If set is closed under some operation, what does that tell us about the structure of the groupoid? 
If set is closed under some operation,
  what does that tell us the about structure
  of the groupoid?

It's a question from one of the previous math exams which was shown to me by one of my friends.
Unfortunately, I can't find a good answer to it. As far as I know, the answer is nothing, because of the definition of groupoid we use. Here's what (among other things) my math textbook has to say about groupoids:

Ordered pair (S,*), where arbitrary
  binary operation closed in  set S
  is marked with *, is called groupoid.

 A: Assuming "the grupoid" means the grupoid (using the definition you quote) made up of that specific set and that specific operation (as opposed to some other operation), then the answer is "Nothing beyond the fact that it is a grupoid." Because any property you care to name that is a property of some but not all grupoids will arise from a particular example of a "set that is closed under some operation," but will also fail to arise from another particular example. That is, absent more information, you can't say anything else. 
A: You settled that the original formulation was a mistake. But another meaning could have been 

If a set inside a groupoid is closed under some operation, what does that tell us about the groupoid?

In that case, one answer would be:
If the inside set is also closed under the same operation as the groupoid operation, then the groupoid
(1) has a non trivial sub-structure (a subgroupoid),
(2) the groupoid can be restricted to this set and form an early pre-ring (a groupoid with two different operations on the same set). 
