We construct non-abelian totally ordered groups of matrices of finite Archimedean rank using the group of o-automorphisms of direct sums of copies of the reals ordered anti-lexicographically. We also prove that each of these o-groups is divisible, and provide, for every \(n>2\), a specific formula to find the n-th root of every element of such group. Finally, we construct an example of a non-commutative totally ordered ring.

中文翻译：

---

某些 o 组矩阵的可分性

我们使用按反字典顺序排序的实数副本的直接和的 o-自同构群来构造有限阿基米德秩矩阵的非阿贝尔全序群。我们还证明了这些 o 群中的每一个都是可分的，并为每个\(n>2\)提供了一个特定的公式来找到该群的每个元素的n次根。最后，我们构造了一个非交换全序环的例子。