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![]() ![]() All definitions tacitly require the homogeneous relation be transitive: for all if and then |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Preorder.png/220px-Preorder.png)
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
[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:
- Reflexivity: for all and
- 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:
- Irreflexivity or anti-reflexivity: not for all that is, is false for all and
- 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.
- Asymptotic order causes a preorder over functions . The corresponding equivalence relation is called asymptotic equivalence.
- Polynomial-time, many-one (mapping) and Turing reductions are preorders on complexity classes.
- Subtyping relations are usually preorders.[2]
- Simulation preorders are preorders (hence the name).
- Reduction relations in abstract rewriting systems.
- The encompassment preorder on the set of terms, defined by if a subterm of t is a substitution instance of s.
- Theta-subsumption,[3] which is when the literals in a disjunctive first-order formula are contained by another, after applying a substitution to the former.
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.
- A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features.
- 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:
- Preference, according to common models.
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:
- Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
- Preorders may be used to define interior algebras.
- Preorders provide the Kripke semantics for certain types of modal logic.
- Preorders are used in forcing in set theory to prove consistency and independence results.[5]
Number of preorders
[edit]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:
- for
- 1 partition of 3, giving 1 preorder
- 3 partitions of 2 + 1, giving preorders
- 1 partition of 1 + 1 + 1, giving 19 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
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]- Partial order – preorder that is antisymmetric
- Equivalence relation – preorder that is symmetric
- Total preorder – preorder that is total
- Total order – preorder that is antisymmetric and total
- Directed set
- Category of preordered sets
- Prewellordering
- Well-quasi-ordering
Notes
[edit]- ^ 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.
- ^ Pierce, Benjamin C. (2002). Types and Programming Languages. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff. ISBN 0-262-16209-1.
- ^ 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.
- ^ In this context, "" does not mean "set difference".
- ^ Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, the Netherlands: Elsevier.