  A `whole-part' theory is developed for a set of finite quantum systems
$\Sigma (n)$ with variables in ${\mathbb Z}(n)$. The partial order `subsystem'
is defined, by embedding various attributes of the system $\Sigma (m)$ (quantum
states, density matrices, etc) into their counterparts in the supersystem
$\Sigma (n)$ (for $m|n$). The compatibility of these embeddings is studied. The
concept of ubiquity is introduced for quantities which fit with this structure.
It is shown that various entropic quantities are ubiquitous. The sets of
various quantities become $T_0$-topological spaces with the divisor topology,
which encapsulates fundamental physical properties. These sets can be converted
into directed-complete partial orders (dcpo), by adding `top elements'. The
continuity of various maps among these sets is studied.
