order ideal

order ideal

Order Ideals and Filters

Let

P

be a poset. A subset

I

of

P

is said to be an

order ideal

if

•

I

is a

lower set

:

↓

I

=

I

, and

•

I

is a

directed set

:

I

is non-empty, and every pair of

elements

in

I

has an

upper bound

in

I

.

An order ideal is also called an ideal for short. An ideal is said to be

principal

if it has the form

↓

x

for some

x

∈

P

.

Given a subset

A

of a poset

P

, we say that

B

is the

ideal generated by

A

if

B

is the smallest order ideal (of

P

) containing

A

.

B

is denoted by

⟨

A

⟩

. Note that

⟨

A

⟩

exists iff

A

is a directed set. In particular, for any

x

∈

P

,

↓

x

is the ideal generated by

x

. Also, if

P

is an

upper semilattice

, then for any

A

⊆

P

, let

A

′

be the set of finite joins of elements of

A

, then

A

′

is a directed set, and

⟨

A

⟩

=

↓

A

′

.

Dually, an

order filter

(or simply a

filter

) in

P

is a non-empty subset

F

which is both an upper set and a filtered set (every pair of elements in

F

has a

lower bound

in

F

). A

principal filter

is a filter of the form

↑

x

for some

x

∈

P

.

Remark

.
This is a

generalization

of the notion of a filter (

http://planetmath.org/Filter

) in a set. In fact, both ideals and filters are generalizations of ideals and filters in semilattices and lattices.

Examples in a Semilattice

A subset

I

in an upper semilattice

P

is a

semilattice ideal

if

1.

if

a

,

b

∈

I

, then

a

∨

b

∈

I

(condition for being an upper subsemilattice)

2.

if

a

∈

I

and

b

≤

a

, then

b

∈

I

Then the two definitions are

equivalent

: if

P

is an upper semilattice, then

I

⊆

P

is a semilattice ideal iff

I

is an order ideal of

P

: if

I

is a semilattice ideal, then

I

is clearly a lower and directed (since

a

∨

b

is an upper bound of

a

and

b

); if

I

is an order ideal, then condition 2 of a semilattice ideal is satisfied. If

a

,

b

∈

I

, then there is a

c

∈

I

that is an upper bound of

a

and

b

. Since

I

is lower, and

a

∨

b

≤

c

, we have

a

∨

b

∈

I

.

Going one step further, we see that if

P

is a

lattice

, then a lattice ideal is exactly an order ideal: if

I

is a lattice ideal, then it is clearly an upper subsemilattice, and if

b

≤

a

∈

I

, then

b

=

a

∧

b

∈

I

also, so that

I

is a semilattice ideal. On the other hand, if

I

is a semilattice ideal, then

I

is an upper subsemilattice, as well as a lower subsemilattice, for if

a

∈

I

, then

a

∧

b

∈

I

as well since

a

∧

b

≤

a

. This shows that

I

is a lattice ideal.

Dually, we can define a

filter

in a

lower semilattice

, which is equivalent to an order filter of the underly poset. Going one step futher, we also see that a lattice filter in a lattice is an order filter of the underlying poset.

Remark

. An alternative but equivalent

characterization

of a semilattice ideal

I

in an upper semilattice

P

is the following:

a

,

b

∈

I

iff

a

∨

b

∈

I

.

Title

order ideal

Canonical name

OrderIdeal

Date of creation

2013-03-22 17:01:14

Last modified on

2013-03-22 17:01:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Classification

msc 06A12

Synonym

filter

Synonym

ideal

Related topic

Filter

Related topic

LatticeFilter

Related topic

LatticeIdeal

Defines

order filter

Defines

semilattice ideal

Defines

semilattice filter

Defines

subsemilattice

Defines

principal ideal

Defines

principal filter

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