How would I apply Dilworth's Theorem to the following set: $S=\{ 0,1,4,6,7,8,9\}$ where any element $a$ is less than or equal to $b$?

Question

This is the set $S=\{ 0,1,4,6,7,8,9\}$ under the order defined by divisibility. I know we have to find antichains and chains, and that the maximum number of partitions of $S$ into chains should equal the cardinality of the antichains, but I don't understand union of chains and how that would fit in it all.

Answer 1

A chain is a subset of a partially ordered set in which any two elements are comparable, and an antichain is a subset where any two distinct elements are incomparable.

The size of a maximum antichain in this poset is 4 (for example, all elements of the set $\{6,7,8,9\}$ are mutually indivisible). The minimum number of chains in a partition of this poset into chains therefore is 4 by Dilworth's theorem. An example of such a minimal partition is $\{1,4,8,0\}$ (because $1|4|8|0$), $\{6\}$, $\{7\}$, $\{9\}$.