source : https://www.cse.iitb.ac.in/~nutan/courses/cs207-12/notes/lec7.pdf
Can we also say that partial order relation can be written as union of n anti chains ?
@Sherrinford check this. http://mathworld.wolfram.com/Antichain.html
@MiNiPanda We won’t consider empty subset, partitions of set are disjoint and non-empty
CHAIN : Let (A ,⋨) be a Poset .A subset of A is called Chain if every two elements in the subset are related.
ANTICHAIN: A subset of A is called Antichain if no two distinct elements in the subset are related.
THEOREM: Let (A ,⋨) be a Poset.Suppose the length of the longest chains in A is n.Then the elements in A can be partitioned into n disjoint Antichains.
@Vicky rix Won't {{1}} and {{2}} also be antichains, in addition to {{1},{2}}.
Edit: Found this source http://mathworld.wolfram.com/Antichain.html
In a POSET, if an element a is related to some other element b (ie there is an upward path), we call these two elements comparable.
And if not; incomparable.
A chain is nothing but a sequence of distinct comparable elements.
An antichain is a sequence of distinct incomparable elements. Note than a lone element is an antichain despite being comparable to itself.
If the length of the longest chain is n, there are are least n antichains.
Source: https://www.researchgate.net/figure/Two-possible-antichain-partitions-on-the-Hasse-diagram_fig3_313376303
Here, the longest chain is: $o\leq a\leq d\leq f\leq g\leq I$. Length is 5.
Antichain partitions are 6.
If the longest chain in a partial order is of length n, then the partial order can be written as a partition of n antichains.