Preorder
Transitive binary relations | |||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| |||||||||||||||||||||||||||||||
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 in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation be transitive: for all if and then |
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 "
Preorders are closely related to
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:
- 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
Given a preorder on one may define an equivalence relation on such that
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
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
Relationship to strict partial orders
If reflexivity is replaced with
- Irreflexivityor anti-reflexivity: not for all that is, is false for all and
- 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
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:
- 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 total preorder.
If then The converse holds (that is, ) if and only if whenever then or
Examples
Graph theory
- 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-minorrelation is also a preorder.
Computer science
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 preordersare preorders (hence the name).
- Reduction relations in abstract rewriting systems.
- The encompassment preorder on the set of terms, defined by if asubstitution instanceof 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
- A objectscorrespond 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
Further examples:
- Every finite topological space gives rise to a preorder on its points by defining if and only if x belongs to every one-to-one correspondencebetween finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.
- A upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered setswithout losing important features.
- The relation defined by if where f is a function into some preorder.
- The relation defined by if there exists some surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.
- The embedding relation for countable total orderings.
Example of a
- Preference, according to common models.
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
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
A
Uses
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
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
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
- 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
- MR 1127325.
- ISBN 0-262-16209-1.
- 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.
References
- 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