In any lattice-ordered group, we say that two elements are orthogonal if their meet is 1. I've been thinking of groups who have only "trivial" orthogonal relations, i.e. $x\perp y\implies x=1$ or $y=1$.
My first thought is that these are the totally ordered groups, but this is sufficient yet not necessary.
Next I thought that maybe these are the Archimedian groups. But $\mathbb Z^2$ lexically ordered has this property and is clearly non-Archimedian.
So I'm wondering:
- Is there a more common name for this than "trivially orthogonal"?
- Is there an alternative classification of these groups, or can we say anything interesting about them?