I have a quick question on what's called the right lexicographical ordering: the text I'm reading defines it as follows ($s \prec t$ means $s$ is an initial segment of $t$):
Consider $\omega^{<\omega}$ and define the right lexicographical ordering $<_r$ as follows: for finite strings $s,t$ we set $s <_r t$ iff $t \prec s$ or, if that fails, $s(i) < t(i)$ where $i$ is the least index at which $s$ and $t$ differ.
This definition seems very unintuitive to me. In particular, what does the term "right" refer to here? It is clear that the empty string is $<_r$-maximal. Further, the ordering says that the shorter the string, the greater it is w.r.t. $<_r$. And if two strings disagree somewhere, then the string with the smaller value at the disagreement is considered smaller w.r.t. $<_r$. This last point doesn't make sense to me intuitively.
Edit: Now with William's comment I am wondering: is this actually the reverse dictionary order? I don't think it is as (as mentioned in my comment) in the usual dictionary order we have $\langle 0,1,2 \rangle < \langle 0,1,3 \rangle$, but we also have $\langle 0,1,2 \rangle <_r \langle 0,1,3 \rangle$. Is there a typo somewhere, or am I just misunderstanding the ordering?