Boolean ideal

Boolean ideal

Let

A

be a

Boolean algebra

and

B

a subset of

A

. The following are

equivalent

:

1.

If

A

is interpreted as a

Boolean ring

,

B

is a ring ideal.

2.

If

A

is interpreted as a Boolean lattice,

B

is a

lattice ideal

.

Before proving this equivalence, we want to mention that a Boolean ring is equivalent to a Boolean lattice, and a more commonly used terminology is a Boolean algebra, which is also valid, as it is an algebra over the

ring of integers

. The standard way of characterizing the ring

structure

from the

lattice

structure is by defining

a

+

b

:=

(

a

′

∧

b

)

∨

(

a

∧

b

′

)

(called the

symmetric difference

) and

a

⋅

b

=

a

∧

b

. From this, we can “solve” for

∨

in terms of

+

and

⋅

:

a

∨

b

=

a

+

b

+

a

⋅

b

.

Proof.

First, suppose

B

is an ideal of the “ring”

A

. If

a

,

b

∈

B

, then

a

∨

b

=

a

+

b

+

a

⋅

b

∈

B

. Suppose now that

a

∈

B

and

c

∈

A

with

c

≤

a

. Then

c

=

c

∧

a

=

c

⋅

a

∈

B

as well. So

B

is a lattice ideal of

A

.

Next, suppose

B

is an ideal of the “lattice”

A

. If

a

,

b

∈

B

, then both

a

′

∧

b

and

a

∧

b

′

are in

B

since the first one is less than or equal to

b

and the second less than or equal to

a

, so their join is in

B

as well, this means that

a

-

b

=

a

+

b

=

(

a

′

∧

b

)

∨

(

a

∧

b

′

)

∈

B

. Furthermore, if

a

∈

B

and

c

∈

A

, then

c

⋅

a

=

c

∧

a

≤

a

∈

B

as well. As a result,

B

is a ring ideal of

A

.
∎

A subset of a Boolean algebra satisfying the two equivalent conditions above is called a

Boolean ideal

, or an

ideal

for short. A

prime Boolean ideal

is a prime lattice ideal, and a

maximal Boolean ideal

is a maximal lattice ideal. Again, these notions and their ring theoretic counterparts match exactly. In fact, one can say more about these ideals in the case of a Boolean algebra:

prime ideals

are precisely the

maximal ideals

. If

A

is a Boolean ring, and

M

is a maximal ideal of

A

, then

A

/

M

is

isomorphic

to

{

0

,

1

}

.

Remark

. The dual notion of a Boolean ideal is a

Boolean filter

, or a filter for short. A Boolean filter is just a

lattice filter

of the Boolean algebra when considered as a lattice. To see the connection between a Boolean ideal and a Boolean lattice, let us define, for any subset

S

of a Boolean algebra

A

, the set

S

′

:=

{

a

′

∣

a

∈

S

}

. It is easy to see that

S

′′

=

S

. Now, if

I

is an ideal, then

I

′

is a filter.

Conversely

, if

F

is a filter,

F

′

is a ideal. In fact, given any Boolean algebra, there is a

Galois connection

I

↦

I

′

,

F

↦

F

′

between the set of Boolean ideals and the set of Boolean filters in

A

. In

addition

,

I

is prime iff

I

′

is. As a result, a filter is prime iff it is an

ultrafilter

(maximal filter).

Title

Boolean ideal

Canonical name

BooleanIdeal

Date of creation

2013-03-22 17:01:59

Last modified on

2013-03-22 17:01:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G05

Classification

msc 03G10

Related topic

BooleanRing

Defines

Boolean filter

Defines

prime Boolean ideal

Defines

maximal Boolean ideal

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