Preorder

 ( Properties Of Binary Relations ) 

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name is meant to suggest that preorders are almost , but not quite, as they are not necessarily antisymmetric.

A natural example of a preorder is the divides relation "x divides y" between . This relation is reflexive as every integer divides itself. It is also transitive. But it is not antisymmetric, because e.g. $1$ divides $-1$ and $-1$ divides $1$. It is to this preorder that "least" refers in the phrase "least common multiple" (in contrast, using the natural order on integers, e.g. $4$ and $6$ have the common multiples $24$, $12$, $0$, $-12$, $-24$, ..., 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 $X$ can equivalently be defined as an equivalence relation on $X$, 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 $\backslash ,\backslash lesssim\backslash ,$ or $\backslash ,\backslash leq\backslash ,$.

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Definition A binary relation $\backslash ,\backslash lesssim\backslash ,$ on a set $X$ is called a ' or ' if it is reflexive and transitive; that is, if it satisfies:

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

A set that is equipped with a preorder is called a preordered set (or proset).For "proset", see e.g. .

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Preorders as partial orders on partitions Given a preorder $\backslash ,\backslash lesssim\backslash ,$ on $X$ one may define an equivalence relation $\backslash ,\backslash sim\backslash ,$ on $X$ by $$a\; \backslash sim\; b\; \backslash quad\; \backslash text\{\; if\; \}\; \backslash quad\; a\; \backslash lesssim\; b\; \backslash ;\; \backslash text\{\; and\; \}\; \backslash ;\; b\; \backslash lesssim\; a.$$ The resulting relation $\backslash ,\backslash sim\backslash ,$ is reflexive since the preorder $\backslash ,\backslash lesssim\backslash ,$ is reflexive; transitive by applying the transitivity of $\backslash ,\backslash lesssim\backslash ,$ twice; and symmetric by definition.

Using this relation, it is possible to construct a partial order on the quotient set $X\; /\; \backslash sim$ of the equivalence, by defining $x\; \backslash leq\; y$ if $x\; \backslash lesssim\; y.$ That this is well-defined, meaning that it does not depend on the particular choice of representatives $x$ and $y$, follows from the definition of $\backslash ,\backslash sim\backslash ,$.

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

Let $X$ be the set of all (valid or invalid) sentences in some subfield of mathematics, like geometry. Define $p\; \backslash Leftarrow\; q$ if $p$ is a logical consequence of $q$.

Then $\backslash Leftarrow$ is a preorder on $X$: every sentence $p$ can be proven from itself (reflexivity), and if $p$ can be proven from $q$, and $q$ from $r$, then $p$ can also be proven from $r$ (transitivity). The corresponding equivalence relation is usually denoted $p\; \backslash Leftrightarrow\; q$, and defined as $p\; \backslash Leftarrow\; q$ and $q\; \backslash Leftarrow\; p$; in this case $p$ and $q$ are called "logically equivalent". The equivalence class of a sentence $p$ is the set of all sentences $q\; \backslash in\; X$ that are logically equivalent to $p$; formally: $p\; =\; \backslash \{\; q\; \backslash mid\; p\; \backslash Leftrightarrow\; q\; \backslash \}$. The preordered set $(X,\; \backslash Leftarrow)$ is a directed set: given two sentences $p,\; q\; \backslash in\; X$, their logical conjunction $p\; \backslash wedge\; q$, pronounced "both $p$ and $q$", is a common upper bound of them, since $p$ is a consequence of $p\; \backslash wedge\; q$, and so is $q$. The partially ordered set $\backslash left(X\; /\; \backslash Leftrightarrow,\; \backslash Leftarrow\backslash right)$ 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 $X$. For this reason, the term is sometimes used for a strict partial order. That is, this is a binary relation on $X$ that satisfies:

1. Irreflexivity or anti-reflexivity: for all $a\; \backslash in\; X;$ that is, is for all $a\; \backslash in\; X,$ and
2. Transitivity: if for all $a,\; b,\; c\; \backslash in\; X.$

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Strict partial order induced by a preorder Any preorder $\backslash ,\backslash lesssim\backslash ,$ gives rise to a strict partial order defined by if and only if $a\; \backslash lesssim\; b$ and not $b\; \backslash lesssim\; a$. Using the equivalence relation $\backslash ,\backslash sim\backslash ,$ introduced above, if and only if $a\; \backslash lesssim\; b\; \backslash text\{\; and\; not\; \}\; a\; \backslash sim\; b;$ and so the following holds The relation is a strict partial order and strict partial order can be constructed this way.

the preorder $\backslash ,\backslash lesssim\backslash ,$ is antisymmetric (and thus a partial order) then the equivalence $\backslash ,\backslash sim\backslash ,$ is equality (that is, $a\; \backslash sim\; b$ if and only if $a\; =\; b$) and so in this case, the definition of  can be restated as:
     

But importantly, this new condition is used as (nor is it equivalent to) the general definition of the relation (that is, is defined as: if and only if $a\; \backslash lesssim\; b\; \backslash text\{\; and\; \}\; a\; \backslash neq\; b$) because if the preorder $\backslash ,\backslash lesssim\backslash ,$ 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 "$\backslash lesssim$" instead of the "less than or equal to" symbol "$\backslash leq$", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that $a\; \backslash leq\; b$ implies

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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 $\backslash ,\backslash sim\backslash ,$ 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 $a\; \backslash leq\; b$ 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 $\backslash ,=,$ so the symbols $\backslash ,\backslash lesssim\backslash ,$ and $\backslash ,\backslash sim\backslash ,$ are not needed.
• Define $a\; \backslash lesssim\; b$ as "" (that is, take the inverse complement of the relation), which corresponds to defining $a\; \backslash sim\; b$ as "neither "; these relations $\backslash ,\backslash lesssim\backslash ,$ and $\backslash ,\backslash sim\backslash ,$ are in general not transitive; however, if they are then $\backslash ,\backslash sim\backslash ,$ 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 $a\; \backslash leq\; b$ then $a\; \backslash lesssim\; b.$ The converse holds (that is, $\backslash ,\backslash lesssim\backslash ;\backslash ;\; =\; \backslash ;\backslash ;\backslash leq\backslash ,$) if and only if whenever $a\; \backslash neq\; b$ 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 $x\; \backslash lesssim\; y$ 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 with $x\; \backslash lesssim\; y$). 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.

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

• Asymptotic order causes a preorder over functions $f:\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$. The corresponding equivalence relation is called asymptotic equivalence.
• Polynomial-time, many-one (mapping) and are preorders on complexity classes.
• Subtyping relations are usually preorders.
  Benjamin C. Pierce (2025). 9780262162098, The MIT Press. ISBN 9780262162098
• Simulation preorders are preorders (hence the name).
• Reduction relations in abstract rewriting systems.
• The encompassment preorder on the set of terms, defined by $s\; \backslash lesssim\; t$ 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.

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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 category. Here the objects correspond to the elements of $X,$ 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 $2\; =\; (0\; \backslash to\; 1).$

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Other Further examples:

• Every finite topological space gives rise to a preorder on its points by defining $x\; \backslash lesssim\; y$ 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 set 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 $x\; \backslash lesssim\; y$ if $f(x)\; \backslash lesssim\; f(y),$ where f is a function into some preorder.
• The relation defined by $x\; \backslash lesssim\; y$ 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 .

Example of a total preorder:

• Preference, according to common models.

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Constructions Every binary relation $R$ on a set $X$ can be extended to a preorder on $X$ 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 $R\; \backslash backslash\; R\; =\; \backslash overline\{R^\backslash textsf\{T\}\; \backslash circ\; \backslash overline\{R\}\}$ forms a preorder called the left residual,In this context, "$\backslash backslash$" does not mean "set difference". where $R^\backslash textsf\{T\}$ denotes the converse relation of $R,$ and $\backslash overline\{R\}$ denotes the complement relation of $R,$ while $\backslash circ$ denotes relation composition.

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

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

A preorder is Total preorder if $a\; \backslash lesssim\; b$ or $b\; \backslash lesssim\; a$ for all $a,\; b\; \backslash in\; X.$

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:

• 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 .
• 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..

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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:

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Interval For $a\; \backslash lesssim\; b,$ the interval $a,$ is the set of points x satisfying $a\; \backslash lesssim\; x$ and $x\; \backslash lesssim\; b,$ also written $a\; \backslash lesssim\; x\; \backslash lesssim\; b.$ 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 and also written An open interval may be empty even if

Also $a,$ can be defined similarly.

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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

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Notes

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

Categories: Properties Of Binary Relations, Order Theory

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