In order 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 bound in P, that is,
In the case where P is ordered by inclusion, and closed under subsets, but does not contain the empty set, this is simply a family of pairwise disjoint sets.
A strong upwards antichainB is a subset of P in which no two distinct elements have a common upper bound in P. Authors will often omit the "upwards" and "downwards" term and merely refer to strong antichains. Unfortunately, there is no common convention as to which version is called a strong antichain. In the context of forcing, authors will sometimes also omit the "strong" term and merely refer to antichains. To resolve ambiguities in this case, the weaker type of antichain is called a weak antichain.
If (P, ≤) is a partial order and there exist distinct x, y ∈ P such that {x, y} is a strong antichain, then (P, ≤) cannot be a lattice (or even a meet semilattice), since by definition, every two elements in a lattice (or meet semilattice) must have a common lower bound. Thus lattices have only trivial strong antichains (i.e., strong antichains of cardinality at most 1).
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 strongantichain 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 strongantichain is an...
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...
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...
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...
monotonicity is replaced by the strict order < {\displaystyle <} , one obtains a stronger requirement. A function with this property is called strictly increasing...
problem Hitchcock transportation problem Assignment problem Chains and antichains in partially ordered sets System of distinct representatives Covers and...
(antisymmetric). a ≤ b {\displaystyle a\leq b} or b ≤ a {\displaystyle b\leq a} (strongly connected, formerly called total). Reflexivity (1.) already follows from...
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...
n)} , where μ {\displaystyle \mu } is the number of edges between its strongly connected components. More recent research has explored efficient ways...
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...
x\lesssim y{\text{ and }}y\lesssim z} then x ≲ z . {\displaystyle x\lesssim z.} Strong connectedness: For all x and y , {\displaystyle x{\text{ and }}y,} x ≲...
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...
y , z } } {\displaystyle \{\{\,\},\{x\},\{x,y,z\}\}} is a chain. An antichain is a subset of a poset in which no two distinct elements are comparable...
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