# 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.

• Doesn't it exactly say that the set with that operation forms a groupoid? – Yuval Filmus Nov 18 '10 at 17:49
• @Yuval Filmus It does. If the question was something like "If set is closed under some operation, what does that tell us about algebraic stricture?", I'd say it forms a groupoid. But it says "...what does that tell us about the structure of the groupoid?". We only know that it is indeed a groupoid, as assumed by question, but we can't actually (as far as I can see, that's why I'm asking) say anything about structure of the groupoid. – AndrejaKo Nov 18 '10 at 17:53
• I think the preferred term is "magma." "Groupoid" has a second meaning in category theory which I think is now more widely used than this meaning. In any case, I can't make sense of this question. Are you sure it's stated correctly? – Qiaochu Yuan Nov 18 '10 at 19:11
• @Qiaochu Yuan We don't use term "magma" here. I just translated question as directly as possible. As far as I can see, magma could be used in this question instead of groupoid. I'm 100% sure that question is correctly translated. In its original form, it doesn't (to me at least) make much sense either. – AndrejaKo Nov 18 '10 at 19:13

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.

• Turns out that this is the correct answer. According to some people who were on the exam that day, the question was meant to be something like "If set is closed under some operation, what does that tell us about algebraic stricture?" and groupoid was supposed to be answer, but whoever was typing questions made a mistake and put groupoid in the question, so they accepted "If we know it's a groupoid, we know it's a groupoid" answers. – AndrejaKo Nov 18 '10 at 19:35

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).