Is a Cancellative Semigroup the Same as a Group?

[prashantgolu]

if a set is closed and associative with respect to an operation * and both cancllation laws hold...prove that the set is a group wrt *.

 

[adriank]

Is this homework? Have you made an attempt at the problem? Show us what you have.

 

[prashantgolu]

thnx..this one is done...

 

[HallsofIvy]

It looks to me like you are trying to prove something that is NOT TRUE. For example, the set of positive integers is closed under ordinary mulitplication which is associative and both cancellation laws hold. But this is not a group.

 

[adriank]

Ah, indeed. And if you take the integers ≥ 2 under multiplication, then you don't even get a monoid. Apparently such a thing is called a cancellative semigroup.