Let be a semigroup, and let , , and denote the sets of left- right- and two-sided ideals of . Let and let be nonempty. Show that and that if , then . Further, show that if is finite, then is not empty (and so is an ideal).
Suppose is a nonempty collection of left ideals of . Certainly . If , then we have for some . Now if , then . So , and thus is a left ideal of . Now suppose , and say . So for all . If , then for all , and so . Thus is a left ideal.
Likewise, the results hold for . Since every two-sided ideal is also a left and a right ideal, the results also follow for .
Now suppose is a finite collection of ideals of . Now for each , so that . Since each is nonempty, is nonempty, and so is nonempty.