Problems in algebra inspired by universal algebraic geometry

Abstract

Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We consider also a special categorical invariant $K_\Theta (H)$ of this geometry. The classical algebraic geometry deals with the variety $\Theta=Com-P$ 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 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. he 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. We start with the short overview of main definitions and results and then consider the list of unsolved problems.