  We prove the following results: (i) Every absolutely weakly compact set in a
Banach lattice is absolutely weakly sequentially compact. (ii) The converse of
(i) holds if $E$ is separable or $B_{E^{**}}$ is absolutely weak$^*$ compact.
(iii) Every absolutely weakly compact subset of a Banach lattice is contained
in the closed convex hull of an absolutely weakly null sequence if and only if
the Banach lattice has the positive Schur property. Examples and applications
are provided.
