Nullstellensatz for relative existentially closed groups

Document Type : Research Paper


Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman


We prove that in every variety of $G$-groups‎, ‎every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations‎. ‎This will generalize  Theorem G of [J‎. ‎Algebra,  219 (1999) ‎16--79]‎. ‎As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{\omega}$-compact‎, ‎they are geometrically equivalent‎.


Main Subjects

[1] G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov, Algebraic geometry over groups: I. Algebraic sets and
ideal theory, J. Algebra, 219 (1999) 16–79.
[2] E. Yu. Daniyarova, A. Myasnikov and V. Remeslenniko,. Algebraic geometry over algebraic structures, II: Funda-
tions, J. Math. Sci. 185 no. 3 (2012) 389–416.
[3] O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. I: irreducibility of quadratic
equations and nullstellensatz, J. Algebra, 200 (1998) 472–516.
[4] O. Kharlampovich and A. Myasnikov, Tarski’s problem about the elementary theory of free groups has a psitive
solution, Electron. Res. Announc. Amer. Math. Soc., 4 (1998) 101–108.
[5] A. Myasnikov and V. Remeslennikov, Algebraic geometry over groups: II. Logical Fundations, J. Algebra, 234
(2000) 225–276.
[6] M. Shahryari, Existentially closed structures and some emebedding theorems, Math. Notes, 101 (2017) 1023–1032.