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If $A_1, \ldots, A_n$ are well ordered sets, then so is the Cartesian product $\prod_{i=1}^n A_i$ under the dictionary order. Am I right?

Is this finite product also well ordered in the anti-dictionary order?

Now suppose that $J$ is an infinite set of indices, and suppose $\left\{ A_\alpha \right\}_{\alpha \in J}$ is a collection of well ordered sets. Then is the set $$A \colon= \prod_{\alpha \in J} A_\alpha$$ also well ordered in the dictionary order? under the anti-dictionary order?

If so, then how to prove this rigorously?

If not, then how to construct a counter example?

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  • $\begingroup$ What is the anti-dictionary order? $\endgroup$ – Ben Grossmann Feb 14 '17 at 20:56
  • $\begingroup$ I think this depends on the ordering given for $J$. Depending on how you order $J$, the sequence $BAAAAA\ldots, ABAAAA\ldots, AABAAA\ldots, \ldots$ may be a counterexample. $\endgroup$ – Tanner Swett Feb 14 '17 at 21:01
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    $\begingroup$ @Omnomnomnom the anti-dictionary order is defined as follows: Let $\left(A_1, <_1 \right), \ldots, \left(A_n, <_n \right)$ be ordered sets, and let $A$ be their Cartesian product. Then the order $<$ defined on $A$ as follows: $\left( x_1, \ldots, x_n \right) < \left( y_1, \ldots, y_n \right)$ if $x_i < y_i$ for some $i$ and $x_j = y_j$ for every $j > i$. There is some exercise in Munkres about this order as well. Maybe I can dig it up for you later. $\endgroup$ – Saaqib Mahmood Feb 14 '17 at 21:12
  • $\begingroup$ @SaaqibMahmuud So it's just the dictionary order on the product $(A_n, <_n), . . . , (A_1, <_1)$ (that is, reverse the order of the factors)? $\endgroup$ – Noah Schweber Feb 14 '17 at 21:15
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The result does not hold for infinite products.

Here's a counterexample: take each $A_i$ to be $\{0, 1\}$ ordered the usual way, $i\in\mathbb{N}$. Then let $e_i$ be the string with a $1$ in the $i$th place and a $0$ everywhere else; what can you say about $e_i$ versus $e_j$ if $i<j$?


Also, it's worth noting that the dictionary order doesn't really make sense if $J$ isn't well-ordered: how else do you know that "the first place where $F$ and $G$ disagree" even exists, if $J$ isn't well-ordered?

For a concrete example of this, consider $J=\{. . . , -3, -2, -1, 0\}$ and let $A_i=\{0, 1\}$ (ordered as usual) for each $i\in J$. Now consider the sequences $$F=( . . . , 0, 1, 0, 1),\quad G=( . . . , 1, 0, 1, 0).$$ Is $F<G$ or is $G<F$?

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  • $\begingroup$ in the dictionary order, $e_i$ would precede $e_j$. Am I right? $\endgroup$ – Saaqib Mahmood Feb 14 '17 at 21:15
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    $\begingroup$ @SaaqibMahmuud No: $(1, 0, 0, ...)>(0, 1, 0, . . .)$ since at their first point of disagreement, $1$ is bigger than $0$. $\endgroup$ – Noah Schweber Feb 14 '17 at 21:16
  • $\begingroup$ oh yes, you're right. Sorry. Yes, I now do understand that the set of all the $e_i$ is non-empty but has no smallest element. What about the anti-dictionary order? $\endgroup$ – Saaqib Mahmood Feb 14 '17 at 21:18
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    $\begingroup$ @SaaqibMahmuud In the antidictionary order, the product isn't even linearly ordered: there's no way to compare elements in general! Which is smaller, $(0, 1, 0, 1, . . . )$ or $(1, 0, 1, 0, . . . )$? You look for the "last" point of disagreement, but there isn't one. This is basically the counterexample in the second part of my answer. $\endgroup$ – Noah Schweber Feb 14 '17 at 21:20

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