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If the longest chain in a partial order is of length $n$, then the partial order can be written as a _____ of $n$ antichains.
asked in Set Theory & Algebra by Veteran (68.8k points) | 551 views

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+9 votes
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Suppose the length of the longest chain in a partial order is n. Then the elements in the poset can be partitioned into n disjoint antichains.
answered by Veteran (35.8k points)
could you please explain with an example ?
@Arjun Sir ...not getting this Q/A... antichain  ??

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.

each element of chain must be in different antichains so if chain having n elements so there must exist n disjoint antichains .
+4 votes

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.

answered by Veteran (15.4k points)
Ex : number of anti chains in [P(A),subset] where A = {1,2} is 3 (i.e)

                        1) {} ,  

                        2) {1,2} ,

                        3) {1} and {2} as {1} and {2} are not related they can form a single anti-chain.


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