We show that there is a computably enumerable function (i.e. computably approximable from below) which dominates almost all functions and its join with any incomplete computably enumerable set is incomplete. Our main methodology is the LR equivalence relation on reals: two oracles are LR equivalent if the notions of Martin-Löf randomness relative to them coincide. We also show that there are c.e. sets which cannot be split into two c.e. sets of the same LR degree. Moreover a c.e. set is low for random iff it computes no c.e. set with this property.