzky, B. Plotkin, and E. Plotkin, “Automorphism of categories of free algebras of varieties,” ERA AMS, 8 (2002) (to appear).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Plotkin, B. Some Problems in Nonclassical Algebraic Geometry. Ukrainian Mathematical Journal 54, 1019–1026 (2002). https://doi.org/10.1023/A:1021720521917
Issue Date:
DOI: https://doi.org/10.1023/A:1021720521917