GATE CSE 1991 | Question: 01,xiv

Question

If the longest chain in a partial order is of length $n$, then the partial order can be written as a _____ of $n$ antichains.

Comments

source : https://www.cse.iitb.ac.in/~nutan/courses/cs207-12/notes/lec7.pdf

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Chains and antichains are there in GATE 2022 syllabus??

Answer 1

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.

Comments

could you please explain with an example ?

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@Arjun Sir ...not getting this Q/A... antichain  ??

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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 Partition of n antichains.

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

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https://www.cse.iitb.ac.in/~nutan/courses/cs207-12/notes/lec7.pdf

​​​​Reference for more details.

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Arjun sir not getting this question please explain sir ...

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"Subset A of S is called anti chain if no 2 elements in A are comparable".

If i take a total ordered poset(Z,<=)  containing n elements then we know every element is comparable to every other. If we take any two of these n, they will also be comparable. So we can take atmost 1 element in a subset A. Total number of such subsets would be n.

Basically in any poset(not only totally ordered) having chain of length n, no two of these n elements can go to the same subset otherwise it won't be an anti chain because "no 2 elements in a subset is comparable". If they belong to same subset then these 2 will be comparable hence won't form an anti chain. So atleast n different subsets for these n elements.

Doubt: Will it be n+1 (1 because of empty subset)? There maybe more such anti chains containing combinations of elements other than these n. But should empty set be considered as an anti chain?

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Just extra information about What is antichain?
A chain in S is a subset C of S in which each pair of elements is comparable; that is, C is totally ordered. An antichain in S is a subset A of S in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in A.



Reference:https://en.wikipedia.org/wiki/Antichain

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

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@Sherrinford check this.
http://mathworld.wolfram.com/Antichain.html

Answer 2

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.

Comments

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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@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

Answer 3

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.