  The purpose of this short paper is to further develop the theory of
transverse generalized complex structures. We focus on proving some equivalent
conditions to the basic $dd^{\mathcal{J}}$ -lemma. We justify our approach by
describing the transverse symplectic structure in this language and relating
the basic $dd^{\mathcal{J}}$-lemma to the surjectivity of the Lefschetz map. We
also present a non-trivial example of a foliation endowed with a transverse
generalized complex structure. Transverse generalized complex structures do not
rely heavily on the existence of a bundle-like metric, which makes them a
convienient tool to study some non-Riemmanian foliations.
