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.