Prove that a semigroup is a group if and only if it is both left and right simple. Exhibit a left simple semigroup which is not right simple.

We begin with a lemma.

Lemma: Let be a semigroup. Then is left simple (right simple) [simple] if and only if () [] for all . Proof: Suppose is left simple, and let . Certainly , so that is a left ideal. Thus for all . Conversely, suppose for all , and let be a left ideal. Now is nonempty by definition; say . Then , and so . Thus , and in fact is the only left ideal of . So is left simple. The proofs for right simple and simple are similar.

Now let be a group and let . If , then we have ; in particular, . So for all . By the lemma, is left simple. Similarly, is right simple.

Now suppose the semigroup is both left and right simple. Let . Since , there exists an element such that . Now let be arbitrary. Since , there exists such that . Now , so in fact is a left identity element of . Similarly, there is a right identity element , and we have , so that is a two-sided identity.

Now let be arbitrary. Since , there exist elements such that . Now , and we have . Thus has a two sided inverse with respect to . Since was arbitrary, every element has an inverse, and so is a group.

Now consider the semigroup with for all . (That is, is the left zero semigroup on two elements.) Suppose is a left ideal. Now for all and , we have . Thus , and so is left simple. However, is not a group, and so (by the previous argument) cannot be right simple.

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