antichain

anti- +‎ chain.

antichain (plural antichains)

1. (set theory, order theory) A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.
   • 1997, Winfried Just, Martin Weese, Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 140:

     First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.

   • 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, LNCS 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 ${\displaystyle N=\{1,2,3,\dots ,n\}}$. Call a family ${\displaystyle {\mathcal {F}}}$ of subsets of ${\displaystyle N}$^{[i.e., F ⊆ the power set P(N), which has partial order ⊆]} an antichain if no set of ${\displaystyle {\mathcal {F}}}$ contains another set of the family ${\displaystyle {\mathcal {F}}}$. What is the size of a largest antichain? Clearly, the family ${\displaystyle {\mathcal {F}}_{k}}$ of all ${\displaystyle k}$-sets satisfies the antichain property with ${\displaystyle \textstyle |{\mathcal {F}}_{k}|={\binom {n}{k}}}$. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size ${\displaystyle \textstyle {\binom {n}{[n/2]}}=\max _{k}{\binom {n}{k}}}$. Sperner's theorem now asserts that there are no larger ones.

• maximal antichain

• anti-China