Given a set of numbers identified by multiple subscripts, $e_{ij}$, is there a standard notation for indicating the maximum with respect to one of these subscripts? For example, how would one indicate the maximum for a particular value of $j$? I thought of something like
$\max_j(\{e_{ij}\})$
but I realize that that is ambiguous (it could mean "maximum for a given value of $j$" or "maximize over all $j$").
Details added for clarification:
In the particular example I am interested in, I am considering results from a set of experiments, and $e_{ij}$ indicates the $i$th datapoint from the $j$th experiment (more generally, the $i$th thing from the $j$th set of things). Note that not all the experiments have the same number of datapoints (i.e., the $\{e_{ij}\}$ do not necessarily represent a rectangular 2D array).
The thing I am calculating, and which I want to write down, is
$$ \sum_{j=1}^N \sum_{i=1}^{n_j} f \left( \frac{e_{ij}}{\max\{e_{ij}\}} \right) = \sum_{i=1}^{n_1} f \left( \frac{e_{i1}}{\max\{e_{i1}\}} \right) + \sum_{i=1}^{n_2} f \left( \frac{e_{i2}}{\max\{e_{i2}\}} \right) + \cdots, $$
where $N$ is the number of experiment, $n_j$ is the number of datapoints from the $j$th experiment, and the details of the function $f$ are unimportant. The point is, in the expression on the left-hand side, I want "max" to indicate the maximum for a given value of $j$ (i.e., the maximum value from the $j$th experiment), as opposed to the maximum of all the $\{e_{ij}\}$.
$\max_j\{e_{ij}\}$ is perfectly fine. If you want to be more specific just write $\max_{j=1,\dots,n}\{e_{ij}\}$.