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An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest ...
As another example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not
An antichain in is a subset of in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in . (However, some authors use the term "antichain" to mean strong antichain , a subset such that there is no element of the poset smaller than two distinct elements of the antichain.)
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 ...
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 ...
A nontrivial poset satisfying the descending chain condition is said to have deviation 0. Then, inductively, a poset is said to have deviation at most α (for an ordinal α) if for every descending chain of elements a 0 > a 1 >... all but a finite number of the posets of elements between a n and a n+1 have deviation less than α. The deviation ...
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements. A subset L {\displaystyle L} of a partially ordered set P {\displaystyle P} is said to be a lower set of P {\displaystyle P} if it is downward closed: if y ∈ L {\displaystyle y\in L} and x ≤ y {\displaystyle x\leq y} then x ∈ L ...