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In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the minimum number of chains needed to cover all elements. This number is called the width of the partial order.
A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes ...
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Formally, let P = (S, ≤) be a partially ordered set. Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties: x and y are incomparable in P, for every z in S, if z < x then z < y, and; for every z in S, if y < z then x < z.
Two elements and of a partially ordered set are called comparable if . If two elements are not comparable, they are called incomparable; that is, x {\displaystyle x} and y {\displaystyle y} are incomparable if neither x ≤ y nor y ≤ x . {\displaystyle x\leq y{\text{ nor }}y\leq x.}
In a partially ordered set there may be some elements that play a special role. The most basic example is given by the least element of a poset. For example, 1 is the least element of the positive integers and the empty set is the least set under the subset order. Formally, an element m is a least element if: m ≤ a, for all elements a of the ...
Let n be a positive integer, and let P be the partial order on the elements a i and b i (for 1 ≤ i ≤ n) in which a i ≤ b j whenever i ≠ j, but no other pairs are comparable. In particular, a i and b i are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.
For the other case, assume that there is some m in M with m ∨ a in F. Now if any element n in M is such that n ∨ b is in F, one finds that (m ∨ n) ∨ b and (m ∨ n) ∨ a are both in F. But then their meet is in F and, by distributivity, (m ∨ n) ∨ (a ∧ b) is in F too. On the other hand, this finite join of elements of M is clearly ...