lattice ideal

lattice ideal

Let

L

be a

lattice

. An

ideal

I

of

L

is a non-empty subset of

L

such that

1.

I

is a sublattice of

L

, and

2.

for any

a

∈

I

and

b

∈

L

,

a

∧

b

∈

I

.

Note the similarity between this definition and the definition of an ideal (

http://planetmath.org/Ideal

) in a ring (except in a ring with 1, an ideal is almost never a

subring

)

Since the fact that

a

∧

b

∈

I

for

a

,

b

∈

I

in the first condition is already implied by the second condition, we can replace the first condition by a weaker one:

for any

a

,

b

∈

I

,

a

∨

b

∈

I

.

Another

equivalent

characterization

of an ideal

I

in a lattice

L

is

1.

for any

a

,

b

∈

I

,

a

∨

b

∈

I

, and

2.

for any

a

∈

I

, if

b

≤

a

, then

b

∈

I

.

Here’s a quick proof. In fact, all we need to show is that the two second conditions are equivalent for

I

. First assume that for any

a

∈

I

and

b

∈

L

,

a

∧

b

∈

I

. If

b

≤

a

, then

b

=

a

∧

b

∈

I

.

Conversely

, since

a

∧

b

≤

a

∈

I

,

a

∧

b

∈

I

as well.

Special Ideals

. Let

I

be an ideal of a lattice

L

. Below are some common types of ideals found in

lattice theory

.

•

I

is

proper

if

I

≠

L

.

•

If

L

contains

0

,

I

is said to be

non-trivial

if

I

≠

0

.

•

I

is a

prime ideal

if it is proper, and for any

a

∧

b

∈

I

, either

a

∈

I

or

b

∈

I

.

•

I

is a

maximal ideal

of

L

if

I

is proper and the only ideal having

I

as a

proper subset

is

L

.

•

ideal generated by

a set

. Let

X

be a subset of a lattice

L

. Let

S

be the set of all ideals of

L

containing

X

. Since

S

≠

∅

(

L

∈

S

), the

intersection

M

of all

elements

in

S

, is also an ideal of

L

that contains

X

.

M

is called the

ideal generated by

X

, written

(

X

]

. If

X

is a singleton

{

x

}

, then

M

is said to be a

principal ideal

generated by

x

, written

(

x

]

. (Note that this construction can be easily carried over to the construction of a

sublattice generated by

a subset of a lattice).

Remarks

. Let

L

be a lattice.

1.

Given any subset

X

⊂

L

, let

X

′

be the set consisting of all finite joins of elements of

X

, which is clearly a

directed set

. Then

↓

X

′

, the down set of

X

′

, is

(

X

]

. Any element of

(

X

]

is less than or equal to a finite join of elements of

X

.

2.

If

L

is a

distributive lattice

, every

maximal ideal is prime

. Suppose

I

⊆

L

is

maximal

and

a

∧

b

∈

I

with

a

∉

I

. Then the ideal generated by

I

and

a

must be

L

, so that

b

≤

p

∨

a

for some

p

∈

I

. Then

b

=

b

∧

b

≤

(

p

∨

a

)

∧

b

=

(

p

∧

b

)

∨

(

a

∧

b

)

∈

I

, which means

b

∈

I

. So

I

is prime.

3.

If

L

is a

complemented lattice

, every prime ideal is maximal. Suppose

I

⊆

L

is prime and

a

∉

I

. Let

b

be a

complement

of

a

, then

b

∈

I

, for otherwise,

0

=

a

∧

b

∉

I

, a

contradiction

. Let

J

be the ideal generated by

I

and

a

, then

1

≤

b

∨

a

∈

J

, so

J

=

L

.

4.

Combining the two results above, in a

Boolean algebra

, an ideal is prime iff it is maximal.

Examples

. In the lattice

L

below,

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

d

⁢

\ar

⁢

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-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

e

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

f

⁢

\ar

⁢

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-

[

l

⁢

d

]

⁢

&

⁢

0

Besides

L

and

{

0

}

, below are all

proper ideals

of

L

:

•

M

=

{

a

,

c

,

d

,

e

,

f

,

0

}

,

•

N

=

{

b

,

c

,

d

,

e

,

f

,

0

}

,

•

R

=

{

c

,

d

,

e

,

f

,

0

}

,

•

S

=

{

d

,

e

,

f

,

0

}

,

•

T

=

{

e

,

0

}

, and

•

U

=

{

f

,

0

}

.

Out of these,

M

,

N

,

S

,

T

,

U

are prime, and

M

,

N

are maximal. The ideal generated by, say

{

c

,

e

}

, is

R

. Looking more closely, we see that

R

can actually be generated by

c

, and so is principal. In fact, all ideals in

L

are principal, generated by their

maximal elements

. It is not hard to see, that in a lattice

L

with acc (

ascending chain condition

), all ideals are principal:

Proof.

. First, let’s show that an ideal

I

in a lattice

L

with acc has at least one maximal element. Suppose

a

∈

I

. If

a

is not maximal in

I

, there is a

a

1

∈

I

such that

a

≤

a

1

. If

a

1

is not maximal in

I

, repeat the process above so we get a chain

a

≤

a

1

≤

a

2

≤

…

in

I

.

Eventually

this chain terminates

a

n

=

a

n

+

1

=

⋯

. Thus

b

=

a

n

is maximal in

I

. Next, suppose that

I

has two distinct maximal elements. Then their join is again in

I

, contradicting maximality. So

b

is unique and all elements

c

such that

c

≤

b

must be in

I

. Therefore,

I

=

(

b

]

.∎

Finally, an example of a sublattice that is not an ideal is the subset

{

b

,

c

,

d

,

e

,

0

}

.

Title

lattice ideal

Canonical name

LatticeIdeal

Date of creation

2013-03-22 15:48:58

Last modified on

2013-03-22 15:48:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B10

Synonym

prime lattice ideal

Synonym

maximal lattice ideal

Related topic

LatticeFilter

Related topic

UpperSet

Related topic

OrderIdeal

Related topic

LatticeOfIdeals

Defines

ideal

Defines

proper ideal

Defines

prime ideal

Defines

sublattice generated by

Defines

ideal generated by

Defines

principal ideal

Defines

maximal ideal

Defines

non-trivial ideal

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