Examples

• Generation by a single element. Since every positive integer n can be written as the sum of n times 1, the element 1 of the monoid (N,+,0) generates the whole monoid. This implies that (N,+,0) is cyclic.
• Not finitely generated. The monoid (Z, ·, 1) can be generated by the set of all prime numbers together with -1, but not by a proper subset of this set. Actually, any generating set of this monoid should contain either +p or -p for every prime p. Thus (Z, ·, 1) is not finitely generated.