## Characterize the left ideal of a semigroup generated by a subset

Let be a semigroup and let be a nonempty subset. Recall that the left ideal of generated by is the intersection of all the ideals which contain , and similarly for right- and two-sided- ideals.

Prove that the left ideal generated by is and that the two-sided ideal generated by is .

Note that if is a left ideal containing , then . In particular, is contained in every left ideal which also contains , and thus is contained in the left ideal generated by . On the other hand, , and thus is a left ideal of which certainly contains . So the left ideal generated by is contained in , so that .

Similarly, if is an ideal of containing , then . So is contained in the ideal generated by . And on the other hand, since , this set is a left ideal, and likewise a right ideal. So is a two sided ideal containing . Hence .

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