Productivity of the Zariski topology on groups

Commentationes mathematicae Universitatis Carolinae

D. Dikranjan¹ 

关键词: Zariski topology;    (elementary;    additively) algebraic subset;    $\delta$-word;    universal word;    verbal function;    (semi) $\mathfrak Z$-productive pair of groups;    direct product;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

This paper investigates the productivity of the Zariski topology $\mathfrak Z_G$ of a group $G$. If $\mathcal G = \{G_i\mid i\in I\}$ is a family of groups, and $G = \prod_{i\in I}G_i$ is their direct product, we prove that $\mathfrak Z_G\subseteq \prod_{i\in I}\mathfrak Z_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $\mathcal G = \{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., $\mathfrak Z_G = \mathfrak Z_{G_1}\times \mathfrak Z_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta$ of groups $G_2$ such that $G_1\times \{e_{2}\}$ is an elementary algebraic subset of ${G_1\times G_2}$ for every group $G_1$. We show among others, that $\Delta$ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to $\Delta$. In particular, $\Delta$ contains arbitrary direct products of free non-abelian groups.

【 授权许可】

CC BY