In mathematics, especially order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are called partial orders to reflect the fact that not every pair of elements of a poset need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation between certain pairs of elements and allows one to reconstruct the whole partial order structure.
A familiar reallife example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the ancestordescendant relationship, but other pairs bear no such relationship.
A partial order is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that:
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.
For a, b distinct elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. If any two elements of a poset are comparable, the poset is called a chain (e.g. the natural numbers under order) and an antichain if any two elements are incomparable.
Standard examples of posets arising in mathematics include:
There are several notions of "greatest" and "least" element in a poset P, notably:
For example, consider the natural numbers, ordered by divisibility: 1 is a least element, as it divides all other elements, but this set does not have a greatest element nor does it have any maximal elements: any g divides 2g, so 2g is greater than g and g cannot be maximal. If instead we consider only the natural numbers that are greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element. In this poset, 60 is an upper bound (though not the least upper bound) of {2,3,5} and 2 is a lower bound of {4,6,8,12}.
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are:
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.
See also orders on the Cartesian product of totally ordered sets.
In some contexts, the partial order defined above is called a nonstrict (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 1to1 correspondence between all nonstrict and strict partial orders.
If "≤" is a nonstrict partial order, then the corresponding strict partial order "<" is the reflexive reduction given by:
Conversely, if "<" is a strict partial order, then the corresponding nonstrict partial order "<" is the reflexive closure "≤" given by:
This is the reason for using the notation "≤".
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 one 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 labeled elements:
Number of nelement binary relations of different types  

n  all  transitive  reflexive  preorder  partial order  total preorder  total order  equivalence relation 
0  1  1  1  1  1  1  1  1 
1  2  2  1  1  1  1  1  1 
2  16  13  4  4  3  3  2  2 
3  512  171  64  29  19  13  6  5 
4  65536  3994  4096  355  219  75  24  15 
OEIS  A002416  A006905  A053763  A000798  A001035  A000670  A000142  A000110 
The number of strict partial orders is the same as that of partial orders.
If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (A000112).
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. Every partial order can be extended to a total order (orderextension principle).^{[1]}
In computer science, algorithms for finding linear extensions of partial orders are called topological sorting.
Every poset (and every preorder) may be considered as a category in which every homset has at most one element. More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)o(x, y) = (x, z). Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if any, is an initial object, and the largest element, if any, a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphismclosed.
A functor from a poset category (a diagram indexed by a poset category) is a commutative diagram.
If P is a partially ordered set that has also been given the structure of a topological space, then it is customary to assume that is a closed subset of the topological product space . Under this assumption partial order relations are well behaved in limits; if and for all i, then ; see Deshpande (1968).
For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains at least the elements a and b.
Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b.
Sometimes the definitions are extended to allow a > b, in which case the interval is empty.
The halfopen intervals [a,b) and (a,b] are defined similarly.
A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural ordering.
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.
