Pairs of orthogonal countable ordinals

Abstract

We characterize pairs of orthogonal countable ordinals. Two ordinals $\alpha$ and $\beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $\alpha$ and $\beta$ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if $\alpha$ and $\beta$ are two countable ordinals, with $\alpha \le \beta$, then $\alpha$ and $\beta$ are orthogonal if and only if either $\omega +1\le \alpha$ or $\alpha =\omega$ and $\beta <\omega \beta$.

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