Let be a semigroup, and let denote the set of all nonempty subsets of . Show that is a semigroup under the operation . Which of the basic properties of are preserved by ?
We need only show that this operator is associative. Indeed, .
Suppose has a left identity . Then for all , we have . That is, is a left identity in . Similarly if is a right identity in , then is a right identity in . Then if is an identity in , is an identity in .
Suppose has a left zero . Then for all , we have . Thus is a left zero in . Similarly, if is a right zero in then is a right zero in , and if is a zero in then is a zero in .
If is commutative, then for all we have . Thus is also commutative.
Suppose is an identity element and is a unit, with . Then , so that is a unit in .
Suppose is a left ideal. Now let and be nonempty subsets. Now , since , and moreover this set is nonempty since and are nonempty. That is to say, . So is a left ideal. Likewise if is a right ideal in , then is a right ideal in , and if is a two-sided ideal in , then is a two-sided ideal in . More generally, if is a subsemigroup, then is a subsemigroup.
Suppose is a left zero semigroup. That is, for all . If , then . That is, is also a left zero semigroup. Similarly, if is a right zero semigroup then so is . Suppose now that has a zero element , and that is a zero semigroup. Now is a zero in , and for all , we have . So is also a zero semigroup.
If is simple, need not be simple, as we show. Consider the group . We can easily verify that the only two-sided ideal in is itself, so that is simple as a semigroup. Now , and evidently, is an ideal in , so that is not simple.