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Set Partition Refinement Lattice

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The set of all partitions of a set can be partially ordered by refinement. A partition
π
1
is a refinement of partition
π
2
if every subset inside
π
1
fits inside a subset of
π
2
. For example,
{{a},{b},{c,d}}
is a refinement of
{{a,b},{c,d}}
; but
{{a},{b,c},{d}}
is not because the subset
{b,c}
is itself not contained in either subset of
{{a,b},{c,d}}
. This Demonstration shows the lattice formed by all the sets of partitions of a given set ordered by refinement.
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