受泛algebraic geometry启发的代数问题
Problems in algebra inspired by universal algebraic geometry
摘要：
设$Θ$是一个代数簇。在每个$Θ$和每个来自$Θ$的代数$H$中，可以考虑$H$上的$Θ$中的代数几何。我们还考虑了这个几何的一个特殊的范畴不变量$K_Θ(H)$。经典代数几何研究的是基域$P$上所有结合代数和交换代数的簇$Θ=Com-P$。在此设置中，代数$H$是基域$P$的扩展。群中的几何与变体$\Grp$和$\Grp-G$有关，其中$G$是一组常数。情况$\Grp -F$其中$F$是一个自由群，是有关塔斯基的问题，致力于逻辑的一个自由群。他描述了一般的洞察力代数几何在不同品种的代数启发了一些新的问题，代数和代数几何。这类问题在很大程度上决定了泛代数几何的内容。我们从主要定义和结果的简短概述开始，然后考虑未解决的问题列表。
英文摘要：
Let $Θ$ be a variety of algebras. In every $Θ$ and every algebra $H$ from $Θ$ one can consider algebraic geometry in $Θ$ over $H$. We consider also a special categorical invariant $K_Θ(H)$ of this geometry. The classical algebraic geometry deals with the variety $Θ=Com-P$ 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.
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