polyadic algebra with equality

polyadic algebra with equality

Let

A

=

(

B

,

V

,

∃

,

S

)

be a

polyadic algebra

. An

equality predicate

on

A

is a

function

E

:

V

×

V

→

B

such that

1.

S

⁢

(

f

)

∘

E

⁢

(

x

,

y

)

=

E

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

for any

f

:

V

→

V

and any

x

,

y

∈

V

2.

E

⁢

(

x

,

x

)

=

1

for every

x

∈

V

, and

3.

E

⁢

(

x

,

y

)

∧

a

≤

S

⁢

(

x

/

y

)

⁢

a

, where

a

∈

B

, and

(

x

/

y

)

denotes the function

V

→

V

that maps

x

to

y

, and

constant

everywhere else.

Heuristically, we can interpret the conditions above as follows:

1.

if

x

=

y

and if we replace

x

by, say

x

1

, and

y

by

y

1

, then

x

1

=

y

1

.

2.

x

=

x

for every

variable

x

3.

if we have a

propositional function

a

that is true, and

x

=

y

, then the

proposition

obtained from

a

by replacing all

occurrences

of

x

by

y

is also true.

The second condition is also known as the

reflexive property

of the equality predicate

E

, and the third is known as the

substitutive property

of

E

A

polyadic algebra with equality

is a pair

(

A

,

E

)

where

A

is a polyadic algebra and

E

is an equality predicate on

A

. Paul Halmos introduced this

concept

and called this simply an

equality algebra

.

Below are some basic properties of the equality predicate

E

in an equality algebra

(

A

,

E

)

:

•

(

symmetric

property)

E

⁢

(

x

,

y

)

≤

E

⁢

(

y

,

x

)

•

(

transitive property

)

E

⁢

(

x

,

y

)

∧

E

⁢

(

y

,

z

)

≤

E

⁢

(

x

,

z

)

•

E

⁢

(

x

,

y

)

∧

a

=

E

⁢

(

x

,

y

)

∧

S

⁢

(

x

,

y

)

⁢

a

, where

(

x

,

y

)

in the

S

is the

transposition

on

V

that swaps

x

and

y

and leaves everything else fixed.

•

if a variable

x

∈

V

is not in the

support

of

a

∈

A

, then

a

=

∃

(

x

)

⁢

(

E

⁢

(

x

,

y

)

∧

S

⁢

(

y

/

x

)

⁢

a

)

.

•

∃

(

x

)

⁢

(

E

⁢

(

x

,

y

)

∧

a

)

∧

∃

(

x

)

⁢

(

E

⁢

(

x

,

y

)

∧

a

′

)

=

0

for all

a

∈

A

and all

x

,

y

∈

V

whenever

x

≠

y

.

•

∃

(

x

)

⁢

(

E

⁢

(

x

,

y

)

∧

E

⁢

(

x

,

z

)

)

=

E

⁢

(

y

,

z

)

for all

x

,

y

,

z

∈

V

where

x

∉

{

y

,

z

}

.

Remarks

•

The degree and local finiteness of a polyadic algebra

(

A

,

E

)

are defined as the degree and the local finiteness and degree of its underlying polyadic algebra

A

.

•

It can be shown that every

locally finite

polyadic algebra of

infinite

degree can be embedded (as a

polyadic

subalgebra

) in a locally finite polyadic algebra with equality of infinite degree.

•

Like

cylindric algebras

, polyadic algebras with equality is an attempt at “converting” a

first order logic

(with

equality

) into algebraic form, so that the logic can be studied using

algebraic

means.

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 with equality

Canonical

name

PolyadicAlgebraWithEquality

Date of creation

2013-03-22 17:51:37

Last modified on

2013-03-22 17:51:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Synonym

equality algebra

Related topic

CylindricAlgebra

Defines

equality predicate

Defines

substitutive

Defines

reflexive

Defines

symmetric

Defines

transitive

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