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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?

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

**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.

### 2 Comments

@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

partitionof n antichains.