Let g be a simple Lie algebra with a Borel subalgebra b and Ab the set of abelian ideals of b. Let ∆ + be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of Ab parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if g = sl n . We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of ∆ + that are modular lattices.2010 Mathematics Subject Classification. 17B20, 17B22, 06A07, 20F55.