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

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I am not sure to which extent these names are used, but Google Books returns some results for trivial partition for the second one and discrete partition or singleton partition for the first one. – Martin Sleziak Jun 3 '12 at 14:23
Okay. Thanks! Maybe this is formal name. – John Smith Jun 3 '12 at 14:27

I do not think that there is terminology, which would be universally agreed. However, Google search suggests that some people us 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.