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 Fundam. Prikl. Mat., 2004, Volume 10, Issue 3, Pages 181–197 (Mi fpm777)

Problems in algebra inspired by universal algebraic geometry

B. I. Plotkin

Hebrew University of Jerusalem

Abstract: Let $\Theta$ be a variety of algebras. In every variety $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We also consider a special categorical invariant $K_\Theta(H)$ of this geometry. The classical algebraic geometry deals with the variety $\Theta=\mathrm{Com-}P$ of all associative and commutative algebras over the ground field of constants $P$. An algebra $H$ in this setting is an extension of the ground field $P$. Geometry in groups is related to the varieties $\mathrm{Grp}$ and $\mathrm{Grp-}G$, where $G$ is a group of constants. The case $\mathrm{Grp-}F$, where $F$ is a free group, is related to Tarski's problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras $H_1$ and $H_2$ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety $\Theta$ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) $K_\Theta(H_1)$ and $K_\Theta(H_2)$ are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let $\Theta^0$ be the category of all algebras $W=W(X)$ free in $\Theta$, where $X$ is finite. Consider the groups of automorphisms $\operatorname{Aut}(\Theta^0)$ for different varieties $\Theta$ and also the groups of autoequivalences of $\Theta^0$. The problem is to describe these groups for different $\Theta$.

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English version:
Journal of Mathematical Sciences (New York), 2006, 139:4, 6780–6791

Bibliographic databases:

UDC: 512.7

Citation: B. I. Plotkin, “Problems in algebra inspired by universal algebraic geometry”, Fundam. Prikl. Mat., 10:3 (2004), 181–197; J. Math. Sci., 139:4 (2006), 6780–6791

Citation in format AMSBIB
\Bibitem{Plo04} \by B.~I.~Plotkin \paper Problems in algebra inspired by universal algebraic geometry \jour Fundam. Prikl. Mat. \yr 2004 \vol 10 \issue 3 \pages 181--197 \mathnet{http://mi.mathnet.ru/fpm777} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2123349} \zmath{https://zbmath.org/?q=an:1072.08002} \elib{http://elibrary.ru/item.asp?id=9068315} \transl \jour J. Math. Sci. \yr 2006 \vol 139 \issue 4 \pages 6780--6791 \crossref{https://doi.org/10.1007/s10958-006-0390-5} \elib{http://elibrary.ru/item.asp?id=14134984} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33750522806} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Katsov, Y, “On geometrically equivalent S-ACTS”, International Journal of Algebra and Computation, 17:5–6 (2007), 1055
2. Plotkin, B, “Some results and problems related to universal algebraic geometry”, International Journal of Algebra and Computation, 17:5–6 (2007), 1133
3. A. G. Pinus, “Geometric scales for varieties of algebras and quasi-identities”, Siberian Adv. Math., 20:3 (2010), 217–222
4. A. G. Pinus, “On the Geometrically Complete Varieties of Algebras”, J. Math. Sci., 205:3 (2015), 440–444
5. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometric equivalence”, Algebra and Logic, 56:4 (2017), 281–294
6. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Universal geometrical equivalence of the algebraic structures of common signature”, Siberian Math. J., 58:5 (2017), 801–812
7. Shahryari M. Shevlyakov A., “Direct Products, Varieties, and Compactness Conditions”, Groups Complex. Cryptol., 9:2 (2017), 159–166
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