Partially ordered set

Transitive binary relations v t e
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all ${\displaystyle a,b}$ and ${\displaystyle S\neq \varnothing :}$ ${\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle aRa}$ ${\displaystyle {\text{not }}aRa}$ ${\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}$
Y 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 Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation ${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$ A term's definition may require additional properties that are not listed in this table.

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.

Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair ${\displaystyle P=(X,\leq )}$ consisting of a set ${\displaystyle X}$ (called the ground set of ${\displaystyle P}$) and a partial order ${\displaystyle \leq }$ on ${\displaystyle X}$. When the meaning is clear from context and there is no ambiguity about the partial order, the set ${\displaystyle X}$ itself is sometimes called a poset.

Partial order relations

The term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders. Strict and non-strict partial orders can be put into a one-to-one correspondence, so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.

Partial orders

A reflexive, weak, or non-strict partial order, commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set ${\displaystyle P}$ that is reflexive, antisymmetric, and transitive. That is, for all ${\displaystyle a,b,c\in P,}$ it must satisfy:

1. Reflexivity: ${\displaystyle a\leq a}$, i.e. every element is related to itself.
2. Antisymmetry: if ${\displaystyle a\leq b}$ and ${\displaystyle b\leq a}$ then ${\displaystyle a=b}$, i.e. no two distinct elements precede each other.
3. Transitivity: if ${\displaystyle a\leq b}$ and ${\displaystyle b\leq c}$ then ${\displaystyle a\leq c}$.

A non-strict partial order is also known as an antisymmetric preorder.

Strict partial orders

An irreflexive, strong, or strict partial order is a homogeneous relation < on a set ${\displaystyle P}$ that is irreflexive, asymmetric and transitive; that is, it satisfies the following conditions for all ${\displaystyle a,b,c\in P:}$

1. Irreflexivity: ${\displaystyle \neg \left(a<a\right)}$, i.e. no element is related to itself (also called anti-reflexive).
2. Asymmetry: if ${\displaystyle a<b}$ then not ${\displaystyle b<a}$.
3. Transitivity: if ${\displaystyle a<b}$ and ${\displaystyle b<c}$ then ${\displaystyle a<c}$.

A transitive relation is asymmetric if and only if it is irreflexive. So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

A strict partial order is also known as a strict preorder.

Correspondence of strict and non-strict partial order relations

Strict and non-strict partial orders on a set ${\displaystyle P}$ are closely related. A non-strict partial order ${\displaystyle \leq }$ may be converted to a strict partial order by removing all relationships of the form ${\displaystyle a\leq a;}$ that is, the strict partial order is the set ${\displaystyle <\;:=\ \leq \ \setminus \ \Delta _{P}}$ where ${\displaystyle \Delta _{P}:=\{(p,p):p\in P\}}$ is the identity relation on ${\displaystyle P\times P}$ and ${\displaystyle \;\setminus \;}$ denotes set subtraction. Conversely, a strict partial order < on ${\displaystyle P}$ may be converted to a non-strict partial order by adjoining all relationships of that form; that is, ${\displaystyle \leq \;:=\;\Delta _{P}\;\cup \;<\;}$ is a non-strict partial order. Thus, if ${\displaystyle \leq }$ is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by $${\displaystyle a<b{\text{ if }}a\leq b{\text{ and }}a\neq b.}$$ Conversely, if < is a strict partial order, then the corresponding non-strict partial order ${\displaystyle \leq }$ is the reflexive closure given by: $${\displaystyle a\leq b{\text{ if }}a<b{\text{ or }}a=b.}$$

Dual orders

The dual (or opposite) ${\displaystyle R^{\text{op}}}$ of a partial order relation ${\displaystyle R}$ is defined by letting ${\displaystyle R^{\text{op}}}$ be the converse relation of ${\displaystyle R}$, i.e. ${\displaystyle xR^{\text{op}}y}$ if and only if ${\displaystyle yRx}$. The dual of a non-strict partial order is a non-strict partial order, and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.

Notation

Given a set ${\displaystyle P}$ and a partial order relation, typically the non-strict partial order ${\displaystyle \leq }$, we may uniquely extend our notation to define four partial order relations ${\displaystyle \leq ,}$ ${\displaystyle <,}$ ${\displaystyle \geq ,}$ and ${\displaystyle >}$, where ${\displaystyle \leq }$ is a non-strict partial order relation on ${\displaystyle P}$, ${\displaystyle <}$ is the associated strict partial order relation on ${\displaystyle P}$ (the irreflexive kernel of ${\displaystyle \leq }$), ${\displaystyle \geq }$ is the dual of ${\displaystyle \leq }$, and ${\displaystyle >}$ is the dual of ${\displaystyle <}$. Strictly speaking, the term partially ordered set refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, ${\displaystyle (P,\leq )}$ or ${\displaystyle (P,<)}$, or, in rare instances, the non-strict and strict relations together, ${\displaystyle (P,\leq ,<)}$.

The term ordered set is sometimes used as a shorthand for partially ordered set, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than ${\displaystyle \leq }$ such as ${\displaystyle \sqsubseteq }$ or ${\displaystyle \preceq }$ to distinguish partial orders from total orders.

When referring to partial orders, ${\displaystyle \leq }$ should not be taken as the complement of ${\displaystyle >}$. The relation ${\displaystyle >}$ is the converse of the irreflexive kernel of ${\displaystyle \leq }$, which is always a subset of the complement of ${\displaystyle \leq }$, but ${\displaystyle >}$ is equal to the complement of ${\displaystyle \leq }$ if, and only if, ${\displaystyle \leq }$ is a total order.

Alternative definitions

Another way of defining a partial order, found in computer science, is via a notion of comparison. Specifically, given ${\displaystyle \leq ,<,\geq ,{\text{ and }}>}$ as defined previously, it can be observed that two elements x and y may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable. This can be represented by a function ${\displaystyle {\text{compare}}:P\times P\to \{<,>,=,\vert \}}$ that returns one of four codes when given two elements. This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality.

Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial orders as subtypes.

A finite poset can be visualized through its Hasse diagram. Specifically, taking a strict partial order relation ${\displaystyle (P,<)}$, a directed acyclic graph (DAG) may be constructed by taking each element of ${\displaystyle P}$ to be a node and each element of ${\displaystyle <}$ to be an edge. The transitive reduction of this DAG is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.

Examples

Standard examples of posets arising in mathematics include:

• The real numbers, or in general any totally ordered set, ordered by the standard less-than-or-equal relation ≤, is a partial order.
• On the real numbers ${\displaystyle \mathbb {R} }$, the usual less than relation < is a strict partial order. The same is also true of the usual greater than relation > on ${\displaystyle \mathbb {R} }$.
• By definition, every strict weak order is a strict partial order.
• The set of subsets of a given set (its power set) ordered by inclusion (see Fig. 1). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.
• The set of natural numbers equipped with the relation of divisibility. (see Fig. 3 and Fig. 6)
• The vertex set of a directed acyclic graph ordered by reachability.
• The set of subspaces of a vector space ordered by inclusion.
• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, ${\displaystyle \left(a_{n}\right)_{n\in \mathbb {N} }\leq \left(b_{n}\right)_{n\in \mathbb {N} }}$ if and only if ${\displaystyle a_{n}\leq b_{n}}$ for all ${\displaystyle n\in \mathbb {N} }$; that is, a componentwise order.
• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all ${\displaystyle x\in X.}$
• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...
• The set of events in special relativity and, in most cases, general relativity, where for two events X and Y, X ≤ Y if and only if Y is in the future light cone of X. An event Y can be causally affected by X only if X ≤ Y.

One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.

Orders on the Cartesian product of partially ordered sets

Fig. 4a Lexicographic order on ${\displaystyle \mathbb {N} \times \mathbb {N} }$

Fig. 4b Product order on ${\displaystyle \mathbb {N} \times \mathbb {N} }$

Fig. 4c Reflexive closure of strict direct product order on ${\displaystyle \mathbb {N} \times \mathbb {N} .}$ Elements covered by (3, 3) and covering (3, 3) are highlighted in green and red, respectively.