poset

poset

\xyoption

all

A

poset

, or

partially ordered set

, consists of a set

P

and a

binary relation

≤

on

P

which

satisfies

the following
properties:

•

≤

is

reflexive

(

http://planetmath.org/Reflexive

), so

a

≤

a

always
holds;

•

≤

is

antisymmetric

, so if

a

≤

b

and

b

≤

a

hold, then

a

=

b

; and

•

≤

is

transitive

(

http://planetmath.org/Transitive3

), so if

a

≤

b

and

b

≤

c

hold, then

a

≤

c

also holds.

The

relation

≤

is called a

partial order

on

P

. In
practice,

(

P

,

≤

)

is usually conflated with

P

; if a distinction is
needed,

P

is called the

ground set

or

underlying set

of

(

P

,

≤

)

.
The binary relation

<

defined by removing the diagonal from

≤

(i.e.

a

<

b

iff

a

≤

b

and

a

≠

b

) satisfies the following properties:

•

<

is

irreflexive

, so if

a

<

b

holds, then

b

<

a

does not
hold; and

•

<

is

transitive

.

Since

≤

is reflexive, it can be uniquely recovered from

<

by adding
the diagonal. For this reason, an irreflexive and transitive binary
relation

<

(called a

strict partial order

) also defines a poset, by means
of the associated relation

≤

described above (which is called

weak partial order

).

Since every partial order is reflexive and transitive, every poset is
a

preorder

. The notion of partial order is stricter than that of
preorder, Let

Q

be the

structure

with ground set

Q

=

{

a

,

b

}

and
binary relation

⪯

=

{

(

a

,

a

)

,

(

a

,

b

)

,

(

b

,

a

)

,

(

b

,

b

)

}

. A diagram
of this structure, omitting loops, is displayed below.

\xymatrix

⁢

b

⁢

\ar

⁢

@

/

/

[

d

]

⁢

a

⁢

\ar

⁢

@

/

/

[

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

≠

b

. So the binary relation on

Q

is not
antisymmetric, 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 weaker
than 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

≤

b

or

b

≤

a

. Let

P

be the structure with ground set

{

a

,

b

,

c

}

and
binary relation

≤

=

{

(

a

,

a

)

,

(

a

,

b

)

,

(

a

,

c

)

,

(

b

,

b

)

,

(

c

,

c

)

}

. A
diagram of this structure, omitting loops, is displayed below.

\xymatrix

⁢

b

⁢

&

⁢

&

⁢

c

⁢

&

⁢

a

⁢

\ar

⁢

[

u

⁢

l

]

⁢

\ar

⁢

[

u

⁢

r

]

⁢

&

Observe that the binary relation on

P

is reflexive, antisymmetric,
and transitive, so

P

is a poset. On the other hand, neither

b

≤

c

nor

c

≤

b

holds in

P

. Thus

P

fails to satisfy the
trichotomy law and is not a total order.

The failure of the trichotomy law for posets motivates the following
terminology. Let

P

be a poset. If

a

≤

b

or

b

≤

a

holds in

P

, we say that

a

and

b

are

comparable

; otherwise, we say
they are

incomparable

. We use the notation

a

∥

b

to indicate that

a

and

b

are incomparable.

If

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

are posets, then a

function

φ

:

P

→

Q

is said to be

order-preserving

, or

monotone

, provided that it preserves inequalities. That is,

φ

is order-preserving if whenever

a

≤

P

b

holds, it follows
that

φ

⁢

(

a

)

≤

Q

φ

⁢

(

b

)

also holds. The

identity function

on
the ground set of a poset is order-preserving. If

(

P

,

≤

P

)

,

(

Q

,

≤

Q

)

, and

(

R

,

≤

R

)

are posets and

φ

:

P

→

Q

and

ψ

:

Q

→

R

are order-preserving functions, then the

composition

ψ

∘

φ

:

P

→

R

is also order-preserving.

Posets together with order-preserving functions form a category, which
we denoted by

𝐏𝐨𝐬𝐞𝐭

. Thus an order-preserving function
between the ground sets of two posets is sometimes also called a

morphism of posets

. The category of posets has
arbitrary

products

(

http://planetmath.org/ProductofPosets

). Moreover, every
poset can itself be viewed as a category, and it turns out that a
morphism 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

are
comparable, and the

antichain

, in which no two elements are
comparable. A poset with a singleton underlying set is necessarily
both a chain and an antichain, but a poset with a larger underlying
set cannot be both.

Example 1

.

Let

ℕ

be the set of natural numbers. Inductively define a
binary relation

≤

on

ℕ

by the following rules:

•

for any

n

∈

ℕ

, the relation

0

≤

n

holds; and

•

whenever

m

≤

n

, the relation

m

+

1

≤

n

+

1

also holds.

Then

(

ℕ

,

≤

)

is a chain, hence a poset. This structure can
be naturally embedded in the larger chains of the integers, the

rational 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

≤

be the
diagonal of

P

. Thus

≤

represents

equality

, which is trivially a
partial order relation (which is also the

intersection

of all partial
orderings on

P

). By construction,

a

≤

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. Most
posets are neither. The following construction gives many such
examples.

Example 3

.

If

X

is any set, the

powerset

P

=

P

⁢

(

X

)

of

X

is partially ordered
by

inclusion

, that is, by the relation

A

≤

B

if and only if

A

⊆

B

.

There are important structure

theorems

for posets concerning chains
and antichains. One of the foundational results is Dilworth’s
theorem. This theorem was massively generalized by Greene and
Kleitman.

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

Example 4

.

Let

P

be a poset ordered by

≤

. The

dual poset

of

P

is defined as
follows: it has the same underlying set as

P

, whose order is defined by

a

≤

′

b

iff

b

≤

a

. It is easy to see that

≤

′

is a partial order. The dual of

P

is usually denoted by

P

∂

.

Graph-theoretical view of posets

Let

P

be a poset with strict partial order

<

. Then

P

can be
viewed as a

directed graph

with

vertex set

the ground set of

P

and

edge set

<

. For example, the following diagram displays the

Boolean
algebra

B

2

as a directed graph.

\xymatrix

⁢

&

⁢

{

0

,

1

}

⁢

&

⁢

{

0

}

⁢

\ar

⁢

[

u

⁢

r

]

⁢

&

⁢

&

⁢

{

1

}

⁢

\ar

⁢

[

u

⁢

l

]

⁢

&

⁢

∅

⁢

\ar

⁢

[

u

⁢

l

]

⁢

\ar

⁢

[

u

⁢

u

]

⁢

\ar

⁢

[

u

⁢

r

]

⁢

&

If

P

is a sufficiently complicated poset, then drawing all of the
edges of

P

can obscure rather than reveal the structure of

P

. For
this 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

from
which 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 we
say that

b

covers

a

if

a

<

b

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

if
one covers another. Define a binary relation

<

:

on

P

by

a

<

:

b

if and only if

b

covers

a

.

By construction, the binary relation

<

:

is a subset of

<

. Since

<

is transitive, the

transitive
closure

(

http://planetmath.org/ClosureOfASetViaRelations

) of

<

:

is also

contained

in

<

.

Proposition

.

Suppose every interval of

P

has finite height. Then

<

is the
transitive closure of

<

:

.

Proof.

We prove this by

induction

on height. By definition of

<

:

, if

a

<

b

and the height of

[

a

,

b

]

is 1, then

a

<

:

b

.

Assume for induction that whenever

a

<

b

and the height of

[

a

,

b

]

is
at most

n

, then

(

a

,

b

)

is in the transitive closure of

<

:

.
Suppose that

a

<

b

and that the height of

[

a

,

b

]

is

n

+

1

. Since
every chain in

[

a

,

b

]

is finite, it contains an element

c

which is
strictly larger than

a

and

minimal

(

http://planetmath.org/MaximalElement

) with
respect to this property. Therefore

[

a

,

c

]

=

{

a

,

c

}

, from which we
conclude that

a

<

:

c

. Since the interval

[

c

,

b

]

is a proper
subinterval of

[

a

,

b

]

, it has height at most

n

, so by the induction

assumption

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

a

<

b

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 graph
to

<

:

. The graph is usually called the

Hasse diagram

of the poset.
Below we display the graph associated to the

cover relation

<

:

of

B

2

.

\xymatrix

⁢

&

⁢

{

0

,

1

}

⁢

&

⁢

{

0

}

⁢

\ar

⁢

[

u

⁢

r

]

⁢

&

⁢

&

⁢

{

1

}

⁢

\ar

⁢

[

u

⁢

l

]

⁢

&

⁢

∅

⁢

\ar

⁢

[

u

⁢

l

]

⁢

\ar

⁢

[

u

⁢

r

]

⁢

&

For simplicity, the Hasse diagram of a poset is usually drawn as an

undirected graph

. Elements which are higher in the partial order
relation are drawn physically higher. Since a strict partial order is

acyclic

, this can be done uniquely and the partial order can be
recovered from the drawing.

References

1

Grätzer, G.,

General

lattice theory

, 2nd ed., Birkhäuser, 1998.

2

Stanley, R.,

Enumerative Combinatorics

, vol. 1, 2nd ed., Cambridge
University Press, Cambridge, 1996.

Title

poset

Canonical name

Poset

Date of creation

2013-03-22 11:43:41

Last modified on

2013-03-22 11:43:41

Owner

mps (409)

Last modified by

mps (409)

Numerical id

22

Author

mps (409)

Entry type

Definition

Classification

msc 06A99

Synonym

partially ordered set

Related topic

Relation

Related topic

PartialOrder

Related topic

Semilattice

Related topic

StarProduct

Related topic

HasseDiagram

Related topic

GreatestLowerBound

Related topic

NetsAndClosuresOfSubspaces

Related topic

OrderPreservingMap

Related topic

DisjunctionPropertyOfWallman

Defines

comparable

Defines

incomparable

Defines

cover

Defines

covering

Defines

order-preserving function

Defines

monotone

Defines

monotonic

Defines

order morphism

Defines

morphism of posets

Defines

dual poset

Defines

consecutive

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