Find the height h of the poset P X P shown below as well as

Find the height h of the poset P = (X, P) shown below as well as a maximum chain and a partition of X into h antichains. (Explain every step.)

The height of the poset represents the maximal number of element present in a maximum chain.

In the given poset (X,P)

a) We need to find the maximally connected chain

Height of the Poset = 8 (since there contains two chains of maximal chain size 8 in the given poset)

First chain of size 8 -> {14,26,7,9,24,4,3}

Second chain of size 8 - {14,6,19,20,5,4,3,22}

b) For calculating the number of partitions required, since we have found that the size of chain is 8

Hence using Folklore theorem

A poset P of height h can be partitioned into h antichains.

Hence the chain can be partitioned into 8 antichains