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

This article is cited in 10 scientific papers (total in 10 papers)

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
\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
\jour J. Math. Sci.
\yr 2006
\vol 139
\issue 4
\pages 6780--6791

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    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  crossref  mathscinet  zmath  isi
    2. Plotkin, B, “Some results and problems related to universal algebraic geometry”, International Journal of Algebra and Computation, 17:5–6 (2007), 1133  crossref  mathscinet  zmath  isi
    3. A. G. Pinus, “Geometric scales for varieties of algebras and quasi-identities”, Siberian Adv. Math., 20:3 (2010), 217–222  mathnet  crossref  mathscinet
    4. A. G. Pinus, “On the Geometrically Complete Varieties of Algebras”, J. Math. Sci., 205:3 (2015), 440–444  mathnet  crossref
    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  mathnet  crossref  crossref  isi
    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  mathnet  crossref  crossref  isi  elib  elib
    7. Shahryari M. Shevlyakov A., “Direct Products, Varieties, and Compactness Conditions”, Groups Complex. Cryptol., 9:2 (2017), 159–166  crossref  mathscinet  zmath  isi  scopus
    8. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraicheskaya geometriya nad algebraicheskimi sistemami. VIII. Geometricheskie ekvivalentnosti i osobye klassy algebraicheskikh sistem”, Fundament. i prikl. matem., 22:4 (2019), 75–100  mathnet
    9. A. G. Pinus, “O predstavlenii reshetok algebraicheskikh mnozhestv universalnykh algebr”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 29 (2019), 98–106  mathnet  crossref
    10. A. G. Pinus, “Algebraicheskie mnozhestva shirokikh algebr”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 32 (2020), 94–100  mathnet  crossref
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