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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.

Chains and antichains are there in GATE 2022 syllabus??

Can we also say that partial order relation can be written as union of n anti chains ?

not  always

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.

Can someone please give a valid example of Chain and demonstrating creation of Anti chains.

@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.

by

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.

@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.

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.

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Here 9 antichain partitions also possible right?

when all single elements are considered.

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