polyadic algebra

polyadic algebra

A

polyadic algebra

is a quadruple

(

B

,

V

,

∃

,

S

)

, where

(

B

,

V

,

∃

)

is a

quantifier algebra

, and

S

is a

function

from the set of functions on

V

to the set of

endomorphisms

on the

Boolean algebra

B

, in other words

S

:

V

V

→

End

⁡

(

B

)

such that

1.

S

⁢

(

1

V

)

=

1

B

,

2.

S

⁢

(

f

∘

g

)

=

S

⁢

(

f

)

∘

S

⁢

(

g

)

,

3.

S

⁢

(

f

)

∘

∃

(

I

)

=

S

⁢

(

g

)

∘

∃

(

I

)

if

f

⁢

(

V

-

I

)

=

g

⁢

(

V

-

I

)

,

4.

∃

(

I

)

∘

S

⁢

(

f

)

=

S

⁢

(

f

)

∘

∃

(

f

-

1

⁢

(

I

)

)

if

f

is

one-to-one

when restricted to

f

-

1

⁢

(

I

)

.

Explanation of notations:

1

V

,

1

B

are

identity functions

on

V

,

B

respectively;

f

,

g

are functions on

V

, and

I

is a subset of

V

. The circle

∘

represents

functional

compositions.

The

degree

and

local finiteness

of a polyadic algebra are defined as the degree and local finiteness of the underlying quantifier algebra.

Heuristically, the function

S

can be thought of as changes to

propositional functions

due to a “

substitution

” of

variables

(elements of

V

). Let us see some examples. Let

V

=

{

x

0

,

x

1

,

…

}

be a countably indexed set of variables. For any propositional function

φ

, define

S

⁢

(

f

)

⁢

(

φ

)

to be the propositional function

φ

1

obtained by replacing each variable

x

that

occurs in

it by

f

⁢

(

x

)

. Below are two examples illustrating how

S

⁢

(

f

)

changes propositional functions:

•

Let

f

:

V

→

V

be the function given by

f

⁢

(

x

0

)

=

f

⁢

(

x

1

)

=

x

0

and

f

⁢

(

x

i

)

=

x

i

+

1

for all

i

>

1

. If

φ

is the propositional function

x

0

2

-

x

1

+

x

2

/

x

3

, then

S

⁢

(

f

)

⁢

(

φ

)

is the propositional function

x

0

2

-

x

0

+

x

3

/

x

4

.

•

Let

f

:

V

→

V

be the function given by

f

⁢

(

x

0

)

=

x

2

, and

f

⁢

(

x

i

)

=

x

i

for all

i

≠

0

. Then the propositional function “

∃

x

0

,

x

1

,

x

2

(

x

0

≠

x

1

∧

x

1

≠

x

2

∧

x

2

≠

x

0

)

” becomes “

∃

x

2

,

x

1

,

x

2

(

x

2

≠

x

1

∧

x

1

≠

x

2

∧

x

2

≠

x

2

)

” under

S

⁢

(

f

)

.

It is not hard to see from the examples above that

S

⁢

(

f

)

respects Boolean operations

∧

and

′

, which is why we want to make

S

⁢

(

f

)

an endomorphism on

B

. Furthermore, the four conditions above can be interpreted as

1.

if there are no substitutions of variables, then there should be no corresponding changes to the propositional functions

2.

applying substitutions

f

∘

g

of varaibles in a propositional function

φ

should have the same effect as applying substitutions

g

of variables in

φ

, followed by substitutions

f

of variables in

S

⁢

(

g

)

⁢

(

φ

)

3.

a substitution

f

of variables should have no effect to a propositional function beginning with

∃

if every variable bound by

∃

is fixed by

f

. For example, if we replace

f

in the second example above by

f

⁢

(

x

3

)

=

x

2

and

f

⁢

(

x

i

)

=

x

i

otherwise, then

`

`

∃

x

0

,

x

1

,

x

2

(

x

0

≠

x

1

∧

x

1

≠

x

2

∧

x

2

≠

x

0

)

"

is unchanged by

S

⁢

(

f

)

, since

x

0

,

x

1

,

x

2

are all fixed by

f

.

4.

Let

φ

=

∃

I

⁢

ψ

⁢

(

I

,

J

)

be a propositional function, where

I

,

J

are sets of variables with

I

bound by

∃

and

J

free. If no two variables

I

get mapped to the same variable, and no

free variable

(in

J

) becomes bound (in

f

⁢

(

I

)

) under the substitution, then

∃

I

⁢

ψ

⁢

(

f

⁢

(

I

)

,

f

⁢

(

J

)

)

and

∃

f

⁢

(

I

)

⁢

ψ

⁢

(

f

⁢

(

I

)

,

f

⁢

(

J

)

)

are semantically the same, which is exactly the statement in the condition.

Remarks

.

•

Paul Halmos first introduced the notion of polyadic algebras. In his

Algebraic Logic

, a compilation of articles on the subject, he called a function on the set

V

of variables a

transformation

, and the triple

(

B

,

V

,

S

)

satisfying the first two conditions above a

transformation algebra

. Therefore, a polyadic algebra is a quadruple

(

B

,

V

,

∃

,

S

)

where

(

B

,

V

,

∃

)

is a quantifier algebra and

(

B

,

V

,

S

)

is a transformation algebra, such that conditions 3 and 4 above are satisfied.

•

The notion of polyadic algebras generalizes the notion of

monadic algebras

. Indeed, a monadic algebra is a polyadic algebra where

V

is a

singleton

.

•

Just as a

Lindenbaum-Tarski algebra

is the algebraic counterpart of a classical

propositional logic

, a polyadic algebra is the algebraic counterpart of a

classical first order logic

without

equality

. A variant of the polyadic algebra is what is known as a

cylindric algebra

, which algebratizes a classical first order logic with equality.

References

1

P. Halmos,

Algebraic Logic

, Chelsea Publishing Co. New York (1962).

2

B. Plotkin,

Universal Algebra

, Algebraic Logic, and

Databases

, Kluwer Academic Publishers (1994).

Title

polyadic algebra

Canonical

name

PolyadicAlgebra

Date of creation

2013-03-22 17:50:39

Last modified on

2013-03-22 17:50:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Related topic

QuantifierAlgebra

Related topic

MonadicAlgebra

Related topic

CylindricAlgebra

Defines

transformation algebra

Generated on Thu Feb 8 20:28:25 2018 by

LaTeXML