I have some ordered tuples $a,b,c$, and I am interested in the following relation:
$ a\succ b \Leftrightarrow \max_i \{a_i-b_i\} >\max_i\{b_i-a_i\} $
That is, I'm interested in the maximum difference between elements of the vectors. (I hope the notation is clear: $a_i$ is the $i^{th}$ element of the tuple.
It's clear that the relation is irreflexive, but what I want to know is: is it transitive or acyclic? That is, can you have vectors such that $a\succ b \succ c \succ a$? If there are such cycles, is there some property you could demand of your vectors such that they would be ruled out?