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 $alpha leq beta$, then $alpha$ and $beta$ are orthogonal if and only if either $omega + 1leq alpha$ or $alpha =omega$ and $beta < omega beta$.
Let $D=(V,A)$ be an acyclic digraph. For $xin V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V rightarrow [1,|V|] $ that has the property that
for all $x,yin V$ if $(x,y)in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: [sum_{xin V} e_{_{D}}(x)cdot g(x) ~geq~ frac{1}{2} sum_{xin V}[e_{_{D}}(x)]^2~.] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets of order dimension two.
