The librarian said: The students may not borrow more than 3 books in one month.
So, Is there a possibility for students to borrow 4 books per month? and why?
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While one meaning of "may" is "be in some degree likely to", in this case the meaning is probably "have permission to". Assuming the latter definition, the statement means that students are not allowed to borrow more than three books in one month.
The usage of 'may' as an order is archaic and not commonly used except by royalty. 'May this be done.' is a complete sentence and an example of Imperative mood, but not used often in everyday language.
For more information of other possible usages, there is a good discussion in wikipedia of usages of May and Might.