example of polyadic algebra

example of polyadic algebra

Recall that the

canonical

example of a

monadic algebra

is that of a

functional monadic algebra

, which is a pair

(

B

,

∃

)

such that

B

is the set of all

functions

from a non-empty set

X

to a

Boolean algebra

A

such that, for each

f

∈

B

, the

supremum

and the

infimum

of

f

⁢

(

X

)

exist, and

∃

is a function on

B

that maps each

element

f

to

f

∃

, a

constant

element whose range is a

singleton

consisting of the supremum of

f

⁢

(

X

)

.

The canonical example of a

polyadic algebra

is an

extension

(

generalization

) of a functional monadic algebra, known as the

functional polyadic algebra

. Instead of looking at functions from

X

to

A

, we look at functions from

X

I

(where

I

is some set), the

I

-fold cartesian power of

X

, to

A

. In this entry, an element

x

∈

X

I

is written as a

sequence

of

elements of

A

:

(

x

i

)

i

∈

I

where

x

i

∈

A

, or

(

x

i

)

for short.

Before constructing

the

functional polyadic algebra based on the sets

X

,

I

and the Boolean algebra

A

, we first introduce the following notations:

•

for any

J

⊆

I

and

x

∈

X

I

, define the subset (of

X

I

)

[

x

]

J

:=

{

y

∈

X

I

∣

x

i

=

y

i

⁢

for every

⁢

i

∉

J

}

,

•

for any function

τ

:

I

→

I

and any

f

:

X

I

→

A

, define the function

f

τ

from

X

I

to

A

, given by

f

τ

⁢

(

x

i

)

:=

f

⁢

(

x

τ

⁢

(

i

)

)

.

Now, let

B

be the set of all functions from

X

I

to

A

such that

1.

for every

f

∈

B

, every

J

⊆

I

and every

x

∈

X

I

, the arbitrary join

⋁

f

⁢

(

[

x

]

J

)

exists.

Before stating the next condition, we introduce, for each

f

∈

B

, a function

f

∃

J

:

X

I

→

A

as follows:

f

∃

J

⁢

(

x

)

:=

⋁

f

⁢

(

[

x

]

J

)

.

Now, we are ready for the next condition:

2.

if

f

∈

B

, then

f

∃

J

∈

B

,

3.

if

f

∈

B

, then

f

τ

∈

B

for

τ

:

I

→

I

.

Note that if

A

were a

complete Boolean algebra

, we can take

B

to be

A

X

I

, the set of all functions from

X

I

to

A

.

Next, define

∃

:

P

⁢

(

I

)

→

B

B

by

∃

(

J

)

⁢

(

f

)

=

f

∃

J

, and let

S

be the semigroup of functions on

I

(with

functional

compositions as

multiplications

), then we call the quadruple

(

B

,

I

,

∃

,

S

)

the

functional polyadic algebra

for the triple

(

A

,

X

,

I

)

.

Remarks

. Let

(

B

,

I

,

∃

,

S

)

be the functional polyadic algebra for

(

A

,

X

,

I

)

.

•

(

B

,

I

,

∃

,

S

)

is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.

•

If

I

is a singleton, then

(

B

,

I

,

∃

,

S

)

can be identified with the functional monadic algebra

(

B

,

∃

)

for

(

A

,

X

)

, for

S

is just

I

, and

X

I

is just

X

.

•

If

I

is

∅

, then

(

B

,

I

,

∃

,

S

)

can be identified with the Boolean algebra

A

, for

S

=

∅

and

X

I

is a singleton, and hence the set of functions from

X

I

to

A

is identified with

A

.

References

1

P. Halmos,

Algebraic Logic

, Chelsea Publishing Co. New York (1962).

Title

example of polyadic algebra

Canonical name

ExampleOfPolyadicAlgebra

Date of creation

2013-03-22 17:53:20

Last modified on

2013-03-22 17:53:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Example

Classification

msc 03G15

Defines

functional polyadic algebra

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