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Preorder

Reflexive and transitive binary relation From Wikipedia, the free encyclopedia

Preorder
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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.

Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total,
Semiconnex
Anti-
reflexive
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions,
for all and
Green tickY indicates that the column's property is always true for 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 Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

Thumb
x R y defined by x//4≤y//4 is a preorder on the natural numbers. It corresponds to the equivalence relation x E y defined by x//4=y//4. The set of equivalence classes is partially ordered, and thus can be shown as a Hasse diagram (depicted).

A natural example of a preorder is the divides relation "x divides y" between integers. This relation is reflexive as every integer divides itself. It is also transitive. But it is not antisymmetric, because e.g. divides and divides . It is to this preorder that "least" refers in the phrase "least common multiple" (in contrast, using the natural order on integers, e.g. and have the common multiples , , , , , ..., but no least one).

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, cf. picture. 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.

A preorder is often denoted or .

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Definition

A binary relation on a set is called a preorder or quasiorder if it is reflexive and transitive; that is, if it satisfies:

  1. Reflexivity: for all and
  2. Transitivity: if for all

A set that is equipped with a preorder is called a preordered set (or proset).[1]

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Preorders as partial orders on partitions

Given a preorder on one may define an equivalence relation on by 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, by defining if That this is well-defined, meaning that it does not depend on the particular choice of representatives and , follows from the definition of .

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 the set of all (valid or invalid) sentences in some subfield of mathematics, like geometry. Define if is a logical consequence of . Then is a preorder on : every sentence can be proven from itself (reflexivity), and if can be proven from , and from , then can also be proven from (transitivity). The corresponding equivalence relation is usually denoted , and defined as and ; in this case and are called "logically equivalent". The equivalence class of a sentence is the set of all sentences that are logically equivalent to ; formally: . The preordered set is a directed set: given two sentences , their logical conjunction , pronounced "both and ", is a common upper bound of them, since is a consequence of , and so is . The partially ordered set is hence also a directed set. See Lindenbaum–Tarski algebra for a related example.

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Relationship to strict partial orders

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:

  1. Irreflexivity or anti-reflexivity: not for all that is, is false for all and
  2. 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 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

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

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Examples

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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-minor relation is also a preorder.

Computer science

In computer science, one can find examples of the following preorders.

Category 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. 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

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.
  • 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:

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Constructions

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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,[5] where denotes the converse relation of and denotes the complement relation of while denotes relation composition.

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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.

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Uses

Preorders play a pivotal role in several situations:

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Number of preorders

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More information Elem­ents, Any ...

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
    I.e., together, 29 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
    I.e., together, 355 preorders.
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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

Notes

References

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