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. 



What is known about semigroups that are generated by (cyclic) subgroups?
In the non-commutative case you can't say much. The monoid of all maps on n letters is generated by the symmetric group and an idempotent. In fact semigroups generated by idempotents can be quite wild. The singular nxn matrices over a field are generated by idempotents. 


In the commutative case much more can be said. In the finite case your semigroup will be a commutative inverse semigroup and so be a semilattice of groups (also called a Clifford semigroup). I guess this should also be true in the infinite case too. 


