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If $M_1$ and $M_2$ are well ordered then $ M = M_1 \times M_2$ is well ordered too.

with the orders $$ (a_1,b_1) \leq (a_2,b_2) \ iff \ b_1 \leq b_2 \ in \ M_2 \ and \ a_1 \leq a_2 \ in \ M_1 \ ,\ b_1=b_2$$

I prove that this is a total order, I prove trichotomy, transitive, but how can I prove that there existe a first element in M.

I tried this by contradiction if $(a_0,b_0) \in M$ is first element but I don't know how continue this idea.

Someone can help me please. Thanks for your time and help.

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  • $\begingroup$ To show that $M$ is well-ordered you need to show that any non-empty subset has a least element, not just $M$ itself. $\endgroup$
    – Rob Arthan
    Mar 16, 2017 at 23:37
  • $\begingroup$ Could you clarify the definition of the order in the product, please? Specifically, what is the role of that $b_1 = b_2$ at the end of the line? $\endgroup$
    – amrsa
    Mar 17, 2017 at 12:53

1 Answer 1

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The "and" in your definition should be "or", in which case it works. Let $a_0$ be the first element in $M_1$ and $b_0$ be the first element in $M_2$. You are guaranteed they exist because $M_1,M_2$ are well ordered. Then $(a_0,b_0)$ is the first element. You can use your definition to show it is less than ore equal to any element of $M$. For a well order every subset of $M$ must have a least element, but you take any collection, take the elements with the smallest second items, and take the one of those that has the smallest first item.

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  • $\begingroup$ yea I have that but I don't know how to relate with the order in $M$ $\endgroup$
    – Rosa Maria Gtz.
    Mar 16, 2017 at 23:18
  • $\begingroup$ You use your definition. Given a point $(a,b)$ you always have $(a_0,b_0) \le (a,b)$ because you have both $a_0 \le a, b_0 \le b$ $\endgroup$
    – Ross Millikan
    Mar 16, 2017 at 23:24
  • $\begingroup$ but what happen if b_0=b ? $\endgroup$
    – Rosa Maria Gtz.
    Mar 16, 2017 at 23:37
  • $\begingroup$ But shouldn't a well-ordered set be also totally ordered? (See the comment of Rob Arthan above.) If both $M_1$ and $M_2$ have more than one element, and $a_0 < a_1 \in M_1$ and $b_0 < b_1 \in M_2$, what is the first element of the set $\{ (a_0, b_1), (a_1, b_0) \}$? $\endgroup$
    – amrsa
    Mar 17, 2017 at 12:43
  • $\begingroup$ @amrsa: yes. I took the definition of the order to be a bit poorly written and be lexicographic order starting with the second element. More correctly it would be $(a_1,b_1) \lt (a_2,b_2) if either $b_1 \lt b_2 or ($b_1=b_2$ and $a_1 \lt a_2$) This is a total order and a well order as long as the underlying orders are well orders. $\endgroup$
    – Ross Millikan
    Mar 17, 2017 at 13:53

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