I have been mucking round with orders, and this is the order that I found I needed. I would like to know if it is defined somewhere else, it is kind of difficult to search for such things.
*EDIT*
The only application I have used it for at the moment is a totally ordered domain (e.g. '$a,b \in R $'), but I think it would extend to a partially ordered lattice.
Consider two intervals (a,b) (a',b'), if one is clearly better than the other, i.e. b<=a' b≤a' then a<=b<=a'<=b' a≤b≤a'≤b' it is better. But if the interval has the same least upper bound where b=b', then only the greatest lower bound matters, so a <= ≤ a'.
The order that this creates is why I defined it. for instance, given 3 reals {1,2,3} we use a function that defines an interval on each where f(1) = [0,2] f(2) = [1,3] and f(3) = [2,4]. Using this order we can then state that f(1) || f(2) and f(2) || f(3) but f(1) < f(3).
The goal of this order is that it should create a lattice as well.
Thanks for your help.
EDIT
I missed a really important part, sorry. This is based on intervals so a <= ≤ b and a' <= ≤ b' must be true.
(a,b) <= ≤ (a',b') iff b <= ≤ a' or (b' = b and a < a') where a <= ≤ b AND a' <= ≤ b'
