Preorder

Last updated
Transitive   binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Green check.svgYGreen check.svgY
Preorder (Quasiorder) Green check.svgY
Partial order Green check.svgYGreen check.svgY
Total preorder Green check.svgYGreen check.svgY
Total order Green check.svgYGreen check.svgYGreen check.svgY
Prewellordering Green check.svgYGreen check.svgYGreen check.svgY
Well-quasi-ordering Green check.svgYGreen check.svgY
Well-ordering Green check.svgYGreen check.svgYGreen check.svgYGreen check.svgY
Lattice Green check.svgYGreen check.svgYGreen check.svgYGreen check.svgY
Join-semilattice Green check.svgYGreen check.svgYGreen check.svgY
Meet-semilattice Green check.svgYGreen check.svgYGreen check.svgY
Strict partial order Green check.svgYGreen check.svgYGreen check.svgY
Strict weak order Green check.svgYGreen check.svgYGreen check.svgY
Strict total order Green check.svgYGreen check.svgYGreen check.svgYGreen check.svgY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green check.svgY 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 Green check.svgY 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.

Contents

Hasse diagram of the preorder x R y defined by x//4<=y//4 on the natural numbers. Equivalence classes (sets of elements such that x R y and y R x) are shown together as a single node. The relation on equivalence classes is a partial order. Preorder.png
Hasse diagram of the preorder x R y defined by x //4≤y //4 on the natural numbers. Equivalence classes (sets of elements such that x R y and y R x) are shown together as a single node. The relation on equivalence classes is a partial order.

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 "greatest common divisor" and "lowest common multiple" (except that, for integers, the greatest common divisor is also the greatest for the natural order of the integers).

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. 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.

As a binary relation, a preorder may be denoted or . In words, when one may say that bcoversa or that aprecedesb, or that breduces 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:

  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]

Preorders as partial orders on partitions

Given a preorder on one may define an equivalence relation on such that 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, 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.

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:

If then The converse holds (that is, ) if and only if whenever then or

Examples


Graph theory

Computer science

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

Category theory

Other

Further examples:

Example of a total preorder:

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

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.

Uses

Preorders play a pivotal role in several situations:

Number of preorders

Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0111111111
1221211111
216134843322
3512171646429191365
465,5363,9944,0961,024355219752415
n2n22n(n−1)2n(n+1)/2n
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:

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

  1. 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 .
  2. Pierce, Benjamin C. (2002). Types and Programming Languages. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff. ISBN   0-262-16209-1.
  3. 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.
  4. In this context, "" does not mean "set difference".
  5. Kunen, Kenneth (1980), Set Theory, An Introduction to Independence Proofs, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, the Netherlands: Elsevier.

Related Research Articles

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets and is a set of ordered pairs consisting of elements from and from . It encodes the common concept of relation: an element is related to an element , if and only if the pair belongs to the set of ordered pairs that defines the binary relation.

In mathematics, a directed set is a nonempty set together with a reflexive and transitive binary relation , with the additional property that every pair of elements has an upper bound. In other words, for any and in there must exist in with and A directed set's preorder is called a direction.

<span class="mw-page-title-main">Equivalence relation</span> Mathematical concept for comparing objects

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).

<span class="mw-page-title-main">Equivalence class</span> Mathematical concept

In mathematics, when the elements of some set have a notion of equivalence, then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent.

<span class="mw-page-title-main">Partially ordered set</span> Mathematical set with an ordering

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in :

  1. (reflexive).
  2. If and then (transitive).
  3. If and then (antisymmetric).
  4. or .

In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

In mathematics, the transitive closureR+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R+ is the unique minimal transitive superset of R.

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:

<span class="mw-page-title-main">Weak ordering</span> Mathematical ranking of a set

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets and are in turn generalized by (strictly) partially ordered sets and preorders.

In set theory, a prewellordering on a set is a preorder on that is strongly connected and well-founded in the sense that the induced relation defined by is a well-founded relation.

In mathematics, a partial equivalence relation is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation.

In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if and are sets and is a relation from to then is the relation defined so that if and only if In set-builder notation,

In mathematics, a homogeneous relation on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people.

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.

<span class="mw-page-title-main">Relation (mathematics)</span> Relationship between two sets, defined by a set of ordered pairs

In mathematics, a relation on a set may, or may not, hold between two given members of the set. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3, and likewise between 3 and 4, but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false. As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not.

In order theory, the Szpilrajn extension theorem, proved by Edward Szpilrajn in 1930, states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties.

In mathematics, a relation on a set is called connected or complete or total if it relates all distinct pairs of elements of the set in one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that .

References