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Nullstellensatz for relative existentially closed groups | ||
| International Journal of Group Theory | ||
| دوره 11، شماره 2، شهریور 2022، صفحه 125-130 اصل مقاله (180.88 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2021.125453.1652 | ||
| نویسنده | ||
| Mohammad Shahryari* | ||
| Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Algebraic geometry over groups؛ Nullstellensatz؛ Existentially closed groups؛ Varieties of groups؛ Quasi-varieties | ||
| مراجع | ||
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[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. | ||
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