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A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require transitivity and reflexivity. |
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 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 a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← 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 and then for all
A set that is equipped with a preorder is called a preordered set (or proset).^{[1]} 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 is transitive and also satisfies the following condition:
- Irreflexivity or Anti-reflexivity: is false for all
Every strict partial order is (by definition) a strict preorder; conversely, because every transitive irreflexive relation is necessarily asymmetric, every strict preorder is also a strict partial order. Consequently, a binary relation is a strict preorder if and only if it is a strict partial order. 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 2 = (0 → 1).
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
The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where x ≤ y 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 x ≤ y). 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).
Every finite topological space gives rise to a preorder on its points by defining x ≤ y 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.
Further examples:
- The relation defined by x ≤ y if f(x) ≤ f(y), where f is a function into some preorder.
- The relation defined by x ≤ y 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.
- The graph-minor relation in graph theory.
- 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.
In computer science, one can find examples of the following preorders.
- Many-one and Turing reductions are preorders on complexity classes.
- The 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 s ≤ t if a subterm of t is a substitution instance of s.
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.^{[3]}
Constructions
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R^{+=}. The transitive closure indicates path connection in R: x R^{+} y if and only if there is an R-path from x to y.
Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲ a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.)
Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~. Note that if the preorder is R^{+=}, S / ~ is the set of R-cycle equivalence classes: x ∈ [y] if and only if x = y or x is in an R-cycle with y. In any case, on S / ~, it is possible to define [x] ≤ [y] if and only if x ≲ y. By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.
Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).
For a preorder "≲", a relation "<" can be defined as a < b if and only if (a ≲ b and not b ≲ a), or equivalently, using the equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. If the preorder is antisymmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).
The relation "<" is not defined as: a < b if and only if (a ≲ b and a ≠ b). Doing so would cause problems if the preorder was not antisymmetric, as the resulting relation "<" would not be transitive (think of how equivalent non-equal elements relate).
Conversely, a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation "≲"; "≤" can be confusing for a preorder that is not antisymmetric, it may suggest that a ≤ b implies that a < b or a = b.
With this construction multiple preorders "≲" can give the same relation "<", so without more information, such as the equivalence relation, "≲" cannot be reconstructed from "<". Possible preorders include the following:
- Define a ≤ b as a < b or a = b (i.e., 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 we don't need the notations ≲ and ~.
- Define a ≲ b as "not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a ~ b as "neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a strict weak order. The resulting preorder is connected (formerly called total), that is, a total preorder.
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.
Number of preorders
Elements | Any | Transitive | Reflexive | Preorder | Partial order | Total preorder | Total order | Equivalence relation |
---|---|---|---|---|---|---|---|---|
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |
3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |
4 | 65,536 | 3,994 | 4,096 | 355 | 219 | 75 | 24 | 15 |
n | 2^{n2} | 2^{n2−n} | ∑n k=0 k! S(n, k) |
n! | ∑n k=0 S(n, k) | |||
OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
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 n = 3:
- 1 partition of 3, giving 1 preorder
- 3 partitions of 2 + 1, giving 3 × 3 = 9 preorders
- 1 partition of 1 + 1 + 1, giving 19 preorders
- for n = 4:
- 1 partition of 4, giving 1 preorder
- 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving 7 × 3 = 21 preorders
- 6 partitions of 2 + 1 + 1, giving 6 × 19 = 114 preorders
- 1 partition of 1 + 1 + 1 + 1, giving 219 preorders
Interval
For a ≲ b, the interval [a, b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains at least the points a and b. One may choose to extend the definition to all pairs (a, b). The extra intervals are all empty.
Using the corresponding strict relation "<", one can also define the interval (a, b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.
Also [a, b) and (a, b] 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
- ^ 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.
- ^ Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, 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