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Yvain comments on A Pure Math Argument for Total Utilitarianism - Less Wrong
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If you change the value of "medium" from "1" to "-5" while leaving the other two states the same, your conclusion no longer holds. For example, on your last graph, (very good, medium) would outrank (very good), even though the former has a value of -2 and the latter of +3. This suggests your system doesn't allow negative utilities, which seems bad because intuitively it's possible for utility to sometimes be negative (eg euthanasia arguments).
It must allow negative numbers, or it's not a group, as (R+,+) is not a group. (Each element must has an inverse which returns that element to the identity element, which for this particular free group is "no one alive".)
However, I believe this specific issue is solved by the lattice structure. If "medium" were "-5" instead of "1", when you add "medium" to any universe, you create a lattice element below the original universe, because we know it is worse than the original universe.
This is a good point - I am now regretting not having given more technical details on what it means to be "order preserving".
The requirement is that
X > 0 ==> X + Y > Y. I've generated the graph under the assumption thatMedium > 0, which results in (very good, medium) > (very good). Clearly the antecedent doesn't hold ifMedium < 0, in which case the graph would go the other direction, as you point out.