# Pairwise comparisons of maxima of differences between ordered n-tuples?

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?

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If I understand you correctly, the relation is not reflexive, unless you change the strict inequality defining the relation to a non-strict one. – Henrik Mar 30 '12 at 12:12
Yes, I originally had that, but then you can obviously have cycles where $a=b=c$ but they're not what I'm interested in. So I went for the strict relation. I forgot to refactor the preamble to my actual question. Thanks for pointing that out! – Seamus Mar 30 '12 at 12:20

Sure, just let $a = (10,1,1)$, $b = (1,9,2)$, $c = (8,1,4)$.