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If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements. In our running example, {,,} and {} are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig. 5).
The parallel composition of P and Q, written P || Q, [7] P + Q, [2] or P ⊕ Q, [1] is defined similarly, from the disjoint union of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q having the same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs ...
The reason is that, in general, there are many different possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the minimal elements of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.
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.)
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.
The red subset = {,,,,,} has one greatest element, viz. 30, and one least element, viz. 1. These elements are also maximal and minimal elements , respectively, of the red subset. In mathematics , especially in order theory , the greatest element of a subset S {\displaystyle S} of a partially ordered set (poset) is an element of S {\displaystyle ...
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 ...
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 ...