Problems in algebra inspired by universal algebraic geometry

B. I. Plotkin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 3, pp. 181-197.

Résumé

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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