Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY indicates that the column's property is always true the row's term (at the very left), while indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

Hasse diagram of the preorder x R y defined by x//4≤y//4 on the natural numbers. Equivalence classes (sets of elements such that x R y and y R x) are shown together as a single node. The relation on equivalence classes is a partial order.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder 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 divides and divides . 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 can equivalently be defined as an equivalence relation on , 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 . In words, when one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → is also used.

Definition

Let be a binary relation on a set so that by definition, is some subset of and the notation is used in place of Then is called a preorder or quasiorder if it is reflexive and transitive; that is, if it satisfies:

  1. Reflexivity: for all and
  2. Transitivity: if for all

A set that is equipped with a preorder is called a preordered set (or proset).[1]

Preorders as partial orders on partitions

Given a preorder on one may define an equivalence relation on such that

The resulting relation 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, which is the set of all equivalence classes of If the preorder is denoted by then is the set of -cycle equivalence classes: if and only if or is in an -cycle with In any case, on 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 and are chosen, follows from the definition of It is readily verified that this yields a partially ordered set.

Conversely, from any partial order on a partition of a set it is possible to construct a preorder on itself. There is a one-to-one correspondence between preorders and pairs (partition, partial order).

Example: Let be a formal theory, which is a set of sentences with certain properties (details of which can be found in the article on the subject). For instance, could be a first-order theory (like Zermelo–Fraenkel set theory) or a simpler zeroth-order theory. One of the many properties of is that it is closed under logical consequences so that, for instance, if a sentence logically implies some sentence which will be written as and also as then necessarily (by modus ponens). The relation is a preorder on because always holds and whenever and both hold then so does Furthermore, for any 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 is commonly denoted with its own special symbol and so this symbol may be used instead of The equivalence class of a sentence denoted by consists of all sentences that are logically equivalent to (that is, all such that ). The partial order on 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 and 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 and if denotes the sentence formed by logical conjunction then and where 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 . For this reason, the term strict preorder is sometimes used for a strict partial order. That is, this is a binary relation on that satisfies:

  1. Irreflexivity or anti-reflexivity: not for all that is, is false for all and
  2. Transitivity: if for all

Strict partial order induced by a preorder

Any preorder gives rise to a strict partial order defined by if and only if and not . Using the equivalence relation introduced above, if and only if and so the following holds

The relation is a strict partial order and every strict partial order can be constructed this way. If the preorder is antisymmetric (and thus a partial order) then the equivalence is equality (that is, if and only if ) and so in this case, the definition of can be restated as:
But importantly, this new condition is not used as (nor is it equivalent to) the general definition of the relation (that is, is not defined as: 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 "" instead of the "less than or equal to" symbol "", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that 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 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 then The converse holds (that is, ) if and only if whenever then or

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 on a set can be extended to a preorder on by taking the transitive closure and reflexive closure, The transitive closure indicates path connection in if and only if there is an -path from to

Left residual preorder induced by a binary relation

Given a binary relation the complemented composition forms a preorder called the left residual,[4] where denotes the converse relation of and denotes the complement relation of while denotes relation composition.

Related definitions

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 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

Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0 1 1 1 1 1 1 1 1 1
1 2 2 1 2 1 1 1 1 1
2 16 13 4 8 4 3 3 2 2
3 512 171 64 64 29 19 13 6 5
4 65,536 3,994 4,096 1,024 355 219 75 24 15
n 2n2 2n(n−1) 2n(n+1)/2 n
k=0
k!S(n, k)
n! n
k=0
S(n, k)
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110

Note that S(n, k) refers to Stirling numbers of the second kind.

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 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.

See also

Notes

  1. ^ For "proset", see e.g. Eklund, Patrik; Gähler, Werner (1990), "Generalized Cauchy spaces", Mathematische Nachrichten, 147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.
  2. ^ Pierce, Benjamin C. (2002). Types and Programming Languages. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff. ISBN 0-262-16209-1.
  3. ^ Robinson, J. A. (1965). "A machine-oriented logic based on the resolution principle". ACM. 12 (1): 23–41. doi:10.1145/321250.321253. S2CID 14389185.
  4. ^ In this context, "" does not mean "set difference".
  5. ^ Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, the Netherlands: Elsevier.

References