In order theory, there is a notion of something called a partially-ordered set, or a poset. That is, it is a set S with some operation < such that it is possible to compare some elements a,b, ∈ S : a < b, but it is also possible that some element c ∈ S can not be meaningfully compared to a, b, or both a and b. You could also consider it to be the union (or possibly cartesian product) of a subset Sp with a total order with another set Sq that has a total order, a partial order, or no notion of order.
A total order is a set such that every element can be compared.
As an example, R, the set of all real numbers, has a total order. On the other hand, the complex numbers C can be considered a partial order. You can meaningfully compare the any two real numbers, and any two imaginary numbers, but there is typically no notion of comparing complex numbers in general (not that it can't be applied, it just typically isn't).
The algebra I've been working on can be interpreted as either the cartesian product R×R or R×R×R; it depends on which is most useful. In any case, it is possible to meaningfully compare any two elements in the subalgebras (n,0), (0,n), (-n,-n), as these are isomorphic to the reals (within certain limits). That is, they are the same under the basic arithmetic operations.
The thing is, under the three-dimensional interpretation, any given vector ax̂ is equivalent to -aŷ - aẑ. This means, if a statement like ax̂ > bx̂ is true, it is also true that ax̂ > -bŷ - bẑ, and any basic algebraic manipulation of it like ax̂ + bŷ > -bẑ.
What is driving me insane about this is that the partial order seems to only apply to two vectors of this specific form. I can not figure out a way to meaningfully define ax̂ > bŷ + cẑ, and doing so would necessarily require me to define some sort of new structure relating vectors like (a,b) and (b,a), which seems wonky. Thus, while it really seems like there should be a way to define either a larger partial order or an actual total order, I can't figure it out.