Notation for "incommensurate" elements?

Question

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$.

Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.

Answer 1

The term is usually incomparable; the notation that I have seen for incomparability is $x\parallel y$.

Comparability of $x$ and $y$ is then denoted by $x\perp y$.

Answer 2

If you don't want to use $\parallel$ you might try $\not\lessgtr$ (\not\lessgtr) or something that looks better, maybe $\,|\!\!\!\!\!\lessgtr$. This would complement the usage of $\lessgtr$ for comparability.

Note that $a\perp b$ in the context of lattice ordered groups usually means $|a|\wedge |b| = 1$ (the neutral element, while $|a| = (a\vee 1)^{-1}(a\vee 1)$ ).