  We give an overview of normality and conormality properties of pre-ordered
Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we
investigate normality of $B(X,Y)$ in terms of normality and conormality of the
underlying spaces $X$ and $Y$.
  Furthermore, we define a class of ordered Banach spaces called quasi-lattices
which strictly contains the Banach lattices, and we prove that every strictly
convex reflexive ordered Banach space with a closed proper generating cone is a
quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that
are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we
show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a
quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and
satisfies an identity analogous to the elementary Banach lattice identity
$\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is
used to show that spaces of operators between such ordered Hilbert spaces are
always absolutely monotone and that the operator norm is positively attained,
as is also always the case for spaces of operators between Banach lattices.
