characterization of free submonoids

characterization of free submonoids

Let

A

be an arbitrary set,
let

A

∗

be the

free monoid

on

A

,
and let

e

be the

identity element

(

empty word

) of

A

∗

.
Let

M

be a

submonoid

of

A

∗

and let

mgs

⁢

(

M

)

be its

minimal

generating set

.

We recall the

universal property

of free monoids:
for every

mapping

f

:

A

→

M

with

M

a monoid,
there exists a unique

morphism

ϕ

:

A

∗

→

M

such that

ϕ

⁢

(

a

)

=

f

⁢

(

a

)

for every

a

∈

A

.

Theorem 1

The following are

equivalent

.

1.

M

is a

free submonoid

of

A

∗

.

2.

Any

equation

x

1

⁢

⋯

⁢

x

n

=

y

1

⁢

⋯

⁢

y

m

,

x

1

,

…

,

x

n

,

y

1

,

…

,

y

m

∈

mgs

⁢

(

M

)

(1)

has only the

trivial

solutions

n

=

m

,

x

1

=

y

1

,

…

,

x

n

=

y

n

.

3.

For every

w

∈

A

∗

,
if

p

,

q

∈

M

exist such that

p

⁢

w

,

w

⁢

q

∈

M

,

then

w

∈

M

.

From point

3

of

Theorem

1

follows

Corollary 1

An

intersection

of free submonoids of

A

∗

is a free submonoid of

A

∗

.

As a

consequence

of Theorem

1

,
there is no

Nielsen-Schreier theorem

for monoids.
In fact, consider

A

=

{

a

,

b

}

and

Y

=

{

a

,

a

⁢

b

,

b

⁢

a

}

⊆

A

∗

:
then

mgs

⁢

(

Y

∗

)

=

Y

,
but

x

1

⁢

x

2

=

y

1

⁢

y

2

has a nontrivial solution over

Y

,
namely,

(

a

⁢

b

)

⁢

a

=

a

⁢

(

b

⁢

a

)

.

We now prove Theorem

1

.

Point

2

implies point

1

.

Let

f

:

mgs

⁢

(

M

)

→

B

be a

bijection

.
By the universal property of free monoids,
there exists a unique morphism

ϕ

:

B

∗

→

M

that extends

f

-

1

;
such a morphism is clearly

surjective

.
Moreover, any equation

ϕ

⁢

(

b

1

⁢

⋯

⁢

b

n

)

=

ϕ

⁢

(

b

1

′

⁢

⋯

⁢

b

m

′

)

translates

into an equation of the form (

1

),
which by

hypothesis

has only trivial solutions:
therefore

n

=

m

,

b

i

=

b

i

′

for all

i

, and

ϕ

is

injective

.

Point

3

implies point

2

.

Suppose the

existence

of

p

,

q

∈

M

such that

p

⁢

w

,

w

⁢

q

∈

M

implies

w

∈

A

∗

is actually in

M

.
Consider an equation of the form (

1

)
which is a counterexample to the thesis,
and such that the length of the compared words is minimal:
we may suppose

x

1

is a prefix of

y

1

,
so that

y

1

=

x

1

⁢

w

for some

w

∈

A

∗

.
Put

p

=

x

1

,

q

=

y

2

⁢

⋯

⁢

y

m

:
then

p

⁢

w

=

y

1

and

w

⁢

q

=

x

2

⁢

⋯

⁢

x

n

belong to

M

by construction.
By hypothesis, this implies

w

∈

M

:
then

y

1

∈

mgs

⁢

(

M

)

equals a

product

x

1

⁢

w

with

x

1

,

w

∈

M

—which,
by definition of

mgs

⁢

(

M

)

,
is only possible if

w

=

e

.
Then

x

1

=

y

1

and

x

2

⁢

⋯

⁢

x

n

=

y

2

⁢

⋯

⁢

y

m

:
since we had chosen a counterexample of minimal length,

n

=

m

,

x

2

=

y

2

,

…

,

x

n

=

y

n

.
Then the original equation has only trivial solutions,
and is not a counterexample after all.

Point

1

implies point

3

.

Let

ϕ

:

B

∗

→

M

be an

isomorphism

of monoids.
Then clearly

mgs

⁢

(

M

)

⊆

ϕ

⁢

(

B

)

;
since removing

m

=

ϕ

⁢

(

b

)

from

mgs

⁢

(

M

)

removes

ϕ

⁢

(

b

∗

)

from

M

,
the equality holds.
Let

w

∈

A

∗

and let

p

,

q

∈

M

satisfy

p

⁢

w

,

w

⁢

q

∈

M

:
put

x

=

ϕ

-

1

⁢

(

p

)

,

y

=

ϕ

-

1

⁢

(

q

)

,

r

=

ϕ

-

1

⁢

(

p

⁢

w

)

,

s

=

ϕ

-

1

⁢

(

w

⁢

q

)

.
Then

ϕ

⁢

(

x

⁢

s

)

=

ϕ

⁢

(

r

⁢

y

)

=

p

⁢

w

⁢

q

∈

M

, so

x

⁢

s

=

r

⁢

y

:
this is an equality over

B

∗

,
and is satisfied only by

r

=

x

⁢

u

,

s

=

u

⁢

y

for some

u

.
Then

w

=

ϕ

⁢

(

u

)

∈

M

.

References

1

M. Lothaire.

Combinatorics on words.

Cambridge University Press 1997.

Title

characterization of free submonoids

Canonical name

CharacterizationOfFreeSubmonoids

Date of creation

2013-03-22 18:21:32

Last modified on

2013-03-22 18:21:32

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

7

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 20M05

Classification

msc 20M10

Defines

intersection of free submonoids is free

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