I have a set (of related concepts) that I wish to divide (classify) into an arbitrary number of subsets (categories), but when I divide the set (into categories) there will always be one subset that is equal to the original "unabridged" set (e.g. the category tree root)... What's the English term for the set that is the union of all subsets?
The reason that I am asking is because I need a term for a UML class diagram where there is an object type from which a group of derived types need to be created, but due to the variability of the derived types of the derived types the unifying term of the "super-parent" (e.g. the logical union of all subsets) is abstract to the point that I simply want to use a term from set theory to abstract this meaning of the ultimate "super-parent"... short of calling the super parent something a vanilla as "object" or "class"...
Not clear what you mean by "dividing" a given set $A$.
However, you are correct that given a set $A$, there is one subset which is equivalent to $A$. This is the non-proper "subset" $A= A$.
The powerset of $A$, $\mathcal P(A)$, is the set of all subsets of $A$. Indeed, there will be one element that is the non-proper subset of $A$, which is the set $A$ itself, such that $A \in\mathcal P(A)$.
Every other element of $\mathcal P(A)$, will be a proper subset of $A$.