Boolean quotient algebra

Boolean quotient algebra

Quotient Algebras via Congruences

Let

A

be a

Boolean algebra

. A

congruence

on

A

is an

equivalence relation

Q

on

A

such that

Q

respects the Boolean operations:

•

if

a

⁢

Q

⁢

b

and

c

⁢

Q

⁢

d

, then

(

a

∨

c

)

⁢

Q

⁢

(

b

∨

d

)

•

if

a

⁢

Q

⁢

b

, then

a

′

⁢

Q

⁢

b

′

By de Morgan’s laws, we also have

a

⁢

Q

⁢

b

and

c

⁢

Q

⁢

d

implying

(

a

∧

c

)

⁢

Q

⁢

(

b

∧

d

)

.

When

a

is

congruent

to

b

, we usually write

a

≡

b

(

mod

Q

)

.

Let

B

be the set of congruence classes:

B

=

A

/

Q

, and write

[

a

]

⁢

Q

, or simply

[

a

]

for the congruence class containing the

element

a

∈

A

. Define on

B

the following

operations

:

•

[

a

]

∨

[

b

]

:=

[

a

∨

b

]

•

[

a

]

′

:=

[

a

′

]

Because

Q

respects join and

complementation

, it is clear that these are well-defined operations on

B

. Furthermore, we may define

[

a

]

∧

[

b

]

:=

(

[

a

]

′

∨

[

b

]

′

)

′

=

(

[

a

′

]

∨

[

b

′

]

)

′

=

[

a

′

∨

b

′

]

′

=

[

(

a

′

∨

b

′

)

′

]

=

[

a

∧

b

]

. It is also easy to see that

[

1

]

and

[

0

]

are the top and bottom elements of

B

. Finally, it is straightforward to verify that

B

is a Boolean algebra. The algebra

B

is called the

Boolean quotient algebra

of

A

via the congruence

Q

.

Quotient Algebras via Ideals and Filters

It is also possible to define quotient algebras via

Boolean ideals

and Boolean filters. Let

A

be a Boolean algebra and

I

an ideal of

A

. Define

binary relation

∼

on

A

as follows:

a

∼

b

if and only if

a

⁢

Δ

⁢

b

∈

I

,

where

Δ

is the

symmetric difference operator

on

A

. Then

1.

∼

is an equivalence on

A

, because

–

a

⁢

Δ

⁢

a

=

0

∈

I

, so

∼

is

reflexive

–

b

⁢

Δ

⁢

a

=

a

⁢

Δ

⁢

b

, so

∼

is

symmetric

, and

–

if

a

∼

b

and

b

∼

c

, then

a

∼

c

; to see this, note that

(

a

-

b

)

∨

(

b

-

c

)

=

(

(

a

-

b

)

∨

b

)

∧

(

(

a

-

b

)

∨

c

′

)

=

(

a

∨

b

)

∧

(

(

a

-

b

)

∨

c

′

)

. Since the LHS (and hence the RHS) is in

I

, and that

a

≤

a

∨

b

and

c

′

≤

(

a

-

b

)

∨

c

′

, RHS

≥

a

∧

c

′

=

a

-

c

∈

I

too. Similarly

c

-

a

∈

I

so that

a

∼

c

.

2.

∼

respects

∨

and

′

, because

–

if

a

∼

b

and

c

∼

d

, then

(

a

∨

c

)

-

(

b

∨

d

)

=

(

a

∨

c

)

∧

(

b

∨

d

)

′

=

(

a

∨

c

)

∧

(

b

′

∧

d

′

)

=

(

a

∧

(

b

′

∧

d

′

)

)

∨

(

c

∧

(

b

′

∧

d

′

)

)

≤

(

a

∧

b

′

)

∨

(

c

∧

d

′

)

∈

I

, so that

(

a

∨

c

)

-

(

b

∨

d

)

∈

I

as well. That

(

b

∨

d

)

-

(

a

∨

c

)

∈

I

is proved similarly. Hence

(

a

∨

c

)

∼

(

b

∨

d

)

.

–

a

′

⁢

Δ

⁢

b

′

=

a

⁢

Δ

⁢

b

, so

∼

preserves

′

.

Thus,

∼

is a congruence on

A

. The quotient algebra

A

/

∼

is called the

quotient algebra

of

A

via the ideal

I

, and is often denoted by

A

/

I

.

From this congruence

∼

, one can re-capture the ideal:

I

=

[

0

]

.

Dually, one can obtain a quotient algebra from a Boolean filter. Specifically, if

F

is a filter of a Boolean algebra

A

, define

∼

on

A

as follows:

a

∼

b

if and only if

a

↔

b

∈

F

,

where

↔

is the

biconditional

operator

on

A

. Then it is easy to show that

∼

too is a congruence on

A

, so that one forms the

quotient algebra

of

A

via the filter

F

, denoted by

A

/

F

. Of course, an easier approach to this is to realize that

F

is a filter of

A

iff

F

′

:=

{

a

′

∣

a

∈

F

}

is an ideal of

A

, and the process of forming

A

/

F

′

turns out to be identical to

A

/

F

.

From

∼

, the filter

F

can be recovered:

F

=

[

1

]

.

In fact, given a congruence

Q

, the congruence class

[

0

]

⁢

Q

is a Boolean ideal and the congruence class

[

1

]

⁢

Q

is a Boolean filter, and that the quotient algebras derived from

Q

,

[

0

]

⁢

Q

and

[

1

]

⁢

Q

are all the same:

A

/

Q

=

A

/

[

0

]

⁢

Q

=

A

/

[

1

]

⁢

Q

.

Title

Boolean quotient algebra

Canonical name

BooleanQuotientAlgebra

Date of creation

2013-03-22 17:59:09

Last modified on

2013-03-22 17:59:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

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