论文标题
免费Metabelian团体的订单
Orders On Free Metabelian Groups
论文作者
论文摘要
$ g $上的双订单是总的双层不变订单。如果对于所有$ f \ leqslant g $ in $ s $中的所有$ f \ leqslant g $，则订单组$（g，\ leqslant）中的子集$ s $，每个元素$ h \ in G $满足$ f \ leqslant h \ leqslant h \ leqslant g $属于$ s $ s $。在本文中，我们表明，相对于任何双阶，等级2的自由metabelian组的派生子组是凸的。此外，我们研究了一个自由的Metabelian较高等级的衍生亚组的凸壳。作为一种应用，我们证明非亚伯利亚自由metabelian有限等级的双阶空间对康托尔集是同型。此外，我们表明，这些群体的双阶无法通过常规语言识别。
A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-order of non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.
