De Morgan algebra

De Morgan algebra

A

bounded

distributive lattice

L

is called a

De Morgan algebra

if there exists a unary

operator

∼

:

L

→

L

such that

1.

∼

(

∼

a

)

=

a

and

2.

∼

(

a

∨

b

)

=

(

∼

a

)

∧

(

∼

b

)

.

From the definition, we have the following properties:

•

∼

is a

bijection

, since for any

a

∈

L

,

a

=

∼

(

∼

a

)

.

•

∼

(

a

∧

b

)

=

∼

[

(

∼

(

∼

a

)

)

∧

(

∼

(

∼

b

)

)

]

=

∼

[

∼

(

(

∼

a

)

∨

(

∼

b

)

)

]

=

(

∼

a

)

∨

(

∼

b

)

, which is the dual statement of (2) above. This, together with condition (2), are commonly known as the De Morgan’s laws.

•

∼

0

=

∼

(

0

∧

(

∼

a

)

)

=

(

∼

0

)

∨

a

for all

a

∈

L

, so

∼

0

=

1

. Dually,

∼

1

=

0

. As a result, a De Morgan algebra is an

Ockham algebra

.

•

a

≤

b

iff

b

=

a

∨

b

iff

∼

b

=

∼

(

a

∨

b

)

=

(

∼

a

)

∧

(

∼

b

)

iff

∼

b

≤

∼

a

.

•

A

Boolean algebra

is always a De Morgan algebra, where the

∼

is the

complementation

operator

′

. The

converse

is not true. In general,

∼

a

is not a

complement

of

a

(that is,

(

∼

a

)

∧

a

≠

0

and

(

∼

a

)

∨

a

≠

1

). Otherwise,

L

is a

complemented lattice

and consequently a Boolean algebra.

Furthermore, a

Kleene algebra

is, by definition, a De Morgan algebra. But the converse is false. For example, consider

L

=

𝐧

×

𝐧

, where

𝐧

=

{

0

,

1

,

…

,

n

}

is a chain with the usual

ordering

. Define

∼

on

L

by

∼

(

a

,

b

)

=

(

n

-

b

,

n

-

a

)

. Then

∼

2

(

a

,

b

)

=

(

a

,

b

)

. The De Morgan’s laws follow from the

identity

n

-

(

a

∨

b

)

=

(

n

-

a

)

∧

(

n

-

b

)

applied to each of the two

components

. But

L

is not Kleene in general. Take

n

=

3

, then

∼

(

1

,

2

)

=

(

1

,

2

)

and

∼

(

2

,

1

)

=

(

2

,

1

)

. But

(

1

,

2

)

=

∼

(

1

,

2

)

∧

(

1

,

2

)

and

(

2

,

1

)

=

∼

(

2

,

1

)

∨

(

2

,

1

)

are not

comparable

.

Next, for any

a

,

b

∈

L

, define

a

-

b

:=

a

∧

(

∼

b

)

. Then

-

is a

binary operator

. It has the following properties:

•

a

-

0

=

a

∧

(

∼

0

)

=

a

∧

1

=

a

.

•

0

-

a

=

0

∧

(

∼

a

)

=

0

.

•

a

-

1

=

a

∧

(

∼

1

)

=

a

∧

0

=

0

.

•

1

-

a

=

1

∧

(

∼

a

)

=

∼

a

.

•

(

a

-

b

)

-

c

=

(

a

∧

(

∼

b

)

)

∧

(

∼

c

)

=

a

∧

(

∼

(

b

∨

c

)

)

=

a

-

(

b

∨

c

)

.

Finally, we define for

a

,

b

∈

L

,

a

+

b

=

(

a

-

b

)

∨

(

b

-

a

)

. This is again a binary operator, with the following properties:

•

a

+

b

=

b

+

a

. This is obvious by the

symmetry

in the definition of

+

.

•

a

+

0

=

a

. We have

a

+

0

=

(

a

-

0

)

∨

(

0

-

a

)

=

a

∨

0

=

a

.

•

a

+

1

=

∼

a

, since

a

+

1

=

(

a

-

1

)

∨

(

1

-

a

)

=

0

∨

(

∼

a

)

=

∼

a

. In particular

1

+

1

=

0

.

•

a

+

a

=

(

a

-

a

)

∨

(

a

-

a

)

=

a

-

a

=

a

∧

(

∼

a

)

. If we define

2

⁢

a

:=

a

+

a

, then

a

+

2

a

=

(

a

-

2

a

)

∨

(

2

a

-

a

)

=

(

a

∧

(

a

∨

(

∼

a

)

)

∨

(

(

a

∧

(

∼

a

)

∧

(

∼

a

)

)

=

a

∨

(

(

a

∧

(

∼

a

)

)

=

a

.

•

More generally, we have

a

+

(

a

+

b

)

=

(

a

-

(

a

+

b

)

)

∨

(

(

a

+

b

)

-

a

)

=

(

a

-

(

(

a

-

b

)

∨

(

b

-

a

)

)

)

∨

(

(

(

a

-

b

)

∨

(

b

-

a

)

)

-

a

)

=

(

a

∧

(

∼

a

∨

b

)

∧

(

∼

b

∨

a

)

)

∨

(

(

(

∼

b

∧

a

)

∨

(

∼

a

∧

b

)

)

∧

(

∼

a

)

)

=

(

a

∧

(

∼

a

∨

b

)

)

∨

(

(

(

∼

b

∧

a

)

∧

(

∼

a

)

)

∨

(

∼

a

∧

b

)

)

=

(

(

a

∧

(

∼

a

∨

b

)

)

∨

(

∼

a

∧

b

)

)

∨

(

∼

b

∧

(

∼

a

∧

a

)

)

=

(

(

∼

a

∧

a

)

∨

(

a

∧

b

)

∨

(

∼

a

∧

b

)

)

∨

(

∼

b

∧

2

a

)

=

(

2

a

∨

(

(

∼

a

∧

a

)

∨

b

)

)

∨

(

∼

b

∧

2

a

)

=

(

2

⁢

a

∨

(

2

⁢

a

∨

b

)

)

∨

(

∼

b

∧

2

⁢

a

)

=

(

2

⁢

a

∨

b

)

∨

(

∼

b

∧

2

⁢

a

)

=

2

a

∨

(

∼

b

∧

2

a

)

∨

b

=

2

⁢

a

∨

b

.

Remark

. Since a De Morgan algebra is an Ockham algebra, a

morphism

between any two objects in the category of De Morgan algebras behaves just like an Ockham algebra

homomorphism

: it preserves

∼

.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998)

Title

De Morgan algebra

Canonical name

DeMorganAlgebra

Date of creation

2013-03-22 16:09:25

Last modified on

2013-03-22 16:09:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D30

Classification

msc 03G10

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