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Let $R_i$ is an ordinal for every $i\in n$ (where $n$ is also an ordinal) and $S_{i,j}$ is an ordinal for every $i\in n$ and $j\in m_i$ (where $m_i$ is also an ordinal for every $i\in n$).

What is a formula for the ordinal representing the pair $(R_i;S_{i,j})$ in the lexicographic order of pairs?

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    For fix $i,j$ the pair $(R_i; S_{i,j})$ is represented by their sum.2012-03-18
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    @azarel: "Their sum"? What is "their"? I don't understand.2012-03-18
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    the pair $(R_i;S_{i,j})$ is represented by $R_i+S_{i,j}$.2012-03-18
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    @azarel: $R_i+S_{i,j} < R_{i'}+S_{i',j'}$ isn't equivalent (consider the case of natural numbers) to $R_i (the condition for lexigraphical order). It seems that you err.2012-03-18
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    I guess you mean the order type of $(\{(R_i;S_{i,j}:i\in n,\ j\in m_i\},<_{lex})$.2012-03-18
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    @azarel: You guess right.2012-03-18
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    @azarel: I've said "right". No not right. I mean the type corresponding to the element $(R_i;S_{i,j})$ in that set, not the order type of the entire set.2012-03-18
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    The question has no nice answer unless you know more about the $R_i$ and $S_{i,j}$. Are the $R_i$ increasing with $i$? Are the $S_{i,j}$ increasing with $j$?2012-03-19
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    @Brian M. Scott: There are no reason to assume that these are increasing.2012-03-20

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