问题及解答

记 $\mathbb{Q}^+$ 为所有正有理数构成的集合, 它在乘法下构成一个群. 

Glass 和 Ribenboim [1]证明了 $(\mathbb{Q}^+,\cdot)$ 的保序自同构是平凡自同构.

Theorem.  The only automorphism of the ordered multiplicative group of positive rational numbers is the trivial automorphism.

References:

[1]  A. M. W. Glass and Paulo Ribenboim, Automorphisms of the ordered multiplicative group of positive rational numbers, Proceedings of the American Mathematical Society, (1994) Vol.122, No. 1, 15--18.