  An \textit{ideal} of $N$-tuples of operators is a class invariant with
respect to unitary equivalence which contains direct sums of arbitrary
collections of its members as well as their (reduced) parts. New decomposition
theorems (with respect to ideals) for $N$-tuples of closed densely defined
linear operators acting in a common (arbitrary) Hilbert space are presented.
Algebraic and order (with respect to containment) properties of the class
$CDD_N$ of all unitary equivalence classes of such $N$-tuples are established
and certain ideals in $CDD_N$ are distinguished. It is proved that infinite
operations in $CDD_N$ may be reconstructed from the direct sum operation of a
pair. \textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense)
uniqueness is established. The issue of classification of ideals in $CDD_N$ (up
to isomorphism) is discussed. A model for $CDD_N$ is described and its concrete
realization is presented. A new partial order of $N$-tuples of operators is
introduced and its fundamental properties are established. Extremal importance
of unitary disjointness of $N$-tuples and the way how it `tidies up' the
structure of $CDD_N$ are emphasized.
