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I often see things like \lambda_{\min} or \lambda_{\max} in papers. To me, it seems that the "min" and "max" are not used as operators, but rather as text, so it would be better to do something like \lambda_{\textup{min}} or \lambda_{\textup{max}}.

However, while there's a clear difference between \min and \textup{min} in regular math mode, I haven't really been able to see a difference between how the different options are typset in the subscripts. Are there situations in which there is a difference?

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When you type \lambda_{\min} you get essentially the same as with \lambda_{\mathrm{min}}. They're different semantically: in the former case you're using an operator, in the latter case a textual subscripts.

You may get the same as \mathrm if you use \textup, but not necessarily: compare

$\lambda_{\textup{min}}$ and \textsf{$\lambda_{\textup{min}}$}

where the second “min” is typeset in the prevailing font valid outside the math formula. There wouldn't usually be differences with either \mathrm{min} or \min. Well, there can be!

Suppose you want to use a style that's unfortunately found in some publications, where sans serif type is used in some contexts, for instance tables. If you have math, you should define a suitable math version for sans serif type (look at sansmath.sty for an example). If the math version is correctly set up, then \mathrm would use the standard (upright) sans serif font.

Note also that, even without taking math versions into account, the font chosen by \mathrm may not be the same as the main text font, but I'd consider such cases as disputable, if not outright wrong, setting.

Barring such situations, using \min instead of \mathrm{min} for a textual subscript might be considered a venial sin against consistency in semantics, unless you consider this as a different way to write the result of applying the operator “min”.

You're the judge.

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