A formal name for "smallest" and "largest" partition

I do not think that there is terminology, which would be universally agreed. However, Google search suggests that some people use the following names: trivial partition for the partition $\{A\}$ and discrete partition or singleton partition for the partition the partition $\{\{a\};a\in A\}$.

Interestingly enough ProofWiki Article suggests that trivial partition is used for both of them and calls the partition $\{A\}$ singleton partition, although no reference is given there. The name used for $\{\{a\}; a\in A\}$ at ProofWiki is partition of singletons.

Some examples of the use of the names mentioned above:

• Introduction to Abstract Algebra By W. Keith Nicholson uses the name trivial partition and singleton partition, see p.19

• An Introduction to Measure Theory By Terence Tao uses the name discrete partition, see p.68

Google Books:

• trivial partition
• discrete partition
• singleton partition
• partition of singletons

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.