Suppose one has a partition $P=\{\{a,b,c\},\{d,e\}\}$ of a set $X=\{a,b,c,d,e\}$.
Which of these is a refinement of $P$:
- $\{\{\{a,b\},\{c\}\},\{d,e\}\}$; or
- $\{\{a,b\},\{c\},\{d,e\}\}$?
What (if anything) is the formal term for the other, in respect of $P$?
Is there a term for a "total refinement" of the first example, e.g. $\{\{\{\{a\},\{b\}\},\{c\}\},\{\{d\},\{e\}\}\}$, such that it cannot be "refined" any further?
A partition $Q$ is a refinement of a partition $P$ if every element of $Q$ is a subset of an element of $P$. So for your question 2. is a refinement of $P$. (I do not understand your notation for 2.)
The partition $\{ \{a\}. \{b\}, \{c\}, \{d\}, \{e\} \}$ is called the finest partition.
Incidentally, a slicker notation for a partition is $ab|c|de$ with a simple bar separating elements in the same cell.