cyclically reduced

cyclically reduced

Let

M

⁢

(

X

)

be a

free monoid with involution

-

1

on

X

. A word

w

∈

M

⁢

(

X

)

is said to be

cyclically reduced

if every

cyclic conjugate

of it is

reduced

. In other words,

w

is cyclically reduced iff

w

is a

reduced word

and that if

w

=

u

⁢

v

⁢

u

-

1

for some words

u

and

v

, then

w

=

v

.

For example, if

X

=

{

a

,

b

,

c

}

, then words such as

c

-

1

⁢

b

⁢

c

2

⁢

a

and

a

⁢

b

⁢

a

⁢

c

2

⁢

b

⁢

a

2

are cyclically reduced, where as words

a

2

⁢

b

⁢

c

⁢

a

-

1

and

c

⁢

b

2

⁢

b

3

⁢

c

are not, the former is reduced, but of the form

a

⁢

(

a

⁢

b

⁢

c

)

⁢

a

-

1

, while the later is not even a reduced word.

Remarks

. The

concept

of cyclically reduced words carries over to words in groups. We consider words in a group

G

.

•

If a word is cyclically reduced, so is its

inverse

and all of its cyclic conjugates.

•

A word

v

is a

cyclic reduction

of a word

w

if

w

=

u

⁢

v

⁢

u

-

1

for some word

u

, and

v

is cyclically reduced. Clearly, every word and its cyclic reduction are

conjugates

of each other. Furthermore, any word has a unique cyclic reduction.

•

Every group

G

has a

presentation

⟨

S

|

R

⟩

such that

(a)

R

is cyclically reduced (meaning every element of

R

is cyclically reduced),

(b)

closed under

inverses (meaning if

u

∈

R

, then

u

-

1

∈

R

), and

(c)

closed under cyclic conjugation (meaning any cyclic conjugate of an

element

in

R

is in

R

).

Furthermore, if

G

is finitely presented,

R

above can be chosen to be finite.

Proof.

Every group

G

has a presentation

⟨

S

|

R

′

⟩

. There is an

isomorphism

from

F

⁢

(

S

)

/

N

⁢

(

R

′

)

to

G

, where

F

⁢

(

S

)

is the

free group

freely generated by

S

, and

N

⁢

(

R

′

)

is the

normalizer

of the subset

R

′

⊆

F

⁢

(

S

)

in

F

⁢

(

S

)

. Let

R

′′

be the set of all cyclic reductions of words in

R

′

. Then

N

⁢

(

R

′′

)

=

N

⁢

(

R

′

)

, since any word not cyclically reduced in

R

′

is conjugate to its cyclic reduction in

R

′′

, and hence in

N

⁢

(

R

′′

)

. Next, for each

u

∈

R

′′

, toss in its inverse and all of its cyclic conjugates. The resulting set

R

is still cyclically reduced. Furthermore,

R

satisfies

the remaining conditions above. Finally,

N

⁢

(

R

)

=

N

⁢

(

R

′′

)

, as any cyclic conjugate

v

of a word

w

is clearly a conjugate of

w

. Therefore,

G

has presentation

⟨

S

|

R

⟩

.

If

G

is finitely presented, then

S

and

R

′

above can be chosen to be

finite sets

. Therefore,

R

′′

and

R

are both finite.

R

is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite.
∎

Title

cyclically reduced

Canonical name

CyclicallyReduced

Date of creation

2013-03-22 17:34:04

Last modified on

2013-03-22 17:34:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20F10

Classification

msc 20F05

Classification

msc 20F06

Defines

cyclic reduction

Defines

cyclic conjugation

Generated on Fri Feb 9 19:28:39 2018 by

LaTeXML