Can you please describe all the linear orders (or even just the preorders) on the set $\mathbb{N}^k \setminus \{0 \}$ (with $k>0$) so that for each $a,b \in \mathbb{N}^k \setminus \{ 0 \}$ we have $$\min \{ a,b \} \le a+b \le \max \{ a,b \}$$ where the sum is taken component-wise? In the last pages of https://www.researchgate.net/publication/43796017_On_finest_and_modular_t-stabilities the authors somehow give a vague solution using 'continuous families of irrational hyperplanes', but I don't understand it very well. Thank you very much.
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2$\begingroup$ Note that if $\leq$ is such an order, then $a=\min\{a,a\}\leq a+a=2a\leq \max\{a,a\}=a$, $\forall a\in\mathbb{N}^k\setminus \{0\}$. $\endgroup$– Liviu NicolaescuCommented May 9, 2017 at 14:32
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$\begingroup$ Expanding Liviu Nicolaescu’s argument, we have $a\le na\le a$ for all positive integers $n$. $\endgroup$– Emil JeřábekCommented May 9, 2017 at 14:56
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