cylindric algebra

cylindric algebra

A

cylindric algebra

is a quadruple

(

B

,

V

,

∃

,

d

)

, where

B

is a

Boolean algebra

,

V

is a set whose

elements

we call

variables

,

∃

and

d

are

functions

∃

:

V

→

B

B

and

d

:

V

×

V

→

B

such that

1.

(

B

,

∃

x

)

is a

monadic algebra

for each

x

∈

V

,

2.

∃

x

∘

∃

y

=

∃

y

∘

∃

x

for any

x

,

y

∈

V

,

3.

d

x

⁢

x

=

1

for all

x

∈

V

,

4.

for any

x

,

y

∈

V

with

x

≠

y

, and any

a

∈

B

, we have the

equality

∃

x

⁢

(

a

∧

d

x

⁢

y

)

∧

∃

x

⁢

(

a

′

∧

d

x

⁢

y

)

=

0

5.

for any

x

,

y

,

z

∈

V

with

x

≠

y

and

x

≠

z

, we have the equality

∃

x

⁢

(

d

x

⁢

y

∧

d

x

⁢

z

)

=

d

y

⁢

z

.

where

∃

x

and

d

x

⁢

y

are the abbreviations for

∃

(

x

)

and

d

⁢

(

x

,

y

)

respectively.

Basically, the first two conditions say that the

(

B

,

V

,

∃

)

portion of the cylindric algebra is very

similar

to a

quantifier algebra

, except the domain is no longer the subsets of

V

, but the elements of

V

instead. The function

d

is the algebraic abstraction of equality. Condition 3 says that

x

=

x

is always true, condition 4 says that the

proposition

a

and its

complement

a

′

, where any

occurrences

of the variable

x

are replaced by the variable

y

, distinct from

x

, is always false, while condition 5 says

y

=

z

iff there is an

x

such that

x

=

y

and

x

=

z

.

Below are some

elementary

properties of

d

:

•

(

symmetric

property)

d

x

⁢

y

=

d

y

⁢

x

•

(

transitive property

)

d

x

⁢

y

∧

d

y

⁢

z

≤

d

x

⁢

z

•

∃

x

⁢

(

d

x

⁢

y

)

=

1

•

∃

x

⁢

(

d

y

⁢

z

)

=

d

y

⁢

z

provided that

x

∉

{

y

,

z

}

•

if

x

≠

y

, then

(a)

∃

x

⁢

(

d

x

⁢

y

∧

a

′

)

=

(

∃

x

⁢

(

d

x

⁢

y

∧

a

)

)

′

,

(b)

d

x

⁢

y

∧

a

=

d

x

⁢

y

∧

∃

x

⁢

(

a

∧

d

x

⁢

y

)

.

Remarks

1.

The

dimension

of a cylindric algebra

(

B

,

V

,

∃

,

d

)

is the

cardinality

of

V

.

2.

From the definition above, a cylindric algebra is a two-sorted

structure

, with

B

and

V

as the two distinct

universes

. However, it is often useful to view a cylindric algebra as a one-sorted structure. The way to do this is to dispense with

V

and identify each

∃

x

as a unary

operator

on

B

, and each

d

x

⁢

y

as a

constant

in

B

. As a result, the cylindric algebra

(

B

,

V

,

∃

,

d

)

becomes a Boolean algebra with possibly infinitely many operators:

(

B

,

∃

x

,

d

x

⁢

y

)

x

,

y

∈

V

.

3.

Let

L

be a the

language

of a

first order logic

, and

S

a set of

sentences

in

L

. Define

≡

on

L

so that

φ

≡

ψ

iff

S

⊢

(

φ

↔

ψ

)

.

Then

≡

is an

equivalence relation

on

L

. For each

formula

φ

∈

L

, let

[

φ

]

be the

equivalence class

containing

φ

. Let

V

be a

countably infinite

set of variables available to

L

. Now, define

operations

∨

,

∧

,

′

,

∃

x

,

d

x

⁢

y

as follows:

[

φ

]

∨

[

ψ

]

:=

[

φ

∨

ψ

]

,

(1)

[

φ

]

∧

[

ψ

]

:=

[

φ

∧

ψ

]

,

(2)

[

φ

]

′

:=

[

¬

⁢

φ

]

,

(3)

0

:=

[

¬

⁢

x

=

x

]

,

(4)

1

:=

[

x

=

x

]

,

(5)

∃

x

⁢

[

φ

]

:=

[

∃

x

⁢

φ

]

,

(6)

d

x

⁢

y

:=

[

x

=

y

]

.

(7)

Then it can be shown that

(

L

/

≡

,

V

,

∃

,

d

)

is a cylindric algebra. Thus a cylindric algebra can be thought of as an “algebraization” of first order logic (with equality), much the same way as a Boolean algebra (

Lindenbaum-Tarski algebra

) as the algebraic counterpart of

propositional logic

.

References

1

P. Halmos,

Algebraic Logic

, Chelsea Publishing Co. New York (1962).

2

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

Cylindric Algebras, Part I.

, North-Holland, Amsterdam (1971).

3

J. D. Monk,

Mathematical Logic

, Springer, New York (1976).

4

B. Plotkin,

Universal Algebra

, Algebraic Logic, and Databases

, Kluwer Academic Publishers (1994).

Title

cylindric algebra

Canonical name

CylindricAlgebra

Date of creation

2013-03-22 17:51:21

Last modified on

2013-03-22 17:51:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Classification

msc 06E25

Related topic

PolyadicAlgebra

Related topic

PolyadicAlgebraWithEquality

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