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In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other.
Order theory captures the intuition of orders that arises from such examples in a general setting. ... Then ≤ is a partial order if it is reflexive, antisymmetric ...
Order theory knows many completion procedures to turn posets into posets with additional completeness properties. For example, the ideal completion of a given partial order P is the set of all ideals of P ordered by subset inclusion. This construction yields the free dcpo generated by P.
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).
The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse. Order-embedding . A function f between posets P and Q is an order-embedding if, for all elements x , y of P , x ≤ y (in P ) is equivalent to f ( x ) ≤ f ( y ) (in Q ).
This intuition, in the context of denotational semantics, was the motivation behind the development of domain theory. The dual notion of a directed-complete partial order is called a filtered-complete partial order. However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly.
The Dedekind–MacNeille completion may be exponentially larger than the partial order it comes from, [12] and the time bounds for such algorithms are generally stated in an output-sensitive way, depending both on the number n of elements of the input partial order, and on the number c of elements of its completion.
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.