Effective inseparability, lattices, and preordering relations

We study effectively inseparable (abbreviated as e.i.) prelattices. Solving a problem raised by Montagna and Sorbi (1985) we show that if L is an e.i. prelattice then the preordering relation of L is universal with respect to all c.e. pre-ordering relations. In fact it is locally universal, i.e. in any nonempty interval one can computably embed every c.e. pre-ordering relation. Also the preordering relation of L is uniformly dense. Some consequences and applications of these results are discussed, in particular to derive uniform density and local universality for certain prelattices of sentences arising in logic.