# zbMATH — the first resource for mathematics

On a question of Frege’s about right-ordered groups. (English) Zbl 0632.06022
The authors study groups G with a right-invariant partial ordering $$<$$ under various additional assumptions on $$<$$. A partial ordering $$<$$ of a set S is called an upper semilinear ordering, if (1) for any $$x\in S$$ the set $$\{$$ $$s\in S:$$ $$x<s\}$$ is linearly ordered, and (2) for any two incomparable elements a,b$$\in S$$ there is $$c\in S$$ with $$a<c$$ and $$b<c$$. The main results are: Theorem 2.1. Let G be a free group of rank 2. Then there is a right-invariant, dense, upper semilinear ordering $$<$$ on G which is left-invariant under multiplication with positive elements of G and not a linear ordering. Theorem 3.1. Let G be a group with a right- invariant, archimedean, upper semilinear ordering $$<$$. Then G is abelian and $$<$$ is linear. - Theorem 2.1 answers, in an amplified form, an old interesting question of G. Frege [Die Grundgesetze der Arithmetik, Band II (Jena 1903, reprinted: G. Olms, Hildesheim, 1966).
Reviewer: M.Droste

##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects)
Full Text: