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.
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2 Answers
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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 $|a| = (a\vee 1)^{-1}(a\vee 1)$ ).
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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 $x\perp y$.
answered Sep 2, 2013 at 22:22
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1$\begingroup$ I saw these during a long slog through partial orders (in a combinatorics context) my first year of grad school. $\endgroup$ Sep 2, 2013 at 22:30
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5$\begingroup$ I’ve seen the $x\|y$ notation for incomparability, but not $x\bot y$ for comparability. In the context of partial orders $P$ used in forcing I’ve seen $p\bot q$ used to mean that there is no $r\in P$ such that $r\le p$ and $r\le q$; $p$ and $q$ are then said to be incompatible. $\endgroup$ Sep 3, 2013 at 5:12