TITLE: A formal name for "smallest" and "largest" partition
QUESTION [7 upvotes]: Consider a set $A=\{1,2,3,4,5\}$,
is there any terminology for the following partitions of $A$ ?
(1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$
(2) $A=\{\{1,2,3,4,5\}\}$.

REPLY [6 votes]: Suppose that $A$ is a non-empty set and $P_1,P_2$ are two partitions of $A$. We say that $P_1$ refines $P_2$ if for every $B\in P_1$ there is $C\in P_2$ such that $B\subseteq C$. This means that $P_1$ was a result of partitioning each part of $P_2$.
We also say that $P_1$ is finer than $P_2$; that $P_1$ is a refinement of $P_2$; or that $P_2$ is coarser than $P_1$.
It is a nice exercise to verify that the relation $x\prec y\iff x\text{ refines }y$ is a partial order over the partitions of $A$, in fact it is a lattice. Every two partitions has a least-refinement and a maximal-coarse partitions.
One can now see that the partition to singletons is the maximum of this partial order. Indeed it is the finest partition. Similarly the partition into a single part is the coarsest partition, and it is the minimum of the order.