example of cylindric algebra

example of cylindric algebra

In this example, we give two examples of a

cylindric algebra

, in which the first is a special case of the second. The first example also explains why the

algebra

is termed

cylindric

.

Example 1.

Consider

ℝ

3

, the three-dimensional Euclidean space, and

R

:=

{

(

x

,

y

,

z

)

∈

ℝ

3

∣

x

2

+

y

2

+

z

2

≤

1

}

.

Thus

R

is the closed unit ball, centered at the origin

(

0

,

0

,

0

)

. Project

R

onto the

x

-

y

plane, so its image is

p

z

⁢

(

R

)

=

{

(

x

,

y

)

∈

ℝ

2

∣

x

2

+

y

2

≤

1

}

.

Taking its

preimage

, we get a

cylinder

C

z

⁢

(

R

)

:=

p

z

-

1

⁢

p

z

⁢

(

R

)

=

{

(

x

,

y

,

z

)

∈

ℝ

3

∣

x

2

+

y

2

≤

1

}

.

C

z

⁢

(

R

)

has the following

properties

:

R

⊆

C

z

⁢

(

R

)

.

(1)

Furthermore, it can be characterized as follows

C

z

⁢

(

R

)

=

{

(

x

,

y

,

z

)

∈

ℝ

3

∣

∃

r

∈

ℝ

⁢

such that

⁢

(

x

,

y

,

r

)

∈

R

}

.

C

z

⁢

(

R

)

is called the

cylindrification

of

R

with respect to the

variable

z

.
It is easy to see that the

characterization

above permits us to generalize the notion of cylindrification to any subset of

ℝ

, with respect to any of the three variables

x

,

y

,

z

. We have in

addition

to (1) above the following properties:

C

u

⁢

(

∅

)

=

∅

,

(2)

C

u

⁢

(

R

∩

C

u

⁢

(

S

)

)

=

C

u

⁢

(

R

)

∩

C

u

⁢

(

S

)

,

(3)

C

u

⁢

(

C

v

⁢

(

R

)

)

=

C

v

⁢

(

C

u

⁢

(

R

)

)

,

(4)

where

u

,

v

∈

{

x

,

y

,

z

}

and

R

,

S

⊆

ℝ

3

.

Property (2) is

obvious

. To see Property (3), it is enough to assume

u

=

z

(for the other cases follow similarly). First let

(

a

,

b

,

c

)

∈

C

z

⁢

(

R

∩

C

z

⁢

(

S

)

)

. Then there is an

r

∈

ℝ

such that

(

a

,

b

,

r

)

∈

R

and

(

a

,

b

,

r

)

∈

C

z

⁢

(

S

)

, which means there is an

s

∈

ℝ

such that

(

a

,

b

,

s

)

∈

S

. Since

(

a

,

b

,

r

)

∈

R

, we have that

(

a

,

b

,

c

)

∈

C

z

⁢

(

R

)

, and since

(

a

,

b

,

s

)

∈

S

, we have that

(

a

,

b

,

c

)

∈

C

z

⁢

(

S

)

as well. This shows one

inclusion

. Now let

(

a

,

b

,

c

)

∈

C

z

⁢

(

R

)

∩

C

z

⁢

(

S

)

, then there is an

r

∈

ℝ

such that

(

a

,

b

,

r

)

∈

R

. But

(

a

,

b

,

r

)

∈

C

z

⁢

(

S

)

also, so

(

a

,

b

,

c

)

∈

C

z

⁢

(

R

∩

C

z

⁢

(

S

)

)

. To see Property (4), it is enough to assume

u

=

x

and

v

=

y

. Let

(

a

,

b

,

c

)

∈

C

x

⁢

(

C

y

⁢

(

R

)

)

. Then there is an

r

∈

ℝ

such that

(

r

,

b

,

c

)

∈

C

y

⁢

(

R

)

, and so there is an

s

∈

ℝ

such that

(

r

,

s

,

c

)

∈

R

. This implies that

(

a

,

s

,

c

)

∈

C

x

⁢

(

R

)

, which implies that

(

a

,

b

,

c

)

∈

C

y

⁢

(

C

x

⁢

(

R

)

)

. So

C

x

⁢

(

C

y

⁢

(

R

)

)

⊆

C

y

⁢

(

C

x

⁢

(

R

)

)

. The other inclusion then follows immediately.

Next, we define the diagonal set

D

x

⁢

y

:=

{

(

x

,

y

,

z

)

∈

ℝ

3

∣

x

=

y

}

with respect to

x

and

y

. This is just the plane whose

projection

onto the

x

-

y

plane is the line

x

=

y

. We may define a total of nine possible diagonal sets

D

v

⁢

w

where

v

,

w

∈

{

x

,

y

,

z

}

. However, there are in fact four distinct diagonal sets, since

D

u

⁢

u

=

{

p

∈

ℝ

3

∣

u

=

u

}

=

ℝ

3

,

(5)

D

u

⁢

v

=

D

v

⁢

u

,

(6)

where

u

,

v

∈

{

x

,

y

,

z

}

. For any subset

R

⊆

ℝ

3

, set

R

u

⁢

v

:=

R

∩

D

u

⁢

v

. For

instance

,

R

x

⁢

y

=

{

(

a

,

b

,

c

)

∈

R

∣

a

=

b

}

.

We may consider

C

x

,

C

y

,

C

z

as unary operations on

ℝ

3

, and the diagonal sets as

constants

(nullary

operations

) on

ℝ

3

. Two additional noteworthy properties are

C

u

⁢

(

R

u

⁢

v

)

∩

C

u

⁢

(

R

u

⁢

v

′

)

=

∅

if

u

≠

v

,

(7)

C

u

⁢

(

D

u

⁢

v

∩

D

u

⁢

w

)

=

D

v

⁢

w

if

u

∉

{

v

,

w

}

,

(8)

where

u

,

v

,

w

∈

{

x

,

y

,

z

}

.

To see Property (7), we may assume

u

=

x

and

v

=

y

. Suppose

(

a

,

b

,

c

)

∈

C

x

⁢

(

R

x

⁢

y

)

∩

C

x

⁢

(

R

x

⁢

y

′

)

. Then there is

r

∈

ℝ

such that

(

r

,

b

,

c

)

∈

R

x

⁢

y

, which implies that

r

=

b

, or that

(

b

,

b

,

c

)

∈

R

. On the other hand, there is

s

∈

ℝ

such that

(

s

,

b

,

c

)

∈

R

x

⁢

y

′

, which implies

s

=

b

, or that

(

b

,

b

,

c

)

∈

R

′

, a

contradiction

. To see Property (8), we may assume

u

=

x

,

v

=

w

,

w

=

z

. If

(

a

,

b

,

c

)

∈

C

x

⁢

(

D

x

⁢

y

∩

D

x

⁢

z

)

, then there is

r

∈

ℝ

such that

(

r

,

b

,

c

)

∈

D

x

⁢

y

∩

D

x

⁢

z

. So

r

=

b

and

r

=

c

. Therefore,

(

a

,

b

,

c

)

=

(

a

,

r

,

r

)

∈

D

y

⁢

z

. On the other hand, for any

(

a

,

r

,

r

)

∈

D

y

⁢

z

,

(

r

,

r

,

r

)

∈

D

x

⁢

y

∩

D

x

⁢

z

, and so

(

a

,

r

,

r

)

∈

C

x

⁢

(

D

x

⁢

y

∩

D

x

⁢

z

)

as well.

Finally, we note that a subset of

ℝ

3

is just a

ternary relation

on

ℝ

, and the

collection

of all ternary relations on

R

is just

P

⁢

(

ℝ

3

)

.

Proposition 1

.

P

⁢

(

ℝ

3

)

is a

Boolean algebra

with the usual set-theoretic operations, and together with cylindrification

operators

and the diagonal sets, on the set

V

=

{

x

,

y

,

z

}

, is a cylindric algebra.

Proof.

Write

A

=

P

⁢

(

ℝ

3

)

. It is easy to see that

A

is a Boolean algebra with operations

∪

,

∩

,

′

,

∅

. Next define

∃

:

V

→

A

A

by

∃

v

:=

C

v

where

v

∈

{

x

,

y

,

z

}

, and

d

:

V

×

V

→

A

by

d

x

⁢

y

:=

D

x

⁢

y

. Then Properties (1), (2), and (3) show that

(

A

,

∃

v

)

is a

monadic algebra

, and Properties (4), (5), (7), and (8) show that

(

A

,

V

,

∃

,

d

)

is cylindric.
∎

Example 2 (Cylindric Set Algebras).

Example 1 above may be generalized. Let

A

,

V

be sets, and set

B

=

P

⁢

(

A

V

)

. For any subset

R

⊆

B

and any

x

,

y

∈

V

, define the

cylindrification

of

R

by

C

x

⁢

(

R

)

:=

{

p

∈

A

V

∣

∃

r

∈

R

⁢

such that

⁢

r

⁢

(

y

)

=

p

⁢

(

y

)

⁢

for any

⁢

y

≠

x

}

,

and the

diagonal set

by

D

x

⁢

y

=

{

p

∈

A

V

∣

p

⁢

(

x

)

=

p

⁢

(

y

)

}

.

Now, define

∃

:

V

→

B

B

and

d

:

V

×

V

→

B

by

∃

x

=

C

x

and

d

x

⁢

y

=

D

x

⁢

y

.

Proposition 2

.

(

B

,

V

,

∃

,

d

)

is a cylindric algebra, called a

cylindric set algebra

.

The proof of this can be easily derived based on the discussion in Example 1, and is left for the reader as an exercise.

Remark

. For more examples of cylindric algebras, see the second reference below.

References

1

L. Henkin, J. D. Monk, A. Tarski,

Cylindric Algebras, Part I.

, North-Holland, Amsterdam (1971).

2

J. D. Monk,

Connections Between Combinatorial Theory and Algebraic Logic, Studies in Algebraic Logic

, The

Mathematical Association of America

, (1974).

3

J. D. Monk,

Mathematical Logic

, Springer, New York (1976).

4

B. Plotkin,

Universal Algebra

, Algebraic Logic, and Databases

, Kluwer Academic Publishers (1994).

Title

example of cylindric algebra

Canonical

name

ExampleOfCylindricAlgebra

Date of creation

2013-03-22 17:52:26

Last modified on

2013-03-22 17:52:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Example

Classification

msc 03G15

Defines

cylindrification

Defines

cylindric set algebra

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