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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.
The Hasse diagram of a six-element fence.. In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations:
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.
The subsets of a set form a partially ordered set (poset) for inclusion. Closure operators allow generalizing the concept of closure to any partially ordered set. Given a poset S whose partial order is denoted with ≤ , a closure operator on S is a function C : S → S {\displaystyle C:S\to S} that is
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).
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P op or P d.This dual order P op is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in P op if and only if y ≤ x holds in P.
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is
Order (differential equation) or order of highest derivative, of a differential equation; Leading-order terms; NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline; Order of convergence, a measurement of convergence; Order of derivation; Order of an entire function