  We study orthogonally additive operators between Riesz spaces without the
Dedekind completeness assumption on the range space. Our first result gives
necessary and sufficient conditions on a pair of Riesz spaces $(E,F)$ for which
every orthogonally additive operator from $E$ to $F$ is laterally-to-order
bounded. Second result provides sufficient conditions on a pair of orthogonally
additive operators $S$ and $T$ to have $S \vee T$, as well as to have $S \wedge
T$, and consequently, for an orthogonally additive operator $T$ to have $T^+$,
$T^-$ or $|T|$ without any assumption on the domain and range spaces. Finally
we prove an analogue of Meyer's theorem on the existence of modules of
disjointness preserving operator for the setting of orthogonally additive
operators.
