Pairs of orthogonal countable ordinals

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

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