A characterization of principal ideals in a semigroup

Let S be a semigroup and let a \in S. Let L(a), R(a), and J(a) be the left, right, and two-sided ideals generated by a, respectively. Prove that L(a) = a \cup Sa, R(a) = a \cup aS, and J(a) = a \cup Sa \cup aS \cup SaS.


Recall that by definition, L(a) is the intersection over the class of left ideals which contain a. Suppose L is a left ideal containing a. Then for all s \in S, sa \in L, and thus a \cup Sa \subseteq L. So a \cup Sa \subseteq L(a). On the other hand, we have S(a \cup Sa) = Sa \cup SSa \subseteq Sa \subseteq a \cup Sa. So a \cup Sa is a left ideal containing a, and thus L(a) \subseteq a \cup Sa. So L(a) = a \cup Sa.

The proofs for R(a) and J(a) are similar; show that every (right,two sided) ideal contains the set in question, and that the set in question is a (right, two-sided) ideal.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: