Preorder

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 Template:Em 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 $1$ divides $-1$ and $-1$ divides $1$. 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 $X$ can equivalently be defined as an equivalence relation on $X$, 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 $\lesssim$ or $\le$. In words, when $a\lesssim b,$ one may say that b Template:Em a or that a Template:Em b, or that b Template:Em to a. Occasionally, the notation ← or → is also used.

Definition

Let $\lesssim$ be a binary relation on a set $P,$ so that by definition, $\lesssim$ is some subset of $P\times P$ and the notation $a\lesssim b$ is used in place of $(a,b)\in \lesssim .$ Then $\lesssim$ is called a Template:Em or Template:Em if it is reflexive and transitive; that is, if it satisfies:

1. Reflexivity: $a\lesssim a$ for all $a\in P,$ and
2. Transitivity: if $a\lesssim b\text{ and }b\lesssim c\text{ then }a\lesssim c$ for all $a,b,c\in P.$

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

Preorders as partial orders on partitions

Given a preorder $\lesssim$ on $S$ one may define an equivalence relation $\sim$ on $S$ such that $$a\sim b\text{ if and only if }a\lesssim b\text{ and }b\lesssim a.$$ The resulting relation $\sim$ is reflexive since the preorder $\lesssim$ is reflexive; transitive by applying the transitivity of $\lesssim$ twice; and symmetric by definition.

Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, $S/\sim ,$ which is the set of all equivalence classes of $\sim .$ If the preorder is denoted by $R^{+=},$ then $S/\sim$ is the set of $R$-cycle equivalence classes: $x\in [y]$ if and only if $x=y$ or $x$ is in an $R$-cycle with $y$. In any case, on $S/\sim$ it is possible to define $[x]\le [y]$ if and only if $x\lesssim y.$ That this is well-defined, meaning that its defining condition does not depend on which representatives of $[x]$ and $[y]$ are chosen, follows from the definition of $\sim .$ It is readily verified that this yields a partially ordered set.

Conversely, from any partial order on a partition of a set $S,$ it is possible to construct a preorder on $S$ itself. There is a one-to-one correspondence between preorders and pairs (partition, partial order).

Template:Em: Let $S$ 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, $S$ could be a first-order theory (like Zermelo–Fraenkel set theory) or a simpler zeroth-order theory. One of the many properties of $S$ is that it is closed under logical consequences so that, for instance, if a sentence $A\in S$ logically implies some sentence $B,$ which will be written as $A\Rightarrow B$ and also as $B\Leftarrow A,$ then necessarily $B\in S$ (by modus ponens). The relation $\Leftarrow$ is a preorder on $S$ because $A\Leftarrow A$ always holds and whenever $A\Leftarrow B$ and $B\Leftarrow C$ both hold then so does $A\Leftarrow C.$ Furthermore, for any $A,B\in S,$ $A\sim B$ if and only if $A\Leftarrow B\text{ and }B\Leftarrow A$; that is, two sentences are equivalent with respect to $\Leftarrow$ if and only if they are logically equivalent. This particular equivalence relation $A\sim B$ is commonly denoted with its own special symbol $A⟺B,$ and so this symbol $⟺$ may be used instead of $\sim .$ The equivalence class of a sentence $A,$ denoted by $[A],$ consists of all sentences $B\in S$ that are logically equivalent to $A$ (that is, all $B\in S$ such that $A⟺B$). The partial order on $S/\sim$ induced by $\Leftarrow ,$ which will also be denoted by the same symbol $\Leftarrow ,$ is characterized by $[A]\Leftarrow [B]$ if and only if $A\Leftarrow B,$ where the right hand side condition is independent of the choice of representatives $A\in [A]$ and $B\in [B]$ of the equivalence classes. All that has been said of $\Leftarrow$ so far can also be said of its converse relation $\Rightarrow .$ The preordered set $(S,\Leftarrow )$ is a directed set because if $A,B\in S$ and if $C:=A\wedge B$ denotes the sentence formed by logical conjunction $\wedge ,$ then $A\Leftarrow C$ and $B\Leftarrow C$ where $C\in S.$ The partially ordered set $(S/\sim ,\Leftarrow )$ is consequently also a directed set. See Lindenbaum–Tarski algebra for a related example.

Relationship to strict partial orders

Template:Anchor If reflexivity is replaced with irreflexivity (while keeping transitivity) then we get the definition of a strict partial order on $P$. For this reason, the term Template:Em is sometimes used for a strict partial order. That is, this is a binary relation $<$ on $P$ that satisfies:

1. Irreflexivity or anti-reflexivity: Template:Em $a<a$ for all $a\in P;$ that is, $a<a$ is Template:Em for all $a\in P,$ and
2. Transitivity: if $a<b\text{ and }b<c\text{ then }a<c$ for all $a,b,c\in P.$

Strict partial order induced by a preorder

Any preorder $\lesssim$ gives rise to a strict partial order defined by $a<b$ if and only if $a\lesssim b$ and not $b\lesssim a$. Using the equivalence relation $\sim$ introduced above, $a<b$ if and only if $a\lesssim b\text{ and not }a\sim b;$ and so the following holds $$a\lesssim b\text{ if and only if }a<b\text{ or }a\sim b.$$ The relation $<$ is a strict partial order and Template:Em strict partial order can be constructed this way. Template:Em the preorder $\lesssim$ is antisymmetric (and thus a partial order) then the equivalence $\sim$ is equality (that is, $a\sim b$ if and only if $a=b$) and so in this case, the definition of $<$ can be restated as: $$a<b\text{ if and only if }a\lesssim b\text{ and }a\ne b(\text{assuming }\lesssim \text{ is antisymmetric}).$$ But importantly, this new condition is Template:Em used as (nor is it equivalent to) the general definition of the relation $<$ (that is, $<$ is Template:Em defined as: $a<b$ if and only if $a\lesssim b\text{ and }a\ne b$) because if the preorder $\lesssim$ 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 "$\lesssim$" instead of the "less than or equal to" symbol "$\le$", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that $a\le b$ implies $a<b\text{ or }a=b.$

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 $\sim$ 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 $a\le b$ as $a<b\text{ or }a=b$ (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 $\lesssim$ and $\sim$ are not needed.
• Define $a\lesssim b$ as "$\text{ not }b<a$" (that is, take the inverse complement of the relation), which corresponds to defining $a\sim b$ as "neither $a<b\text{ nor }b<a$"; these relations $\lesssim$ and $\sim$ are in general not transitive; however, if they are then $\sim$ 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 $a\le b$ then $a\lesssim b.$ The converse holds (that is, $\lesssim =\le$) if and only if whenever $a\ne b$ then $a<b$ or $b<a.$

Examples

Graph theory

• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where $x\lesssim y$ 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 Template:Nowrap with $x\lesssim y$). 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).
• The graph-minor relation is also a preorder.

Computer science

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

• Asymptotic order causes a preorder over functions $f:\mathbb{N}\to \mathbb{N}$. 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.
• Simulation preorders are preorders (hence the name).
• Reduction relations in abstract rewriting systems.
• The encompassment preorder on the set of terms, defined by $s\lesssim t$ if a subterm of t is a substitution instance of s.
• Theta-subsumption, which is when the literals in a disjunctive first-order formula are contained by another, after applying a substitution to the former.

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 $P,$ 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 $2=(0\to 1).$

Other

Further examples:

• Every finite topological space gives rise to a preorder on its points by defining $x\lesssim y$ 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.

• The relation defined by $x\lesssim y$ if $f(x)\lesssim f(y),$ where f is a function into some preorder.
• The relation defined by $x\lesssim y$ 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.

Example of a total preorder:

• Preference, according to common models.

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 $y.$

Left residual preorder induced by a binary relation

Given a binary relation $R,$ the complemented composition $R\mathrm{\setminus }R=\stackrel{‾}{R^{\mathsf{\text{T}}}\circ \bar{R}}$ forms a preorder called the left residual, where $R^{\mathsf{\text{T}}}$ denotes the converse relation of $R,$ and $\bar{R}$ denotes the complement relation of $R,$ while $\circ$ denotes relation composition.

Related definitions

If a preorder is also antisymmetric, that is, $a\lesssim b$ and $b\lesssim a$ implies $a=b,$ then it is a partial order.

On the other hand, if it is symmetric, that is, if $a\lesssim b$ implies $b\lesssim a,$ then it is an equivalence relation.

A preorder is total if $a\lesssim b$ or $b\lesssim a$ for all $a,b\in P.$

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:

• 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.

Number of preorders

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: Template:Unordered list

Interval

For $a\lesssim b,$ the interval $[a,b]$ is the set of points x satisfying $a\lesssim x$ and $x\lesssim b,$ also written $a\lesssim x\lesssim b.$ 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

References

• Schmidt, Gunther, "Relational Mathematics", Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press, 2011, Template:Isbn
• Template:Citation