In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name is meant to suggest that preorders are almost partial orders, but not quite, as they are not necessarily antisymmetric. A natural example of a preorder is the divides relation "x divides y" between integers, polynomials, or elements of a commutative ring. For example, the divides relation is reflexive as every integer divides itself. But the divides relation is not antisymmetric, because 1 divides -1 and -1 divides 1. It is to this preorder that "greatest" and "lowest" refer in the phrases "greatest common divisor" and "lowest common multiple" (except that, for integers, the greatest common divisor is also the greatest for the natural order of the integers). Preorders are closely related to equivalence relations and (non-strict) partial orders. Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X can equivalently be defined as an equivalence relation on X, together with a partial order on the set of equivalence class. Like partial orders and equivalence relations, preorders (on a nonempty set) are never asymmetric. A preorder can be visualized as 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. 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. As a binary relation, a preorder may be denoted or ,\leq,. In words, when one may say that b a or that a b, or that b to a. Occasionally, the notation ← or → is also used.

Definition

Let be a binary relation on a set P, so that by definition, is some subset of P \times P and the notation is used in place of Then is called a **' or **' if it is reflexive and transitive; that is, if it satisfies: A set that is equipped with a preorder is called a preordered set (or proset).

Preorders as partial orders on partitions

Given a preorder on S one may define an equivalence relation ,\sim, on S such that The resulting relation ,\sim, is reflexive since the preorder is reflexive; transitive by applying the transitivity of twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / \sim, which is the set of all equivalence classes of ,\sim. If the preorder is denoted by R^{+=}, then S / \sim is the set of R-cycle equivalence classes: x \in [y] if and only if x = y or x is in an R-cycle with y. In any case, on S / \sim it is possible to define if and only if That this is well-defined, meaning that its defining condition does not depend on which representatives of [x] and [y] are chosen, follows from the definition of ,\sim., It is readily verified that this yields a partially ordered set. Conversely, from any partial order on a partition of a set S, it is possible to construct a preorder on S itself. There is a one-to-one correspondence between preorders and pairs (partition, partial order). The relation is a preorder on S because always holds and whenever and both hold then so does Furthermore, for any A, B \in S, A \sim B if and only if ; that is, two sentences are equivalent with respect to if and only if they are logically equivalent. This particular equivalence relation A \sim B is commonly denoted with its own special symbol A \iff B, and so this symbol ,\iff, may be used instead of ,\sim. The equivalence class of a sentence A, denoted by [A], consists of all sentences B \in S that are logically equivalent to A (that is, all B \in S such that A \iff B). The partial order on S / \sim induced by which will also be denoted by the same symbol is characterized by if and only if where the right hand side condition is independent of the choice of representatives A \in [A] and B \in [B] of the equivalence classes. All that has been said of so far can also be said of its converse relation The preordered set is a directed set because if A, B \in S and if denotes the sentence formed by logical conjunction ,\wedge,, then and where C \in S. The partially ordered set is consequently also a directed set. See Lindenbaum–Tarski algebra for a related example.

Relationship to strict partial orders

If**** reflexivity is**** replaced**** with**** irreflexivity (while**** keeping transitivity****)**** then**** we**** get the definition**** of**** a strict**** partial order on**** P.**** For this**** reason****,**** the term**** **** is**** sometimes used**** for a strict**** partial order.**** That is, this is a binary relation ,<, on P that satisfies:

<ol> <li>[Irreflexivity](https://bliptext.com/articles/irreflexive-relation) or anti-reflexivity: a < a for all a \in P; that is, \,a < a is for all a \in P, and</li> <li>[Transitivity](https://bliptext.com/articles/transitive-relation): if for all </li> </ol>

Strict partial order induced by a preorder

Any preorder gives rise to a strict partial order defined by a < b if and only if and not. Using the equivalence relation ,\sim, introduced above, a < b if and only if and so the following holds The relation ,<, is a strict partial order and strict partial order can be constructed this way. the preorder is antisymmetric (and thus a partial order) then the equivalence ,\sim, is equality (that is, a \sim b if and only if a = b) and so in this case, the definition of ,<, can be restated as: But importantly, this new condition is used as (nor is it equivalent to) the general definition of the relation ,<, (that is, ,<, is defined as: a < b if and only if ) because if the preorder is not antisymmetric then the resulting relation ,<, would not be transitive (consider how equivalent non-equal elements relate). This is the reason for using the symbol "\lesssim" instead of the "less than or equal to" symbol "\leq", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that a \leq b implies

Preorders induced by a strict partial order

Using the construction above, multiple non-strict preorders can produce the same strict preorder ,<,, so without more information about how ,<, was constructed (such knowledge of the equivalence relation ,\sim, for instance), it might not be possible to reconstruct the original non-strict preorder from ,<., Possible (non-strict) preorders that induce the given strict preorder ,<, include the following: If a \leq b then The converse holds (that is, ) if and only if whenever a \neq b then a < b or b < a.

Examples

Graph theory

Computer science

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

Category theory

Other

Further examples: Example of a total preorder:

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. Left residual preorder induced by a binary relation Given a binary relation R, the complemented composition forms a preorder called the left residual, where denotes the converse relation of R, and denotes the complement relation of R, while \circ denotes relation composition.

Related definitions

If a preorder is also antisymmetric, that is, and implies a = b, 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 \in P. A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.

Uses

Preorders play a pivotal role in several situations:

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:

Interval

For the interval [a, b] 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 (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.