a) Symmetric chain partition for the power set P([5]) of [5] := {1, 2, 3, 4, 5} under the partial order of set inclusion are: {[1, 2, 3, 4, 5]}, {[1], [2], [3], [4], [5]}, {[1, 2], [3, 4], [5]}, {[1], [2, 3], [4, 5]}, {[1, 2, 3], [4, 5]}, {[1, 2, 4], [3, 5]}, {[1, 2, 5], [3, 4]}, {[1, 3, 4], [2, 5]}, {[1, 3, 5], [2, 4]}, {[1, 4, 5], [2, 3]}, {[1, 2], [3], [4], [5]}, {[2, 3], [1], [4], [5]}, {[3, 4], [1], [2], [5]}, {[4, 5], [1], [2], [3]}, {[1], [2, 3, 4], [5]}, {[1], [2, 3, 5], [4]}, {[1], [2, 4, 5], [3]}, {[1], [3, 4, 5], [2]}, {[2], [3, 4, 5], [1]}, {[1, 2], [3, 4, 5]}, {[1, 3], [2, 4, 5]}, {[1, 4], [2, 3, 5]}, {[1, 5], [2, 3, 4]}, {[1, 2, 3, 4], [5]}, {[1, 2, 3, 5], [4]}, {[1, 2, 4, 5], [3]}, {[1, 3, 4, 5], [2]}, {[2, 3, 4, 5], [1]}.
By using the Hasse diagram, one can verify that each element is included in exactly one set of every symmetric chain partition. Consequently, the collection of all symmetric chain partitions of the power set P([5]) is a partition of the power set P([5]), which partitions all sets according to their sizes. Hence, there are 2n−1 = 16 chains in the power set P([5]).
b) There are 5 maximal clusters, namely antichains of ([5]): {[1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]}.
These maximal antichains are indeed maximal as there is no inclusion relation between any two elements in the same antichain, and adding any other element in the power set to such an antichain would imply a relation of inclusion between some two elements of the extended antichain, which contradicts the definition of antichain. The maximal antichains found are, indeed, maximal.
c) The maximal chains of P([5]) are: {[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5]}, {[1], [1, 2], [1, 2, 3], [1, 2, 3, 5], [1, 2, 3, 4, 5]}, {[1], [1, 2], [1, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4, 5]}, {[1], [1, 2], [1, 2, 4], [1, 2, 4, 5], [1, 2, 3, 4, 5]}, {[1], [1, 3], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5]}, {[1], [1, 3], [1, 2, 3], [1, 2, 3, 5], [1, 2, 3, 4, 5]}, {[1], [1, 4], [1, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4, 5]}, {[1], [1, 4], [1, 3, 4], [1, 3, 4, 5], [1, 2, 3, 4, 5]}, {[1], [1, 5], [1, 4, 5], [1, 3, 4, 5], [1, 2, 3, 4, 5]}, {[1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [2, 3, 4, 5]}, {[1, 2], [1, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4, 5], [2, 3, 4, 5]}, {[1, 3], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [2, 3, 4, 5]}, {[1, 4], [1, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4, 5], [2, 3, 4, 5]}, {[1, 5], [1, 4, 5], [1, 3, 4, 5], [1, 2, 3, 4, 5], [2, 3, 4, 5]}.The minimal antichain partitions of P([5]) are: {{[1], [2], [3], [4], [5]}, {[1, 2], [3, 4], [5]}, {[1, 3], [2, 4], [5]}, {[1, 4], [2, 3], [5]}, {[1, 5], [2, 3, 4]}}, {[1], [2, 3], [4, 5]}, {[2], [1, 3], [4, 5]}, {[3], [1, 2], [4, 5]}, {[4], [1, 2, 3], [5]}, {[5], [1, 2, 3, 4]}}.
The maximal chains are maximal since there is no other chain that extends it. The antichain partitions are minimal since there are no less elements in any other partition.
d) The Möbius function values µ(a, x) near the vertices x on the Hasse diagram of the h8edba poset where x = a, b, c, d, e, f, g, h are:{µ(a, a) = 1}, {µ(a, b) = -1, µ(b, b) = 1}, {µ(a, c) = -1, µ(c, c) = 1}, {µ(a, d) = -1, µ(d, d) = 1}, {µ(a, e) = -1, µ(e, e) = 1}, {µ(a, f) = -1, µ(f, f) = 1}, {µ(a, g) = -1, µ(g, g) = 1}, and {µ(a, h) = -1, µ(h, h) = 1}.
Therefore, symmetric chain partition and maximal clusters of the poset are found. Furthermore, maximal chains and minimal antichain partitions of P([5]) have also been found along with explanations of maximal chains and minimal antichain partitions. Lastly, Möbius function values µ(a,x) near the vertices x on the Hasse diagram of the h8edba poset have been computed.
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