(FUNDAMENTAL AND APPLIED MATHEMATICS)

2004, VOLUME 10, NUMBER 3, PAGES 181-197

## Problems in algebra inspired by universal algebraic geometry

B. I. Plotkin

Abstract

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Let Q be a variety of algebras. In every variety Q and every algebra $H$ from Q one can consider algebraic geometry in Q over $H$. We also consider a special categorical invariant $K$Q(H) of this geometry. The classical algebraic geometry deals with the variety Q = 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 $Grp$ and $Grp-G$, where $G$ is a group of constants. The case $Grp-F$, 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 Q that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) $K$Q(H1) and $K$Q(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Q0 be the category of all algebras $W=W\left(X\right)$ free in Q, where $X$ is finite. Consider the groups of automorphisms $Aut\left($Q0) for different varieties Q and also the groups of autoequivalences of Q0. The problem is to describe these groups for different Q.

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