Talk:Comparability

I have restricted the definition to partial orders. In the previous version there was inconsistency about the question whether an element is comparable to itself. According to it is. Can the definition be extended? In a preorder, are p and q with pRq and qRp called comparable?--Patrick 13:45, 15 May 2007 (UTC)

Self-comparability
Since a partial order P is, a fortiori, a relation (mathematics), elements x and y are comparable with respect to P if and only if $$\{(x,y), (y,x)\} \cap P$$ is nonempty. And the difference between non-strict and strict versions of a partial order is precisely that one is the reflexive closure (resp. reflexive reduction) of the other. So any element x is comparable to itself with respect to P if and only if P is reflexive (non-strict).—PaulTanenbaum 01:20, 6 July 2007 (UTC)


 * 1) Comparability is that which recurs. Human thorugh comparison recognize them as the similarity among the objects which are compared.  —Preceding unsigned comment added by 210.212.249.50 (talk) 06:07, 24 December 2010 (UTC)

Notation
Is there a somewhat authoritative source or tradition for the notation $$\perp$$ and $$\parallel$$ for comparable and incomparable? It would be good to have a reference or else to say that the notation is not standard. It seems that almost always people just use the words and do not introduce symbols for the two concepts. Bjbraams (talk) 21:11, 7 January 2012 (UTC)

In fact, the notation $$\parallel$$ for incomparable seems well established; I find it used by both Blyth and Roman in their textbooks on lattices and order, and also by Graetzer. Birkhoff just uses the word. For comparable I find that Blyth uses $$\not\parallel$$ while Roman and Graetzer only use the word. Bjbraams (talk) 22:27, 7 January 2012 (UTC)

Why my delete
Prev version labeled ⊥ as "perpendicular" and || as "parallel." I don't see that those labels provided any clarity or aided anyone's understanding, but I do imagine that they might have confused novice readers. That's why I undid the addition of those labels. If somebody thinks that there's a good reason to restore the labels, please let us know here what that reason is.—PaulTanenbaum (talk) 23:59, 12 September 2013 (UTC)