Preorder

This article is about the binary relation. For the graph vertex ordering, see Depth-first search. For other uses, see Preorder (disambiguation).

In mathematics, especially in order theory, a preorder or quasi-order is a binary relation that is reflexive and transitive. All partial orders and equivalence relations are preorders, but preorders are more general.

The name 'preorder' comes from the idea that preorders are 'almost' (partial) orders, but not quite; they're neither anti-symmetric nor symmetric. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not anti-symmetric, some of the ordinary intuition that a student may have with regards to the symbol ≤ may not apply. On the other hand, a pre-order can be used, in a straight-forward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worth-while, depending on the problem domain being studied.

In words, when a ≤ b, one may say that b covers a or that b precedes a, or that b reduces to a. Occasionally, the notation ← or $\lesssim$ is used instead of ≤.

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. Note that, in general, the corresponding graphs may be cyclic graphs: preorders may have cycles in them. 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 may have many disconnected components. The diamond lemma is an important result for certain kinds of preorders.

Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.^{[citation needed]}

Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)

if a ≤ b and b ≤ c then a ≤ c (transitivity)

Note that an alternate definition of preorder requires the relation to be irreflexive. However, as this article is examining preorders as a logical extension of non-strict partial orders, the current definition is more intuitive.

A set that is equipped with a preorder is called a preordered set (or proset).

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.

On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.

A preorder which is preserved in all contexts (i.e. respected by all functions on P) is called a precongruence. A precongruence which is also symmetric (i.e. is an equivalence relation) is a congruence relation.

Equivalently, a preorder on the set P can be defined as a category with object set P where every homset has at most one element (one for objects which are related, zero otherwise).

Examples

The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where x ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for every pair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partially ordered sets (preorders satisfying an additional anti-symmetry property).
Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not 1-to-1.
A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features.
The relation defined by $x \le y$ iff $f(x) \le f(y)$ , where f is a function into some preorder.
The relation defined by $x \le y$ iff there exists some injection from x to y. Injection may be replaced by surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.
The embedding relation for countable total orderings.
The graph-minor relation in graph theory.

In computer science, one can find examples of the following preorders.

The subtyping relations are usually preorders.
Simulation preorders are preorders (hence the name).
Reduction relations in abstract rewriting systems.

Example of a total preorder:

Preference, according to common models.

Constructions

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 R: x R⁺y if and only if there is an R-path from x to y.

Given a preorder $\lesssim$ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a $\lesssim$ b and b $\lesssim$ a. (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⁺⁼, S / ~ is the set of R-cycle equivalence classes: x ∈ [y] if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x $\lesssim$ y. 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 " $\lesssim$ ", a relation "<" can be defined as a < b if and only if (a $\lesssim$ b and not b $\lesssim$ a), or equivalently, using the equivalence relation introduced above, (a $\lesssim$ 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 anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).

(Alternatively, for a preorder " $\lesssim$ ", a relation "<" can be defined as a < b if and only if (a $\lesssim$ b and a ≠ b). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before.)

Conversely we have a $\lesssim$ b if and only if a < b or a ~ b. This is the reason for using the notation " $\lesssim$ "; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.

Note that with this construction multiple preorders " $\lesssim$ " can give the same relation "<", so without more information, such as the equivalence relation, " $\lesssim$ " cannot be reconstructed from "<". Possible preorders include the following:

Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations $\lesssim$ and ~.
Define a $\lesssim$ b as "not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a ~ b as "neither a < b nor b < a"; these relations $\lesssim$ and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

Number of preorders

Number of n-element 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

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 n=3:
- 1 partition of 3, giving 1 preorder
- 3 partitions of 2+1, giving 3 × 3 = 9 preorders
- 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

for n=4:
- 1 partition of 4, giving 1 preorder
- 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders
- 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders
- 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

Interval

For a $\lesssim$ b, the interval [a,b] is the set of points x satisfying a $\lesssim$ x and x $\lesssim$ b, also written a $\lesssim$ x $\lesssim$ b. It contains at least the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.

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.

References

Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9