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What are the subsemigroups of $(\mathbb N,+)?$

My abstract algebra professor asked this question for homework, yet upon turning it in he still refused to tell us how to solve this problem. Now I do know it has something to do with a sets greatest common divisor. He said take sets like 3,5 and expand them or 5,8 and try to observe a pattern. He also said to pay attention to the set and when it becomes regular. For example, expanding 3,5,6,8,9,10,11,.. . I approached it through defining a and b belonging to N and them using am+bn since they're both natural numbers. Then I defined c=gcd(a,b) and thus there exist integers r and s such that ar+bs. The I said you could rewrite a=ck and b=cq and then sub'd the values in to am+bn and found that this set is just say set T multiplied by c. My professor said I was on to something, but I am still confused on how to show which sets of natural numbers are closed under addition. I know even, and multiples are but the others like 3,5 and 5,8 have to have this general rule that applies to all. I hope this wasn't too confusing but I need to figure this out and can't find anything online about sets of natural numbers.

  • $\begingroup$ You can find a complete answer into the question body of this topic. $\endgroup$ – user26857 Jan 24 '13 at 15:54