I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read Refereeing a Paper, but it seems my question is different.

The general question is, roughly:

What is the point after that you get nervous when refereeing a paper?

Specifically:

1. I've found many English mistakes. (Well, I'm not a native English and so I can understand. So I am not nervous yet)

2. I've found some maths inaccuracies like "let A be any set"... and then I have discovered that the proof of the first result works only for finite sets. (OK, those are only inaccuracies - I am not nervous yet)

3. There are many references like "we use the notation of [X]", "this result is proved in [X]", where [X] is a BOOK, without specifying a precise section, or the number of the result... should I get this book and read all to find out the correct references? - just thinking of it, makes me a bit nervous..

4. (most importantly). At some point, the author defines a supremum of a set of elements of a group... OK, it's clear that there is something missing, but also correcting intuitively the definition, it turns out that this supremum is not a constant, depending on one of the elements of the group, and so it is (apparently) useless for his/her aims. I am not saying that this is a serious mistake, but just that now... I'm getting nervous!

Now, taking into account that the person who asked me to referee this paper told me: be selective, we accept only 20% of submitted papers,

what should you do in these cases? Reject? Ask for a revision? Not getting nervous and try to see if the rest of the piece is good (I'm quite a good guy and I'm doing that at the moment)?

Of course I will talk with the editor, but I also would like to know more opinions that might be helpful in future.

Thank you in advance,

Valerio