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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

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
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