In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation.

Hasse diagram of the preorder x R y defined by x//4≤y//4 on the natural numbers. Due to the cycles, R is not anti-symmetric. If all numbers in a cycle are considered equivalent, a partial, even linear, order[1] is obtained. See first example below.

The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. 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 antisymmetric, 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 one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → or 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. In general, the corresponding graphs may contain cycles. 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.

Formal definitionEdit

Consider a homogeneous relation   on some given 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).[2] For emphasis or contrast to strict preorders, a preorder may also be referred to as a non-strict preorder.

If reflexivity is replaced with irreflexivity (while keeping transitivity) then the result is called a strict preorder; explicitly, a strict preorder on   is a homogeneous binary relation   on   that satisfies the following conditions:

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

A binary relation is a strict preorder if and only if it is a strict partial order. By definition, a strict partial order is an asymmetric strict preorder, where   is called asymmetric if   for all   Conversely, every strict preorder is a strict partial order because every transitive irreflexive relation is necessarily asymmetric. Although they are equivalent, the term "strict partial order" is typically preferred over "strict preorder" and readers are referred to the article on strict partial orders for details about such relations. In contrast to strict preorders, there are many (non-strict) preorders that are not (non-strict) partial orders.

Related definitionsEdit

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  

The notion of a preordered set   can be formulated in a categorical framework as a thin category; that is, as a category with at most one morphism from an object to another. Here the objects correspond to the elements of   and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can be understood as an enriched category, enriched over the category  

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.

ExamplesEdit

Graph theoryEdit

  • (see figure above) By x//4 is meant the greatest integer that is less than or equal to x divided by 4, thus 1//4 is 0, which is certainly less than or equal to 0, which is itself the same as 0//4.
  • 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 in graph theory.

Computer scienceEdit

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

OtherEdit

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.
  • A category with at most one morphism from any object x to any other object y is a preorder. Such categories are called thin. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.

Example of a total preorder:

UsesEdit

Preorders play a pivotal role in several situations:

ConstructionsEdit

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,[6] where   denotes the converse relation of   and   denotes the complement relation of   while   denotes relation composition.

Preorders and partial orders on partitionsEdit

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.

Preorders and strict preordersEdit

Strict preorder induced by a preorder

Given a preorder   a new relation   can be defined by declaring that   if and only if   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 preorder

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  

Number of preordersEdit

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 2n2n 2n(n+1)/2   n!  
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.

IntervalEdit

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 alsoEdit

NotesEdit

  1. ^ on the set of numbers divisible by 4
  2. ^ 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.
  3. ^ Pierce, Benjamin C. (2002). Types and Programming Languages. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff. ISBN 0-262-16209-1.
  4. ^ 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.
  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.
  6. ^ In this context, " " does not mean "set difference".

ReferencesEdit

  • 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