antichain

Etymology
.

Noun

 * 1)  A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.
 * 2) * 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, 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.
 * 1) * 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 $$\mathcal{F}$$ of subsets of $$N$$[i.e., F ⊆ the power set P(N), which has partial order ⊆] an antichain if no set of $$\mathcal{F}$$ contains another set of the family $$\mathcal{F}$$. What is the size of a largest antichain? Clearly, the family $$\mathcal{F}_k$$ of all $$k$$-sets satisfies the antichain property with $$\textstyle|\mathcal{F}_k|=\binom{n}{k}$$. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size $$\textstyle\binom{n}{[n/2]}=\max_k\binom{n}{k}$$. Sperner's theorem now asserts that there are no larger ones.
 * 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 $$\mathcal{F}$$ of subsets of $$N$$[i.e., F ⊆ the power set P(N), which has partial order ⊆] an antichain if no set of $$\mathcal{F}$$ contains another set of the family $$\mathcal{F}$$. What is the size of a largest antichain? Clearly, the family $$\mathcal{F}_k$$ of all $$k$$-sets satisfies the antichain property with $$\textstyle|\mathcal{F}_k|=\binom{n}{k}$$. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size $$\textstyle\binom{n}{[n/2]}=\max_k\binom{n}{k}$$. Sperner's theorem now asserts that there are no larger ones.

Translations

 * Finnish: antiketju
 * Italian: anticatena
 * Spanish: anticadena