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I was just lookning at the difinition of the OWA operator. It is defined as:

$F(a_1, ..., a_n) = \sum_{j=1}^n w_j b_j$ where $b_j$ is the $j$th largest of the $a_i$.

The part that strikes me odd is "$b_j$ is the $j$th largest". I am not mathematician, so I wanted to know if saying that something is "$j$th largest" considered an ok practice? Strikes me as not being very mathematics-like...

Is there an alternative definition, that does not use "English" language?

Just curious!

P.S.: I have no idea what to tag this question with, so I'd appreciate if someone could re-tag it appropriately.

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It is ok practice as long as the meaning is clear and unambiguous. It means that there is a permutation $i\mapsto k_i$ of $\{1,2,\ldots,n\}$ such that $a_{k_1}\geq a_{k_2}\geq\cdots\geq a_{k_n}$, and then by definition, $b_j=a_{k_j}$.

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    @drozzy: The indices of the $a_i$s in your question are always from the set $\{1,2,\ldots, n\}$, and like I said in my answer, rearranging the $a_i$s in decreasing order can be accomplished by permuting the indices. If $(a_1,a_2,a_3,a_4)=(3,4,4,7)$ as in the example in my comment, then we want to rearrange this in decreasing order, which is $7\geq 4\geq 4\geq 3$. The point of my comment is that there are actually $2$ ways to permute the index set $\{1,2,3,4\}$ in order to do so, but the resulting reordering on the $a_i$s is unique in terms of what actual number lies in each position.2012-01-19