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✗ 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 | indicates that the column's property is always true the row's term (at the very left), while
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.
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 definition
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:
- Reflexivity: for all and
- 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:
- Irreflexivity or Anti-reflexivity: not for all that is, is false for all and
- 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 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
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.
Examples
Graph theory
- (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 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.^{[3]}
- 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,^{[4]} which is when the literals in a disjunctive first-order formula are contained by another, after applying a substitution to the former.
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 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.
- 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:
- Preference, according to common models.
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]}
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,^{[6]} where denotes the converse relation of and denotes the complement relation of while denotes relation composition.
Preorders and 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 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 preorders
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
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 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 | 2^{n2} | 2^{n2−n} | 2^{n(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
- 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
- ^ on the set of numbers divisible by 4
- ^ 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.
- ^ Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, The Netherlands: Elsevier.
- ^ In this context, "" does not mean "set difference".
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