theory, a subset A of a partially ordered set P is a strong downwards antichain if it is an antichain in which no two distinct elements have a common lower...

In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are...

said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and...

adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an...

combinatorial property is that all of the antichains of the forcing poset are countable. An (strong) antichain A {\displaystyle A} of P {\displaystyle \mathbb...

is that every tree of height ω1 either has a branch of length ω1 or an antichain of cardinality ℵ1. The generalized Suslin hypothesis says that for every...

sets is a strict subset of another is called a Sperner family, or an antichain of sets, or a clutter. For example, the family of k-element subsets of...

monotone Boolean functions of n variables. Equivalently, it is the number of antichains of subsets of an n-element set, the number of elements in a free distributive...

strict Sperner poset is a graded poset in which all maximum antichains are rank levels. A strongly Sperner poset is a graded poset which is k-Sperner for all...

inclusion. Antichain principle: Every partially ordered set has a maximal antichain. Equivalently, in any partially ordered set, every antichain can be extended...

characterizing the height of a partially ordered set in terms of partitions into antichains can be formulated as the perfection of the comparability graph of the...

antichain in the order and an independent set in the graph. Thus, a coloring of a comparability graph is a partition of its elements into antichains,...

monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all...

of elements of the set in one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology...

divisibility, is not a well-quasi-order: the prime numbers are an infinite antichain (see Pic.2). ( N k , ≤ ) {\displaystyle (\mathbb {N} ^{k},\leq )} , the...

program P is minimal among the models of P relative to set inclusion. The antichain property If I and J are stable models of the same logic program then I...

poset. A poset is algebraic if it has a base of compact elements. Antichain. An antichain is a poset in which no two elements are comparable, i.e., there...

PFA which instead of arbitrary dense subsets applies only to maximal antichains of size ω 1 {\displaystyle \omega _{1}} . Martin's maximum is the strongest...

here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given...

monotonicity is replaced by the strict order < {\displaystyle <} , one obtains a stronger requirement. A function with this property is called strictly increasing...

theorem implies the statement that there can be no infinite antichains, because an infinite antichain is a set that does not contain any pair related by the...

analytical hierarchy. (This is not the same as an analytic set) antichain An antichain is a set of pairwise incompatible elements of a poset anti-foundation...

problem Hitchcock transportation problem Assignment problem Chains and antichains in partially ordered sets System of distinct representatives Covers and...

lemma was referenced on The Simpsons in the episode "Bart's New Friend". Antichain – Subset of incomparable elements Bourbaki–Witt theorem Chain-complete...

set of maximal elements of a subset S {\displaystyle S} is always an antichain, that is, no two different maximal elements of S {\displaystyle S} are...

(antisymmetric). a ≤ b {\displaystyle a\leq b} or b ≤ a {\displaystyle b\leq a} (strongly connected, formerly called total). Reflexivity (1.) already follows from...

values of both parameters, gives infinitely many incomparable graphs (an antichain in the homomorphism preorder). Other properties, such as density of the...

n)} , where μ {\displaystyle \mu } is the number of edges between its strongly connected components. More recent research has explored efficient ways...

subset of P over M. P satisfies the countable chain condition if every antichain in P is at most countable. This implies that V and V[G] have the same...

x\neq y} then x R y {\displaystyle xRy} or y R x {\displaystyle yRx} . Strongly connected: for all x , y ∈ X , {\displaystyle x,y\in X,} x R y {\displaystyle...