A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the ancestor-descendant relationship but, in general, other pairs bear no such relationship.
In other words, a partial order is an antisymmetric preorder.
A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.
Standard examples of posets arising in mathematics include:
All three can similarly be defined for the Cartesian product of more than two sets.
See also .
In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In these contexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive and transitive, and therefore asymmetric. In other words, asymmetric (hence irreflexive) and transitive.
Thus, for all a, b, and c in P, we have that:
There is a 1-to-1 correspondence between all non-strict and strict partial orders.
If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the reflexive reduction given by:
Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "<" is the "≤" given by:
Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.
The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.
Any of these four relations ≤, <, ≥, and > on a given set uniquely determine the other three.
In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.
Sequence [A001035] in OEIS gives the number of partial orders on a set of n elements:
The number of strict partial orders is the same as that of partial orders.
A total order T is a linear extension of a partial order P if, whenever x ≤ y in P it also holds that x ≤ y in T. In computer science, algorithms for finding linear extensions of partial orders are called topological sorting.
Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.
Also [a,b) and (a,b] can be defined similarly.
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