Abstract
Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of R+. Even the class CongSimp(Q+) of all congruence-simple subsemirings of Q+ has not been classified yet.
We classify all the maximal elements of the congruence-simole saturaated subsemirings of Q+. We classifity all the maximal elements of CongSimp(Q+) and show that every element of CongSimp(Q+) \ {Q+} is contained in at least one of them.