Consider the set S = (A, B, C, D}. Consider the following 4 partitions π1, π2,…
2007
Consider the set S = (A, B, C, D}. Consider the following 4 partitions π1, π2, π3, π4 on S : π1 = {ABCD}' , π2 = {AB}' , {CD}' , π3 = {ABC}' , {D}' , π4 = A' , B' , C' , D'. Let P be the partial order on the set of partitions S' = {π1, π2, π3, π4} defined as follows : πi P πj if and only if πi refines πj. The poset diagram for (S', P ) is :
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Step 1 — Partitions: Write down each partition explicitly.
π1 = {A,B,C,D} (one block); π2 = {A,B}, {C,D} (two blocks); π3 = {A,B,C}, {D} (two blocks); π4 = {A}, {B}, {C}, {D} (four singletons).
Step 2 — Refinement relation: Recall πi P πj means πi refines πj (every block of πi is contained in some block of πj).
π4 refines π2: each singleton is contained in a block of π2 (for example {A}⊆{A,B}, {C}⊆{C,D}, etc.).
π4 refines π3: each singleton is contained in a block of π3 (for example {A},{B},{C}⊆{A,B,C}, {D}⊆{D}).
π2 refines π1 and π3 refines π1 because every block of π2 or π3 is contained in the single block {A,B,C,D}.
π2 and π3 are incomparable: neither refines the other. For instance, the block {C,D} of π2 is not contained in either block of π3, and the block {A,B,C} of π3 is not contained in either block of π2.
Step 3 — Hasse diagram (cover relations):
Covers: π4 is covered by π2 and by π3 (no partition strictly between π4 and those two); π2 and π3 are each covered by π1.
Conclusion: The poset is a diamond: the single-block partition π1 is the top element, the discrete partition π4 is the bottom element, and the two two-block partitions π2 and π3 occupy the two incomparable middle positions. This matches the diagram that places π1 at the top, π4 at the bottom, and π2 and π3 on the two middle corners of a diamond.