# A formal name for “smallest” and “largest” partition

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) $A=\{\{1,2,3,4,5\}\}$.

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

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.