Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
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.

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.

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.

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 edit

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 edit

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 edit

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 edit

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 edit

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:

  • Define   as   (that is, 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 the symbols   and   are not needed.
  • Define   as " " (that is, take the inverse complement of the relation), which corresponds to defining   as "neither  "; these relations   and   are in general not transitive; however, if they are then   is an equivalence; in that case " " is a strict weak order. The resulting preorder is connected (formerly called total); that is, a total preorder.

If   then   The converse holds (that is,  ) if and only if whenever   then   or  

Examples edit

Graph theory edit

  • The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where   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  ). 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 antisymmetry property).
  • The graph-minor relation is also a preorder.

Computer science edit

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

Category theory edit

  • A category with at most one morphism from any object x to any other object y is a preorder. Such categories are called thin. Here the objects correspond to the elements of   and there is one morphism for objects which are related, zero otherwise. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.
  • Alternately, a preordered set can be understood as an enriched category, enriched over the category  

Other edit

Further examples:

  • Every finite topological space gives rise to a preorder on its points by defining   if and only if x belongs to every neighborhood of y. Every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a one-to-one correspondence between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.
  • The relation defined by   if   where f is a function into some preorder.
  • The relation defined by   if 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.

Example of a total preorder:

Constructions edit

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 edit

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 edit

Preorders play a pivotal role in several situations:

Number of preorders edit

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!S(n, k)
n! n
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:

  • for  
    • 1 partition of 3, giving 1 preorder
    • 3 partitions of 2 + 1, giving   preorders
    • 1 partition of 1 + 1 + 1, giving 19 preorders
    I.e., together, 29 preorders.
  • for  
    • 1 partition of 4, giving 1 preorder
    • 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving   preorders
    • 6 partitions of 2 + 1 + 1, giving   preorders
    • 1 partition of 1 + 1 + 1 + 1, giving 219 preorders
    I.e., together, 355 preorders.

Interval edit

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 edit

Notes edit

  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 edit

  • Schmidt, Gunther, "Relational Mathematics", Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press, 2011, ISBN 978-0-521-76268-7
  • Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9