poset

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A poset, or partially ordered set, consists of a set $P$and a binary relation $\le$ on $P$ which satisfies the followingproperties:

• $\le$ is reflexive (http://planetmath.org/Reflexive), so $a\le a$ alwaysholds;

• $\le$ is antisymmetric, so if $a\le b$ and $b\le a$ hold, then$a=b$; and

• $\le$ is transitive (http://planetmath.org/Transitive3), so if $a\le b$and $b\le c$ hold, then $a\le c$ also holds.

The relation $\le$ is called a partial order on $P$. Inpractice, $(P,\le )$ is usually conflated with $P$; if a distinction isneeded, $P$ is called the ground set or underlying set of $(P,\le )$.The binary relation defined by removing the diagonal from $\le$(i.e.  iff $a\le b$ and $a\ne b$) satisfies the following properties:

• is irreflexive, so if holds, then does nothold; and

• is transitive.

Since $\le$ is reflexive, it can be uniquely recovered from by addingthe diagonal. For this reason, an irreflexive and transitive binaryrelation (called a strict partial order) also defines a poset, by meansof the associated relation $\le$ described above (which is called weak partial order).

Since every partial order is reflexive and transitive, every poset isa preorder. The notion of partial order is stricter than that ofpreorder, Let $Q$ be the structure with ground set $Q=\{a,b\}$ andbinary relation $⪯=\{(a,a),(a,b),(b,a),(b,b)\}$. A diagramof this structure, omitting loops, is displayed below.

$$b\mathrm{@}{/}^{/}[d]a\mathrm{@}{/}^{/}[u]$$

Observe that the binary relation on $Q$ is reflexive and transitive,so $Q$ is a preorder. On the other hand, $a⪯b$ and $b⪯a$, while $a\ne b$. So the binary relation on $Q$ is notantisymmetric, implying that $Q$ is not a poset.

Since every total order is reflexive, antisymmetric, and transitive,every total order is a poset. The notion of partial order is weakerthan that of total order. A total order must obey the trichotomy law,which states that for any $a$ and $b$ in the order, either $a\le b$ or$b\le a$. Let $P$ be the structure with ground set $\{a,b,c\}$ andbinary relation $\le =\{(a,a),(a,b),(a,c),(b,b),(c,c)\}$. Adiagram of this structure, omitting loops, is displayed below.

Observe that the binary relation on $P$ is reflexive, antisymmetric,and transitive, so $P$ is a poset. On the other hand, neither $b\le c$ nor $c\le b$ holds in $P$. Thus $P$ fails to satisfy thetrichotomy law and is not a total order.

The failure of the trichotomy law for posets motivates the followingterminology. Let $P$ be a poset. If $a\le b$ or $b\le a$ holds in$P$, we say that $a$ and $b$ are comparable; otherwise, we saythey are incomparable. We use the notation $a\parallel b$to indicate that $a$ and $b$ are incomparable.

If $(P,{\le }_P)$ and $(Q,{\le }_Q)$ are posets, then a function$\phi :P\to Q$ is said to be order-preserving, ormonotone, provided that it preserves inequalities. That is,$\phi$ is order-preserving if whenever $a{\le }_{P}b$ holds, it followsthat $\phi (a){\le }_Q\phi (b)$ also holds. The identity function onthe ground set of a poset is order-preserving. If $(P,{\le }_P)$,$(Q,{\le }_Q)$, and $(R,{\le }_R)$ are posets and $\phi :P\to Q$ and$\psi :Q\to R$ are order-preserving functions, then thecomposition $\psi \circ \phi :P\to R$ is also order-preserving.

Posets together with order-preserving functions form a category, whichwe denoted by $\mathrm{𝐏𝐨𝐬𝐞𝐭}$. Thus an order-preserving functionbetween the ground sets of two posets is sometimes also called amorphism of posets. The category of posets hasarbitrary products (http://planetmath.org/ProductofPosets). Moreover, everyposet can itself be viewed as a category, and it turns out that amorphism of posets is the same as a functor between the two posets.

Examples of posets

The two extreme posets are the chain, in which any two elements arecomparable, and the antichain, in which no two elements arecomparable. A poset with a singleton underlying set is necessarilyboth a chain and an antichain, but a poset with a larger underlyingset cannot be both.

Example 1.

Let $\mathbb{N}$ be the set of natural numbers. Inductively define abinary relation $\le$ on $\mathbb{N}$ by the following rules:

• for any $n\in \mathbb{N}$, the relation $0\le n$ holds; and

• whenever $m\le n$, the relation $m+1\le n+1$ also holds.

Then $(\mathbb{N},\le )$ is a chain, hence a poset. This structure canbe naturally embedded in the larger chains of the integers, therational numbers, and the real numbers.

The next example shows that nontrivial antichains exist.

Example 2.

Let $P$ be a set with cardinality greater than $1$. Let $\le$ be thediagonal of $P$. Thus $\le$ represents equality, which is trivially apartial order relation (which is also the intersection of all partialorderings on $P$). By construction, $a\le b$ in $P$ if and only$a=b$. Thus no two elements of $P$ are comparable.

So far the only posets we have seen are chains and antichains. Mostposets are neither. The following construction gives many suchexamples.

Example 3.

If $X$ is any set, the powerset $P=P(X)$ of $X$ is partially orderedby inclusion, that is, by the relation $A\le B$ if and only if$A\subseteq B$.

There are important structure theorems for posets concerning chainsand antichains. One of the foundational results is Dilworth’stheorem. This theorem was massively generalized by Greene andKleitman.

A final example shows that one can manufacture a poset from an existing one.

Example 4.

Let $P$ be a poset ordered by $\le$. The dual poset of $P$ is defined asfollows: it has the same underlying set as $P$, whose order is defined by $a{\le }^{\prime }b$iff $b\le a$. It is easy to see that ${\le }^{\prime }$ is a partial order. The dual of $P$is usually denoted by $P^{\partial }$.

Graph-theoretical view of posets

Let $P$ be a poset with strict partial order . Then $P$ can beviewed as a directed graph with vertex set the ground set of $P$ andedge set . For example, the following diagram displays the Booleanalgebra $B_2$ as a directed graph.

If $P$ is a sufficiently complicated poset, then drawing all of theedges of $P$ can obscure rather than reveal the structure of $P$. Forthis reason it is convenient to restrict attention to a subrelation of from which can be uniquely recovered.

We describe a method of constructing a canonical subgraph of $P$ fromwhich the partial order can be recovered as long as every interval of$P$ has finite height. If $a$ and $b$ are elements of $P$, then wesay that $b$ covers $a$ if and there are no elements of$P$ strictly larger than $a$ but strictly smaller than $b$, that is, if$[a,b]=\{a,b\}$. Two elements are said to be consecutive ifone covers another. Define a binary relation on $P$ by

By construction, the binary relation is a subset of . Since is transitive, the transitiveclosure (http://planetmath.org/ClosureOfASetViaRelations) of is also contained in .

Proposition.

Suppose every interval of $P$ has finite height. Then is thetransitive closure of .

Proof.

We prove this by induction on height. By definition of , if and the height of $[a,b]$ is 1, then .

Assume for induction that whenever and the height of $[a,b]$ isat most $n$, then $(a,b)$ is in the transitive closure of .Suppose that and that the height of $[a,b]$ is $n+1$. Sinceevery chain in $[a,b]$ is finite, it contains an element $c$ which isstrictly larger than $a$ and minimal (http://planetmath.org/MaximalElement) withrespect to this property. Therefore $[a,c]=\{a,c\}$, from which weconclude that . Since the interval $[c,b]$ is a propersubinterval of $[a,b]$, it has height at most $n$, so by the inductionassumption we conclude that $(c,b)$ is in the transitive closure of. Since $(a,c)$ and $(c,b)$ are in the transitive closure of, so is $(a,b)$. Hence whenever and the height of $[a,b]$is at most $n+1$, then $(a,b)$ is in the transitive closure of .

This completes the proof.∎

In the same way we associated a graph to we can associate a graphto . The graph is usually called the Hasse diagram of the poset.Below we display the graph associated to the cover relation of$B_2$.

For simplicity, the Hasse diagram of a poset is usually drawn as anundirected graph. Elements which are higher in the partial orderrelation are drawn physically higher. Since a strict partial order isacyclic, this can be done uniquely and the partial order can berecovered from the drawing.