Divisibility on certain o-groups of matrices

Abstract

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.