reduced word

reduced word

Let

X

be a set, and

(

X

∐

X

-

1

)

∗

the

free monoid with involution

on

X

. An

element

w

∈

(

X

∐

X

-

1

)

∗

can be uniquely written by

juxtaposition

of elements of

(

X

∐

X

-

1

)

, i.e.

w

=

w

1

⁢

w

2

⁢

…

⁢

w

n

,

w

j

∈

(

X

∐

X

-

1

)

,

then we may improperly say that

w

is a word on

(

X

∐

X

-

1

)

∗

, considering

(

X

∐

X

-

1

)

∗

simply as the

free monoid

on

(

X

∐

X

-

1

)

.

The word

w

=

w

1

⁢

w

2

⁢

…

⁢

w

n

∈

(

X

∐

X

-

1

)

∗

is called

reduced

when

w

j

≠

w

j

+

1

-

1

for each

j

∈

{

1

,

2

,

…

,

n

-

1

}

.

Now, starting from a word

w

=

w

1

⁢

w

2

⁢

…

⁢

w

n

∈

(

X

∐

X

-

1

)

∗

, we can iteratively erase factors

w

j

⁢

w

j

+

1

from

w

whenever

w

j

=

w

j

+

1

-

1

, and this iterative process, that we call

reduction

of

w

, produce a reduced word

w

′

∈

(

X

∐

X

-

1

)

∗

. At each step of the process there may be more than one couple of adiacent letters candidate to be erased, so we may ask if different

sequences

of erasing may produce different reduced words. The following

theorem

answers the question.

Theorem 1

Each couple of reduction of a same word

w

∈

(

X

∐

X

-

1

)

∗

produce the same reduced word

w

′

∈

(

X

∐

X

-

1

)

∗

.

The unique reduced word

w

′

is called the

reduced form

of

w

. Thus there exists a well define map

red

:

(

X

∐

X

-

1

)

∗

→

(

X

∐

X

-

1

)

∗

that send a word

w

to his reduced form

red

⁢

(

w

)

.

We can use the map

red

to build the

free group

on

X

in the following way. Let

FG

⁢

(

X

)

=

red

⁢

(

(

X

∐

X

-

1

)

+

)

be the set of reduced words on

(

X

∐

X

-

1

)

, i.e.

FG

⁢

(

X

)

=

red

⁢

(

(

X

∐

X

-

1

)

+

)

=

{

red

⁢

(

w

)

|

w

∈

(

X

∐

X

-

1

)

+

}

=

{

w

∈

(

X

∐

X

-

1

)

+

|

w

=

red

⁢

(

u

)

}

.

Note that

red

⁢

(

(

X

∐

X

-

1

)

∗

)

=

red

⁢

(

(

X

∐

X

-

1

)

+

)

, being that

red

⁢

(

x

⁢

x

-

1

)

=

ε

, where

ε

denotes the

empty word

. Now, we define a

product

⋅

on

FG

⁢

(

X

)

that makes it a group: given

v

,

w

∈

FG

⁢

(

X

)

we define

v

⋅

w

=

red

⁢

(

v

⁢

w

)

,

i.e.

v

⋅

w

is the reduced form of the juxtaposition of the words

v

and

w

. The we have the following result.

Theorem 2

FG

⁢

(

X

)

with the product

⋅

is a group. Moreover, it is the

free group

on

X

, in the sense that it solves the following universal problem: given a group

G

and a map

Φ

:

X

→

G

, a

group homomorphism

Φ

¯

:

FG

⁢

(

X

)

→

G

exists such that the following

diagram

commutes:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

Φ

⁢

&

⁢

FG

⁢

(

X

)

⁢

\ar

⁢

[

d

⁢

l

]

Φ

¯

⁢

&

⁢

G

⁢

&

where

ι

:

X

→

FG

⁢

(

X

)

is the

inclusion map

.

It is well known from

universal algebra

that

FG

⁢

(

X

)

is unique up to

isomorphisms

. With this construction, the map

red

:

(

X

∐

X

-

1

)

+

→

FG

⁢

(

X

)

[resp.

red

:

(

X

∐

X

-

1

)

∗

→

FG

⁢

(

X

)

] is the

quotient

projection

from
the free semigroup with involution on

X

[resp. the free monoid with involution on

X

] and the free group on

X

.

References

1

J.M. Howie,

Fundamentals of

Semigroup

Theory

, Oxford University Press, Oxford, 1991.

2

R. Lyndon and P. Schupp,

Combinatorial Group Theory

, Springer-Verlag, 1977.

3

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

Title

reduced word

Canonical name

ReducedWord

Date of creation

2013-03-22 16:11:47

Last modified on

2013-03-22 16:11:47

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

16

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20E05

Defines

reduced word

Defines

reduced form

Defines

reduction

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