In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder. An antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation.
The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol
In words, when , one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation
To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.
Consider some set P and a binary relation P. Then preorder, or quasiorder, if it is reflexive and transitive; i.e., for all a, b and c in P, we have that:
A set that is equipped with a preorder is called a preordered set (or proset).^{[1]}
If a preorder is also antisymmetric, that is, and implies , then it is a partial order.
On the other hand, if it is symmetric, that is, if implies , then it is an equivalence relation.
A preorder is total if or for all a, b.
Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category; i.e., as a category with at most one morphism from an object to another. Here the objects correspond to the elements of P, and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can be understood as an enriched category, enriched over the category .
A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.
In computer science, one can find examples of the following preorders.
Example of a total preorder:
Preorders play a pivotal role in several situations:
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R^{+=}. The transitive closure indicates path connection in if and only if there is an R-path from x to y.
Given a preorder ? on S one may define an equivalence relation ~ on S such that if and only if and . (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.)
Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~. Note that if the preorder is R^{+=}, is the set of R-cycle equivalence classes: if and only if or x is in an R-cycle with y. In any case, on we can define if and only if . By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.
Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).
For a preorder "?", a relation "<" can be defined as if and only if ( and not ), or equivalently, using the equivalence relation introduced above, (a ? b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. If the preorder is antisymmetric, hence a partial order "a < b if and only if ( and ).
(We do not define the relation "<" as if and only if ( and ). Doing so would cause problems if the preorder was not antisymmetric, as the resulting relation "<" would not be transitive (think of how equivalent non-equal elements relate).)
Conversely we have if and only if or . This is the reason for using the notation "?"; "a b implies that or .
Note that with this construction multiple preorders "?" can give the same relation "<", so without more information, such as the equivalence relation, "?" cannot be reconstructed from "<". Possible preorders include the following:
Given a binary relation , the complemented composition forms a preorder called the left residual, where denotes the converse relation of , and denotes the complement relation of , while denotes relation composition.
Elements | Any | 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 | 355 | 219 | 75 | 24 | 15 | |||
n | 2^{n2} | 2^{n2-n} | ?n k=0 k! S(n, k) |
n! | ?n k=0 S(n, k) | |||
OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
For , the interval is the set of points x satisfying and , also written . It contains at least the points a and b. One may choose to extend the definition to all pairs . The extra intervals are all empty.
Using the corresponding strict relation "<", one can also define the interval as the set of points x satisfying and , also written . An open interval may be empty even if .
Also and can be defined similarly.