# Preorder

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In mathematics, especially in order theory, a preorder or quasiorder 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 (that are not partial orders) are 'almost' (partial) orders, but not quite; they're neither necessarily 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 necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.

In words, when ab, one may say that b covers a or that a precedes b, 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.

## 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:

aa (reflexivity)
if ab and bc then ac (transitivity)

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

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

On the other hand, if it is symmetric, that is, if ab implies ba, 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 preordered set P can be defined as a category with objects the elements of P, and each hom-set having at most one element (one for objects which are related, zero otherwise).

Alternately, a preordered set can be understood as an enriched category, enriched over the category 2 = (0→1).

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class. Preordered classes can be defined as thin categories, i.e. as categories with at most one morphism from an object to another.

## Examples

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

Example of a total preorder:

## Uses

Preorders play a pivotal role in several situations:

## 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 (ab and ab).

(Alternatively, for a preorder "$\lesssim$ ", a relation "<" can be defined as a < b if and only if (a $\lesssim$ b and ab). 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 ab 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:

## Number of preorders

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

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.