Preorder
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 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 are 'almost' 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 ≲ 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 some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive; i.e., for all a, b and c in P, we have that:A set that is equipped with a preorder is called a preordered set.
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, b.
Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category; i.e., as a category with at most one morphism from an object to another. Here the objects correspond to the elements of P, 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
- The reachability relationship in any directed graph 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. 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.
- 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.
- 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 preorder relation.
- Many-one and Turing reductions are preorders on complexity classes.
- The subtyping relations are usually preorders.
- Simulation preorders are preorders.
- 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.
- Preference, according to common models.
Uses
- 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.
Constructions
Given a preorder ≲ on S one may define an equivalence relation ~ on S such that if and only if and.
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+=, is the set of R-cycle equivalence classes: if and only if or x is in an R-cycle with y. In any case, on we can define if and only if. 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.
For a preorder "≲", a relation "<" can be defined as if and only if, or equivalently, using the equivalence relation introduced above,. 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 if and only if.
. Doing so would cause problems if the preorder was not antisymmetric, as the resulting relation "<" would not be transitive
Conversely we have if and only if or. This is the reason for using the notation "≲"; "≤" can be confusing for a preorder that is not antisymmetric, it may suggest that implies that or.
Note that 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 as or . 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 as "not ", which corresponds to defining as "neither nor "; 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 total, that is, a total preorder.
Number of preorders
As explained above, there is a 1-to-1 correspondence between preorders and pairs. Thus the number of preorders is the sum of the number of partial orders on every partition. For example: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.