  It was recently proved that the dualization in lattices given by
implicational bases is impossible in output-polynomial time unless P=NP. In
this paper, we~show that this result holds even when the premises in the
implicational base are of size at most two. Then we show using hypergraph
dualization that the problem can be solved in output quasi-polynomial time
whenever the implicational base has bounded independent-width, defined as the
size of a maximum set of implications having independent conclusions. Lattices
that share this property include distributive lattices coded by the ideals of
an interval order, when both the independent-width and the size of the premises
equal one.
