rational set

rational set

Given an

alphabet

Σ

, recall that a

regular language

R

is a certain subset of the

free monoid

M

generated by

Σ

, which can be obtained by taking singleton subsets of

Σ

, and perform, in a finite number of steps, any of the three basic

operations

: taking union, string concatenation, and the

Kleene star

.

The construction of a set like

R

is still possible without

M

being

finitely generated

free.

Let

M

be a monoid, and

𝒮

M

the set of all singleton subsets of

M

. Consider the

closure

ℛ

M

of

S

under the operations of union,

product

, and the formation of a

submonoid

of

M

. In other words,

ℛ

M

is the smallest subset of

M

such that

•

∅

∈

ℛ

M

,

•

A

,

B

∈

ℛ

M

imply

A

∪

B

∈

ℛ

M

,

•

A

,

B

∈

ℛ

M

imply

A

⁢

B

∈

ℛ

M

, where

A

⁢

B

=

{

a

⁢

b

∣

a

∈

A

,

b

∈

B

}

,

•

A

∈

ℛ

M

implies

A

*

∈

ℛ

M

, where

A

*

is the submonoid generated by

A

.

Definition

. A

rational set

of

M

is an element of

ℛ

M

.

If

M

is a finite generated free monoid, then a rational set of

M

is also called a

rational language

, more commonly known as a

regular language

.

Like regular languages, rational sets can also be represented by

regular expressions

. A regular expression over a monoid

M

and the set it represents are defined inductively as follows:

•

∅

and

a

are regular expressions, for any

a

∈

M

, representing sets

∅

and

{

a

}

respectively.

•

if

a

,

b

are regular expressions, so are

a

∪

b

,

a

⁢

b

, and

a

*

. Furthermore, if

a

,

b

represent sets

A

,

B

, then

a

∪

b

,

a

⁢

b

, and

a

*

represent sets

A

∪

B

,

A

⁢

B

, and

A

*

respectively.

Parentheses are used, as usual, to avoid ambiguity.

From the definition above, it is easy to see that a set

A

⊆

M

is

rational

iff it can be represented by a regular expression over

M

.

Below are some basic properties of rational sets:

1.

Any rational set

M

is a subset of a finitely generated submonoid of

M

. As a result, every rational set over

M

is finite iff

M

is

locally finite

(meaning every finitely generated submonoid of

M

is actually finite).

2.

Rationality is preserved under

homomorphism

: if

A

is rational over

M

and

f

:

M

→

N

is a homomorphism, then

f

⁢

(

A

)

is rational over

N

.

3.

Conversely

, if

B

∈

f

⁢

(

M

)

is rational over

N

, then there is a rational set

A

over

M

such that

f

⁢

(

A

)

=

B

. Thus if

f

is onto, every rational set over

N

is mapped by a rational set over

M

. If

f

fails to be onto, the statement becomes false. In fact,

inverse

homomorphisms generally do not preserve rationality.

References

1

S. Eilenberg,

Automata

,

Languages

, and

Machines

, Vol. A

, Academic Press (1974).

Title

rational set

Canonical name

RationalSet

Date of creation

2013-03-22 19:01:16

Last modified on

2013-03-22 19:01:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Defines

rational language

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