Preorder

Preorders as partial orders on partitions

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

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

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

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

1. Irreflexivity or anti-reflexivity: not ${\displaystyle a<a}$ for all ${\displaystyle a\in X;}$ that is, ${\displaystyle a<a}$ is false for all ${\displaystyle a\in X,}$ and
2. Transitivity: if ${\displaystyle a<b\text{ and }b<c\text{ then }a<c}$ for all ${\displaystyle a,b,c\in X.}$

Strict partial order induced by a preorder

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

Preorders induced by a strict partial order

Using the construction above, multiple non-strict preorders can produce the same strict preorder ${\displaystyle <,}$ so without more information about how ${\displaystyle <}$ was constructed (such as knowledge of the equivalence relation ${\displaystyle \sim }$ for instance), it might not be possible to reconstruct the original non-strict preorder from ${\displaystyle <.}$ Possible (non-strict) preorders that induce the given strict preorder ${\displaystyle <}$ include the following:

• Define ${\displaystyle a\le b}$ as ${\displaystyle a<b\text{ or }a=b}$ (that is, take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "${\displaystyle <}$" through reflexive closure; in this case the equivalence is equality ${\displaystyle =,}$ so the symbols ${\displaystyle \lesssim }$ and ${\displaystyle \sim }$ are not needed.
• Define ${\displaystyle a\lesssim b}$ as "${\displaystyle \text{ not }b<a}$" (that is, take the inverse complement of the relation), which corresponds to defining ${\displaystyle a\sim b}$ as "neither ${\displaystyle a<b\text{ nor }b<a}$"; these relations ${\displaystyle \lesssim }$ and ${\displaystyle \sim }$ are in general not transitive; however, if they are then ${\displaystyle \sim }$ is an equivalence; in that case "${\displaystyle <}$" is a strict weak order. The resulting preorder is connected (formerly called total); that is, a total preorder.

If ${\displaystyle a\le b}$ then ${\displaystyle a\lesssim b.}$ The converse holds (that is, ${\displaystyle \lesssim =\le }$) if and only if whenever ${\displaystyle a\ne b}$ then ${\displaystyle a<b}$ or ${\displaystyle b<a.}$

Examples

Graph theory

• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where ${\displaystyle x\lesssim y}$ 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 (x, y) with ${\displaystyle x\lesssim y}$). 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 ${\displaystyle f:\mathbb{N}\to \mathbb{N}}$. 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 ${\displaystyle s\lesssim t}$ 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 ${\displaystyle X,}$ 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 ${\displaystyle 2=(0\to 1).}$

Other

Further examples:

• Every finite topological space gives rise to a preorder on its points by defining ${\displaystyle x\lesssim y}$ 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 ${\displaystyle x\lesssim y}$ if ${\displaystyle f(x)\lesssim f(y),}$ where f is a function into some preorder.
• The relation defined by ${\displaystyle x\lesssim y}$ 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 ${\displaystyle R}$ on a set ${\displaystyle X}$ can be extended to a preorder on ${\displaystyle X}$ by taking the transitive closure and reflexive closure, ${\displaystyle R^{+=}.}$ The transitive closure indicates path connection in ${\displaystyle R:x{R}^{+}y}$ if and only if there is an ${\displaystyle R}$-path from ${\displaystyle x}$ to ${\displaystyle y.}$

Left residual preorder induced by a binary relation

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

Related definitions

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

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

A preorder is total if ${\displaystyle a\lesssim b}$ or ${\displaystyle b\lesssim a}$ for all ${\displaystyle a,b\in X.}$

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.

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

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 ${\displaystyle n=3:}$
  • 1 partition of 3, giving 1 preorder
  • 3 partitions of 2 + 1, giving ${\displaystyle 3\times 3=9}$ preorders
  • 1 partition of 1 + 1 + 1, giving 19 preorders
  I.e., together, 29 preorders.
• for ${\displaystyle n=4:}$
  • 1 partition of 4, giving 1 preorder
  • 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving ${\displaystyle 7\times 3=21}$ preorders
  • 6 partitions of 2 + 1 + 1, giving ${\displaystyle 6\times 19=114}$ preorders
  • 1 partition of 1 + 1 + 1 + 1, giving 219 preorders
  I.e., together, 355 preorders.

Interval

For ${\displaystyle a\lesssim b,}$ the interval ${\displaystyle [a,b]}$ is the set of points x satisfying ${\displaystyle a\lesssim x}$ and ${\displaystyle x\lesssim b,}$ also written ${\displaystyle a\lesssim x\lesssim b.}$ It contains at least the points a and b. One may choose to extend the definition to all pairs ${\displaystyle (a,b)}$. The extra intervals are all empty.

Using the corresponding strict relation "${\displaystyle <}$", one can also define the interval ${\displaystyle (a,b)}$ as the set of points x satisfying ${\displaystyle a<x}$ and ${\displaystyle x<b,}$ also written ${\displaystyle a<x<b.}$ An open interval may be empty even if ${\displaystyle a<b.}$

Also ${\displaystyle [a,b)}$ and ${\displaystyle (a,b]}$ can be defined similarly.