  We define a notion of ideals in the category of ordered monoids satisfying
the Cuntz axioms introduced in [2] and termed Cu$^\sim$. We show that the set
of ideals of a Cu$^\sim$-semigroup $S$ has a complete lattice structure. In
fact, we prove that for any separable C*-algebra with stable rank one A, the
assignment $\,$ I $\mapsto$ Cu$_1$(I) defines a complete lattice isomorphism
between Lat(A) and Lat(Cu$_1$(A)). Further, we introduce the notion of quotient
ideals and exactness for the (non abelian) category Cu$^\sim$. We show that
Cu$_1$(A)/Cu$_1$(I) $\simeq$ Cu$_1$(A/I) for any I $\in$ Lat(A) and that the
functor Cu$_1$ is exact. Finally, we link S , S$_+$ and S$_{max}$ in a
split-exact sequence that allows us to extract additional information that lies
within the unitary Cuntz semigroup of a separable C*-algebra of stable rank
one.
