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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 $\Sigma (m)$ (quantumstates, density matrices, etc) into their counterparts in the supersystem$\Sigma (n)$ (for $m|n$). The compatibility of these embeddings is studied. Theconcept of ubiquity is introduced for quantities which fit with this structure.It is shown that various entropic quantities are ubiquitous. The sets ofvarious quantities become $T_0$-topological spaces with the divisor topology,which encapsulates fundamental physical properties. These sets can be convertedinto directed-complete partial orders (dcpo), by adding `top elements'. Thecontinuity of various maps among these sets is studied.

Partial order and a $T_0$-topology in a set of finite quantum systems