"Antichain" Example Sentences
1. A set with no antichains is called chain complete.
2. An antichain is a subset of a partially ordered set.
3. The maximal antichains of a partially ordered set can be computed using the Dilworth's theorem.
4. A finite partially ordered set with n elements has at most 2^(n/2) antichains.
5. The smallest antichain of a partially ordered set is just one element.
6. In a poset, a maximal antichain cannot be contained within any other antichain.
7. A partially ordered set is said to have a finite antichain if it has a largest antichain, and an infinite antichain otherwise.
8. The width of a partially ordered set is the cardinality of its largest antichain.
9. The largest antichains in the Boolean lattice are the complements of singletons.
10. Every finite partially ordered set has a unique antichain decomposition.
11. The size of the largest antichain in a finite partially ordered set is the same as the size of the smallest chain partition.
12. The antichain polytope of a partially ordered set is the convex hull of its characteristic vectors.
13. A partially ordered set is said to be antichain-resistant if it can contain infinite antichains but any subset that is isomorphic to an infinite antichain must contain an infinite antichain.
14. A partially ordered set is antichain-dense if every finite subset has a maximal antichain.
15. The antichain cover number of a poset is the minimum number of antichains whose union covers the set.
16. The Dilworth number of a partially ordered set is the size of its largest antichain, and the minimum number of chains into which the set can be partitioned.
17. A partially ordered set is said to be bounded antichain-complete if every bounded antichain has a least upper bound.
18. The thick antichains of a partially ordered set are the antichains that cannot be enlarged to a strictly larger antichain.
19. A poset is said to be uniform if all of its maximal antichains have the same cardinality.
20. An antichain is said to be maximal if it cannot be extended by adding any other comparable elements to it.
21. A subset of a poset is linearly independent if and only if it forms an antichain.
22. The antichain generating function of a poset is the sum over all antichains of their characteristic functions times x to the size of the antichain.
23. In the poset of subsets of a set, the minimal antichains are the singleton sets.
24. A poset is said to be strongly chain antichain if any two maximal chains intersect in at most one element.
25. The Sperner property of a poset is the condition that every antichain has a least upper bound in the poset.
26. The poset on a finite set in which two elements are comparable if one is a subset of the other is called the Boolean algebra.
27. An antichain is called indispensable if there are no comparable elements besides the ones in the antichain.
28. In a very narrow sense, the word "antichain" refers only to maximal antichains.
29. An antichain is said to be dense if it contains two elements x and y such that every element between x and y is comparable to neither x nor y.
30. The de Bruijn-Erdős theorem states that for every positive integer n, there is a finite partially ordered set of width n and cardinality 2n - 1, which is the maximum possible size for a such set without an antichain of size n+1.
Common Phases
1. Determine the largest
antichain in the given poset;
2. Identify whether the given poset is an
antichain itself;
3. Compute the size of the smallest
antichain that covers the entire poset;
4. Construct a Hasse diagram for the given
antichain;
5. Prove that the given set of subsets is an
antichain by showing that no two elements contain each other;
6. Use Dilworth's Theorem to find the minimum number of chains that cover the entire
antichain;
7. Apply the Erdős–Szekeres Theorem to prove that the given poset contains a chain or an
antichain of a certain size;
8. Show that the given
antichain is maximal by proving that no element can be added to it without violating the
antichain property.