User talk:Dfeuer/Definition:Lexicographic Ordering on Product

Hey Dfeuer I like this generalisation of lexicographic order you did.

Maybe I'm not getting it though.

Is it necessary that $I$ is the underlying set of some well ordered set?

--Jshflynn (talk) 20:35, 18 December 2012 (UTC)


 * The definition on PlanetMath tries to lift this restriction, but I think that's a mistake. Take a look at what they did, mentally fix the piece of the definition they forgot to put in, and then consider the fact that under the repaired definition it's possible for $a_i \le_i b_i$ for each $i \in I$ but $a \not\preceq b$. I also think all "dictionary order" intuition goes out the window if the index set isn't well-ordered. --Dfeuer (talk) 21:26, 18 December 2012 (UTC)


 * Hmm... Let be back off from that slightly. I think it's possible to fix it so that's not true, but it still
 * doesn't feel much like a dictionary. Define $x \prec y$ iff for some $k \in I$, $x_k < y_k$ and
 * $i < k \implies x_i \le y_i$. Then define $\preceq$ in terms of $\prec$ --Dfeuer (talk) 21:34, 18 December 2012 (UTC)


 * More intuitively: if the index set is well-ordered and $x\ne y$, then there must be a "first place" where they differ. If the index set is not well-ordered, this may not be true. --Dfeuer (talk) 21:39, 18 December 2012 (UTC)