A semigroup $(A,\cdot)$ that is idempotent (i.e. $a^2=a$ for every element $a\in A$) is naturally generated by its subgroups (every element on itself constitutes a trivial group). I would like to know what is known about semigroups that are generated by (cyclic) groups? Again, such groups do not need to share a common neutral element. I am interested in the commutative case mainly, but I appreciate any comments.
In particular, I would like to know references to this topic, since I haven't found anything so far. Thank you.