# Set Partition Refinement Lattice

Set Partition Refinement Lattice

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

π

1

π

2

π

1

π

2

{{a},{b},{c,d}}

{{a,b},{c,d}}

{{a},{b,c},{d}}

{b,c}

{{a,b},{c,d}}