  Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak
b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let
$\Delta^+$ be the corresponding set of positive roots. We continue our study of
combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by
the long positive roots. In particular, the union of an arbitrary set of
maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We
also characterise the greatest lower bound of two positive roots, when it
exists, and point out interesting subposets of $\Delta^+$ that are modular
lattices.
