lattice filter

lattice filter

Let

L

be a

lattice

. A

filter

(of

L

) is the dual

concept

of an ideal (

http://planetmath.org/LatticeIdeal

). Specifically, a filter

F

of

L

is a non-empty subset of

L

such that

1.

F

is a sublattice of

L

, and

2.

for any

a

∈

F

and

b

∈

L

,

a

∨

b

∈

F

.

The first condition can be replaced by a weaker one:
for any

a

,

b

∈

F

,

a

∧

b

∈

F

.

An

equivalent

characterization

of a filter

I

in a lattice

L

is

1.

for any

a

,

b

∈

F

,

a

∧

b

∈

F

, and

2.

for any

a

∈

F

, if

a

≤

b

, then

b

∈

F

.

Note that the dualization switches the meet and join

operations

, as well as reversing the

ordering

relationship.

Special Filters

. Let

F

be a filter of a lattice

L

. Some of the common types of filters are defined below.

•

F

is a

proper filter

if

F

≠

L

, and, if

L

contains

0

,

F

≠

0

.

•

F

is a

prime filter

if it is proper, and

a

∨

b

∈

F

implies that either

a

∈

F

or

b

∈

F

.

•

F

is an

ultrafilter

(or

maximal filter

) of

L

if

F

is proper and the only filter properly contains

F

is

L

.

•

filter generated by a set

. Let

X

be a subset of a lattice

L

. Let

T

be the set of all filters of

L

containing

X

. Since

T

≠

∅

(

L

∈

T

), the

intersection

N

of all

elements

in

T

, is also a filter of

L

that contains

X

.

N

is called the

filter generated by

X

, written

[

X

)

. If

X

is a

singleton

{

x

}

, then

N

is said to be a

principal filter

generated by

x

, written

[

x

)

.

Examples

.

1.

Consider the

positive integers

, with meet and join defined by the

greatest common divisor

and the least common

multiple

operations. Then the

positive

even numbers

form a filter, generated by

2

. If we toss in

3

as an additional element, then

1

=

2

∧

3

∈

[

{

2

,

3

}

)

and consequently any positive integer

i

∈

[

{

2

,

3

}

)

, since

1

≤

i

. In general, if

p

,

q

are relatively prime, then

[

{

p

,

q

}

)

=

ℤ

+

. In fact, any proper filter in

ℤ

+

is principal. When the generator is prime, the filter is prime, which is also maximal. So prime filters and ultrafilters coincide in

ℤ

+

.

2.

Let

A

be a set and

2

A

the

power set

of

A

. If the

set inclusion

is the ordering defined on

2

A

, then the

definition

of a filter here coincides with the ususal definition of a filter (

http://planetmath.org/Filter

) on a set in general.

Remark

. If

F

is both a filter and an ideal of a lattice

L

, then

F

=

L

.

Title

lattice filter

Canonical

name

LatticeFilter

Date of creation

2013-03-22 15:49:01

Last modified on

2013-03-22 15:49:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B10

Synonym

ultra filter

Synonym

ultra-filter

Synonym

maximal filter

Related topic

Ultrafilter

Related topic

UpperSet

Related topic

LatticeIdeal

Related topic

OrderIdeal

Defines

filter

Defines

prime filter

Defines

ultrafilter

Defines

filter generated by

Defines

principal filter

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