characteristic monoid

characteristic monoid

Given a

semiautomaton

M

=

(

S

,

Σ

,

δ

)

, the

transition function

is typically defined as a

function

from

S

×

Σ

to

S

. One may instead think of

δ

as a set

C

⁢

(

M

)

of functions

C

(

M

)

:=

{

δ

a

:

S

→

S

∣

a

∈

Σ

}

,

where

δ

a

(

s

)

:=

δ

(

s

,

a

)

.

Since the transition function

δ

in

M

can be extended to the domain

S

×

Σ

*

, so can the set

C

⁢

(

M

)

:

C

(

M

)

:=

{

δ

u

:

S

→

S

∣

u

∈

Σ

*

}

,

where

δ

u

(

s

)

:=

δ

(

s

,

u

)

.

The advantage of this interpreation is the following: for any input words

u

,

v

over

Σ

:

δ

u

∘

δ

v

=

δ

v

⁢

u

,

which can be easily verified:

δ

v

⁢

u

⁢

(

s

)

=

δ

⁢

(

s

,

v

⁢

u

)

=

δ

⁢

(

δ

⁢

(

s

,

v

)

,

u

)

=

δ

⁢

(

δ

v

⁢

(

s

)

,

u

)

=

δ

u

⁢

(

δ

v

⁢

(

s

)

)

=

(

δ

u

∘

δ

v

)

⁢

(

s

)

.

In particular,

δ

λ

is the

identity function

on

S

, so that the set

C

⁢

(

M

)

becomes a monoid, called the

characteristic monoid

of

M

.

The characteristic monoid

C

⁢

(

M

)

of a semiautomaton

M

is related to the

free monoid

Σ

*

generated by

Σ

in the following manner: define a

binary relation

∼

on

Σ

*

by

u

∼

v

iff

δ

u

=

δ

v

. Then

∼

is clearly an

equivalence relation

on

Σ

*

. It is also a

congruence relation

with respect to

concatenation

: if

u

∼

v

, then for any

w

over

Σ

:

δ

u

⁢

w

⁢

(

s

)

=

δ

u

⁢

(

δ

w

⁢

(

s

)

)

=

δ

v

⁢

(

δ

w

⁢

(

s

)

)

=

δ

v

⁢

w

⁢

(

s

)

and

δ

w

⁢

u

⁢

(

s

)

=

δ

w

⁢

(

δ

u

⁢

(

s

)

)

=

δ

w

⁢

(

δ

v

⁢

(

s

)

)

=

δ

w

⁢

v

⁢

(

s

)

.

Putting the two together, we see that if

x

∼

y

, then

u

⁢

x

∼

v

⁢

x

∼

v

⁢

y

. We denote

[

u

]

the congruence class in

Σ

*

/

∼

containing the word

u

.

Now, define a map

ϕ

:

C

(

M

)

→

Σ

*

/

∼

by setting

ϕ

⁢

(

δ

u

)

=

[

u

]

. Then

ϕ

is

well-defined

. Furthermore, under

ϕ

, it is easy to see that

C

⁢

(

M

)

is anti-isomorphic to

Σ

*

/

∼

.

Remark

. In order to avoid using anti-isomorphisms, the usual practice is to introduce a

multiplication

⋅

on

C

⁢

(

M

)

so that

δ

u

⋅

δ

v

:=

δ

v

∘

δ

u

. Then

C

⁢

(

M

)

under

⋅

is

isomorphic

to

Σ

*

/

∼

.

Some

properties

:

•

If

M

and

N

are isomorphic semiautomata with identical input

alphabet

, then

C

⁢

(

M

)

=

C

⁢

(

N

)

.

•

If

N

is a

subsemiautomaton

of

M

, then

C

⁢

(

N

)

is a

homomorphic image

of a

submonoid

of

C

⁢

(

M

)

.

•

If

N

is a homomorphic image of

M

, so is

C

⁢

(

N

)

a homomorphic image of

C

⁢

(

M

)

.

References

1

A. Ginzburg,

Algebraic Theory of Automata

, Academic Press (1968).

2

M. Ito,

Algebraic Theory of Automata and

Languages

, World Scientific, Singapore (2004).

Title

characteristic monoid

Canonical

name

CharacteristicMonoid

Date of creation

2013-03-22 19:01:10

Last modified on

2013-03-22 19:01:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 20M35

Classification

msc 03D05

Classification

msc 68Q45

Generated on Sat Feb 10 12:47:55 2018 by

LaTeXML