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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.
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 ).
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 mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which
Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set can equivalently be defined as an equivalence relation on , together with a
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.