polynomials in algebraic systems

polynomials in algebraic systems

Polynomials in an Algebraic System

Let

(

A

,

O

)

be an

algebraic system

. Given any non-negative integer

n

, let

S

be a set of

functions

from

A

n

to

A

, satisfying the following criteria:

1.

the

projection

π

i

:

A

n

→

A

is an element of

S

for each

i

between

1

and

n

.

2.

for every

0

-ary operator symbol

c

∈

O

, the corresponding

element

c

A

∈

A

is an element of

S

.

3.

for any

m

-ary

operator

ω

A

on

A

, and any

p

1

,

…

,

p

m

∈

S

, the function

ω

A

⁢

(

p

1

,

…

,

p

m

)

:

A

n

→

A

, given by

ω

A

⁢

(

p

1

,

…

,

p

m

)

⁢

(

a

1

,

…

,

a

n

)

:=

ω

A

⁢

(

p

1

⁢

(

a

1

,

…

,

a

n

)

,

…

,

p

m

⁢

(

a

1

,

…

,

a

n

)

)

is an element of

S

.

Take the

intersection

of all such sets and call it

𝐏

n

⁢

(

A

)

. Then

𝐏

n

⁢

(

A

)

is the smallest set satisfying the above two conditions. We call an element of

𝐏

n

⁢

(

A

)

an

n

-ary

polynomial

of the

algebra

A

. A

polynomial

of

A

is some

n

-ary polynomial of

A

. Two polynomials

p

,

q

are said to be

equivalent

if they have the same arity

n

, and for any

𝐚

∈

A

n

,

p

⁢

(

𝐚

)

=

q

⁢

(

𝐚

)

.

Examples

.

•

Let

G

be a group. Then

f

⁢

(

x

)

=

x

is a unary polynomial. In general, a unary polynomial of

G

has the form

x

n

,

n

≥

1

. For a binary polynomial, first note that

f

⁢

(

x

,

y

)

=

x

and

g

⁢

(

x

,

y

)

=

y

are both binary. For this, a binary polynomial of

G

looks like

x

n

1

⁢

y

m

1

⁢

⋯

⁢

x

n

k

⁢

y

m

k

,

where

n

1

,

m

k

≥

0

and the rest are positive integers. If

G

is

abelian

, a binary polynomial has the simplified form

x

n

⁢

y

m

, and more generally, an

n

-ary polynomial of an abelian group has the form

x

1

n

1

⁢

⋯

⁢

x

k

n

k

.

•

Let

R

be a ring.

f

⁢

(

x

)

=

x

is a unary polynomial, so are

g

⁢

(

x

)

=

n

⁢

x

and

h

⁢

(

x

)

=

x

m

where

n

,

m

are non-negative integers. In general, a unary polynomial of

R

looks like

n

m

⁢

x

m

+

n

m

-

1

⁢

x

m

-

1

+

⋯

+

n

1

⁢

x

+

n

0

,

where

n

i

are non-negative integers with

n

m

>

0

, and

m

is any positive integer. This is the form of a polynomial that is familiar to most people.

•

Continuing from the example above, note that, however,

a

m

⁢

x

m

+

a

m

-

1

⁢

x

m

-

1

+

⋯

+

a

1

⁢

x

+

a

0

where

a

i

∈

R

is in general

not

a polynomial under this definition. This is an example of an

algebraic function

in an algebraic system.

Remarks

.

•

𝐏

0

⁢

(

A

)

is non-empty iff

O

contains

constant symbols

(nullary function symbols).

•

By virtue of the definition,

𝐏

n

⁢

(

A

)

is an algebraic system for each non-negative integer

n

.

Polynomial Symbols

If we have two algebras

A

,

B

of the same type, then, by the definition above, a polynomial of

A

may be considered as a polynomial of

B

, and vice versa. Putting this formally, we introduce

polynomial symbols

for a class

K

of algebras of the same type

O

. Let

X

=

{

x

1

,

x

2

,

…

}

be a

countably infinite

set of

variables

, or symbols. For any non-negative integer

n

, consider a set

S

consisting of the following elements:

1.

x

1

,

…

,

x

n

,

2.

every nullary operator symbol,

3.

if

ω

∈

O

is

m

-ary and

p

1

,

…

,

p

m

∈

S

, then

ω

⁢

(

p

1

,

…

,

p

m

)

∈

S

.

Take the intersection of all such sets and we end up with a set satisfying the two conditions again, call it

𝐏

n

⁢

(

O

)

. Any element of

𝐏

n

⁢

(

O

)

is called an

n

-ary polynomial symbol

of

K

. A

polynomial symbol

of

K

is just some

n

-ary polynomial symbol. In

model theory

, a polynomial symbol is nothing more than a term (

http://planetmath.org/Term

) over

X

in the

O

-

language

.

Now, suppose

A

∈

K

. For a given non-negative integer

n

, consider the function

f

A

:

𝐏

n

⁢

(

O

)

→

𝐏

n

⁢

(

A

)

, defined recursively by

1.

f

A

⁢

(

x

i

)

:=

π

i

,

2.

f

A

⁢

(

c

)

:=

c

A

,

3.

f

A

⁢

(

ω

⁢

(

p

1

,

…

,

p

n

)

)

:=

ω

A

⁢

(

f

A

⁢

(

p

1

)

,

…

,

f

A

⁢

(

p

n

)

)

.

Then by the constructions of

𝐏

n

⁢

(

O

)

and

𝐏

n

⁢

(

A

)

,

f

A

is a

surjection

. Any polynomial

p

of

A

is said to be

induced

by a polynomial symbol

q

of

K

if

f

A

⁢

(

q

)

=

p

. We usually write

p

=

q

A

to denote that the polynomial

p

(of

A

) is induced by the polynomial symbol

q

(of

K

). It is not hard to see that, between two algebras of the same type, there is a

one-to-one correspondence

between their respective polynomials the same arities. However, equivalence of polynomials in one algebra does not translate to equivalence in another.

Example

.

x

∨

(

y

∧

z

)

and

(

x

∨

y

)

∧

(

x

∨

z

)

are both polynomials symbols in the class of lattices. In the

subclass

of

distributive

lattices, the induced polynomials are equivalent. However, in the subclass of modular lattices, the equivalence no longer holds.

Remark

. Like

𝐏

n

⁢

(

A

)

,

𝐏

n

⁢

(

O

)

is also an algebraic system, called the

n

-ary

polynomial algebra

, or

absolutely free algebra

, of

O

. Furthermore, the function

f

A

above is an algebra homomorphism.

References

1

G. Grätzer:

Universal Algebra

, 2nd Edition, Springer, New York (1978).

2

S. Burris, H.P. Sankappanavar:

A Course in Universal Algebra

, Springer, New York (1981).

Title

polynomials in algebraic systems

Canonical name

PolynomialsInAlgebraicSystems

Date of creation

2013-03-22 16:47:31

Last modified on

2013-03-22 16:47:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A40

Synonym

absolutely free algebra

Related topic

TermAlgebra

Related topic

AlgebraicFunction

Defines

polynomial

Defines

equivalent polynomials

Defines

polynomial symbol

Defines

induced polynomial

Defines

polynomial algebra

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