  We study conditions when a certain type of the Riesz Decomposition Property
(RDP for short) holds in the lexicographic product of two po-groups. Defining
two important properties of po-groups, we extend known situations showing that
the lexicographic product satisfies RDP or even RDP$_1$, a stronger type of
RDP. We recall that a very strong type of RDP, RDP$_2$, entails that the group
is lattice ordered. RDP's of the lexicographic products are important for the
study of lexicographic pseudo effect algebras, or perfect types of pseudo
MV-algebras and pseudo effect algebras, where infinitesimal elements play an
important role both for algebras as well as for the first order logic of valid
but not provable formulas.
