Exhibit a 0-simple semigroup

Give an example of a 0-simple semigroup.

Recall that a semigroup S with zero is called 0-simple if S^2 \neq 0 and S has no ideals other than 0 and S.

Let G be a group, and consider the semigroup S = G_0. (Recall that if T is a semigroup, T_0 is the semigroup obtained by attaching a new element 0 to T which acts like a zero.) So S is a semigroup with zero. Certainly S^2 \neq 0. Now let J \subseteq S be an ideal. If J \neq 0, then there exists an element u \in J such that u \neq 0. Now Gu = G (since G is a group). So G = Gu \subseteq SJ \subseteq J, and certianly 0 \in J. So J = S. Thus S = G_0 is 0-simple.

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