Antichain

English

Etymology

From Template:Prefix.

Noun

1. Template:Lb A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.
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   • 2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, Template:W 7730, page 245,

     We first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals.

   • 2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199,

     In 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set $N=\{1,2,3,\dots ,n\}$. Call a family $ℱ$ of subsets of $N$^{[i.e., F ⊆ the power set P(N), which has partial order ⊆]} an antichain if no set of $ℱ$ contains another set of the family $ℱ$. What is the size of a largest antichain? Clearly, the family ${ℱ}_k$ of all $k$-sets satisfies the antichain property with $|{ℱ}_k|=(\genfrac{}{}{0ex}{}{n}{k})$. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size $(\genfrac{}{}{0ex}{}{n}{[n/2]})=\underset{k}{\max }(\genfrac{}{}{0ex}{}{n}{k})$. Sperner's theorem now asserts that there are no larger ones.

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