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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.
A Hasse diagram of the factors of 60 ordered by the is-a-divisor-of relation. In order theory, a Hasse diagram (/ ˈ h æ s ə /; German:) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q, [7] P * Q, [2] or P ⧀ Q, [1] is the partially ordered set whose elements are the disjoint union of the elements of P and Q. In P; Q, two elements x and y that both belong to P or that both belong to Q have the same order relation that they do in P ...
An ordered set in which every pair of elements is comparable is called totally ordered. Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube. [2] A partially ordered set is series-parallel if and only if it does not have four elements forming a fence. [3]
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 ...
The complement of the comparability graph of an interval order (, ≤) is the interval graph (,). Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).
A partially ordered group G is called integrally closed if for all elements a and b of G, if a n ≤ b for all natural n then a ≤ 1. [1]This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. [2]