Bijective combinatorics of a certain class of monoids

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Bijective combinatorics of a certain class of monoids

Marie Albenque

To cite this version:

Marie Albenque. Bijective combinatorics of a certain class of monoids. Eurocomb 2007, Sep 2007,

Seville, Spain. pp.225-229, �10.1016/j.endm.2007.07.038�. �hal-00160726�

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Marie Albenque

∗

Abstra t

We give a new and bije tive proof for the formula of the growth

fun tionofthepositivebraidmonoidwithrespe ttoArtingenerators.

Introdu tion

Consider

n + 1

strands numbered from 1 to

n + 1

and

n

elementary moves

σ

1 , . . . , σ

n

where

σ

i

representsthe rossingofstrand

i

and

i + 1

withstrand

i

above. Starting from an un rossed onguration we apply sequen es of elementary moves to obtain a braided onguration; two ongurations are

equivalentifone anbeobtainedfromtheotheronlybymovingthestrands

without tou hing thetop and bottom extremities. An equivalen e lass of

ongurationis alleda braid. Thesetof braids, with on atenation oftwo

braidsasinternal law, isamonoid. It ispre iselythepositivebraidmonoid

on

n + 1

strands, denoted

P

, generated by

Σ = {σ

1 , . . . , σ

n

}

( alled Artin generators) and subje tto therelations:

σ

i

σ

j

= σ

j

σ

i

for

|i − j| ≥ 2

(0.1)

σ

i

σ

i+1

σ

i

= σ

i+1

σ

i

σ

i+1

for

1 ≤ i ≤ n − 1

(0.2)

For

u ∈ P

thereis anatural length fun tion dened by:

|u|

Σ

= min{k | ∃u

1 , . . . , u

k

∈ Σ

su hthat

u = u

1 . . . u

k

}

The growth fun tion of the positive braid monoid for the Artin generators

isdened by:

F (t) =

X

b∈P

t

|b|

Σ

∗

LIAFA,CNRS-UniversitéParis7, ase7014,2,pla eJussieu,75251 ParisCedex05,

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braid monoid. In [Bra91 ℄, Brazil use the fa t that, to ea h braid is

asso i-ateda uniquede ompositionusually alleditsnormal form(oritsGarside's

normal form or Thurston's normal form). This set of normal forms is

reg-ular, meaning that it is re ognized by a nite state automaton. From this

automaton's adja en y matrix,we an obtain dire tlythe growth fun tion.

Butit is not a verye ient way to ompute it, asfor braids on

n

strands, this automaton has

n!

states and then it takes a time exponential in the numberofstrandstogettheformula. In[Deh07 ℄,Dehornoygivesamethod

to redu e from

n!

to

p(n)

(where

p(n)

is the numberof partitionsof

n

) the numberofstates ofthis automaton.

Bronfman(see [Bro01 ℄)andKrammer (see hapter17 of[Kra05 ℄)give a

newmethod to ompute thegrowth fun tioninquadrati time. Theirproof

is based on an in lusion-ex lusion prin iple. We give here a dierent and

bije tive proof ofthis result.

To explain our point of view, let look at thehistory of results for tra e

monoids. Tra emonoids(also alledheapsof pie esmonoids or free

par-tially ommutative monoids) denoted

M

aredened bythefollowing semi-grouppresentation:

M

= hΣ | ab = ba

if

(a, b) ∈ Ii,

where

Σ

is a nite set of generators and

I

is a symmetri and antireexive relation of

Σ × Σ

alled the ommutation relation. In 1969, Cartier and Foata omputedthegrowthfun tionofthesemonoidsbyusingan

in lusion-ex lusion prin iple to get a Möbius inversion formula (see [CF69℄). The

proofsofBronfmanand Krammerusethe same kind ofarguments.

In [Vie86 ℄, Viennot gives a new way to ompute the growth fun tion

of heaps of pie es monoids. To perform this, he onsiders the set

G

of all heaps of pie es of height at most one and onstru ts an involution from

G × M

into itself. This involution mat hes elementsof monoids two bytwo andmakesit easy to ount them. However, the onstru tion of this pairing

provides more than a new way to ompute the growth fun tion. It gives

indeedanadditional ombinatorial understandingoftra e monoids. Several

byprodu tsaregiven in[Vie86℄.

In the present paper, we show that Viennot's proof an be extended

to braid monoids, whi h gives a new point of view inthe ombinatori s of

braids. More pre isely, we explain in 1.2 how to dene a set

G

of simple braids andhowto useitto onstru t aninvolution from

(G × P)

to itself.

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ex-monoidsamongothers.

1 Growth fun tion of braid monoids

1.1 Presentation of braid monoids

Wedenoteby

Σ

theset

{σ

1 , . . . , σ

n

}

andby

Σ

∗

thefreemonoid on

Σ

. That isto say

Σ

∗

isthe setof nitewordson thealphabet

Σ

with on atenation asmonoid law. Wedenote by

1

the empty word.

Thepositivebraidmonoid

P

on

n+1

strandshasthefollowingsemigroup presentation:

P

= hσ

1 , . . . , σ

n

/ σ

i

σ

i+1

σ

i

= σ

i+1

σ

i

σ

i+1

and

σ

i

σ

j

= σ

j

σ

i

if

|i − j| ≤ 2i

We denoteby

≺

divisibilityontheleftin

P

(thatis,for

a, b ∈ P

,we have

a ≺ b

ifandonlyifthereexists

c ∈ P

su hthat

ac = b

)anddenote similarly by

≻

divisibility onthe right.

For

u ∈ P

,set

left

(u) = {σ ∈ S

su hthat

σ ≺ u}

right

(u) = {σ ∈ S

su h that

u ≻ σ}

1.2 A few notations

We dene a newset

G

of generators of

P

. For all

j, i ∈ {1, . . . , n}

su h that

j + i ≤ n + 1

,set:

δ

{j,j+1,...,j+i}

= (σ

j+i−1

) . . . (σ

j+1

)(σ

j

)

Theelement

δ

{j,...,j+i}

is thebraid where strand

j + i

movesup to position

j

behindthe braid. Weset then:

∆

_{{j,j+1,...,j+i}}

= δ

_{j,j+1}

· δ

_{j,j+1,j+2}

· . . . · δ

_{j,...,j+i}

(1.1)

= (σ

j

)(σ

j+1

σ

j

) . . . (σ

j+i−1

. . . σ

j+1

σ

j

),

(1.2)

Theelement

∆

{j,...,j+i}

isthehalf-turn ofthestrands

j, j + 1, . . . , j + i

(see Fig.1.2 for examples).

Set

G

to be:

G =

[

J

1 ∪...∪J

p

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1

2

3

4

5

6

7 δ

_{2,3

_}

δ

{2,3,4}

δ

{2,3,4,5}

Figure1: De ompositionin

P

7

of

∆

{2,3,4,5}

asaprodu tof

δ

{2,3}

,

δ

{2,3,4}

and

δ

_{2,3,4,5}

.

where

J

1 , . . . , J

p

aredisjointsubsetsformedby onse utiveintegersof

{1, . . . , n+

1}

,su hthat

J

l

< J

l+1

forall

l ∈ {1, . . . , p−1}

(i.e.: if

i

l

∈ J

l

and

i

l+1

∈ J

l+1

then

i

l

< i

l+1

) and su hthat

|J

l

| ≥ 2

for all

l ∈ {1, . . . , p}

.

We use the onvention that if

J

isa singleton,

∆

J

isthe empty word. But in the following, we always write an element of

G

in the simplest possible way (i.e.: withouttermsequal to theemptyword).

We introdu e a fewnotations:

•

Let

τ : G → N

be thefun tion dened by:

τ (g) =

p

X

i=1

(|J

i

| − 1)

if

g = ∆

J

1 · . . . · ∆

J

p

.

(1.4)

So

τ (g)

is thenumberof dierent Artin generatorswhi h appearin a representative of

g

.

•

Let

u

be a word of

S

∗

su h that

u = v · w

with

v, w ∈ S

∗

, we set:

v

−1

· u = w

and

u · w

−1

_{= v}

.

•

Let

J = {i, i + 1, . . . , i + j}

beasetof onse utiveintegers, wedenote:

1.

m(J) = i + j − 1

2.

J + 1 = J ∪ {i + j + 1} = {i, . . . , i + j, i + j + 1}

3.

J − 1 = J \{i + j} = {i, . . . , i + j − 1}

Remark 1. Let

g = ∆

J

1 . . . ∆

J

p

be anelement of

G

. We an noti e that

g

is the least ommon multiple of

S

p

i=1

(J

i

− 1)

.

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We an nowstate the main result:

Theorem 1. In

Zhh P ii

, the following identityholds:

(

X

g∈G

(−1)

τ(g)

g) · (

X

b∈P

b) = 1

(1.5)

Corollary 1. The growthfun tion of the positive braid monoid isequal to:

F (t) =

X

b

∈P

t

|b|

S

₌





X

g∈G

(−1)

τ(g)

t

|g|

S





−1

(1.6)

Proofs of Theorem 1 and Corollary 1 will be given in se tions 1.4 and

1.5.

Example 1 (Expli it formula on 4 strands). On Figure 2, we an read the

value of

τ

and the length of all elements of

G

on 4 strands. The growth fun tionis then:

F

4 (t) =

1 1 − 3t + t

2 _{+ 2t}

3 _{− t}

6 τ = 0

τ = 1

|.|

S

= 1

τ = 2

|.|

S

= 2

τ = 2

|.|

S

= 3

τ = 3

|.|

S

= 6

|.|

S

= 0

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1.4 Denition of the involution of

(G × P)

We onstru t an involution from

(G × P)

to itself with only

(1, 1)

as xed point. This givesus a natural way to pair elementsof

(G × P)\(1, 1)

whi h willimply great simpli ationsin thefollowing (see subse tion 1.5).

To onstru t theinvolution, given a ouple

(g, b) ∈ (G × P)

we want to move pie es of

g

(respe tively

b

) from

g

to

b

(respe tively from

b

to

g

). Let us make it more pre ise; we dene the set

E

of elements whi h will be allowed tomove:

E = {δ

J

, J

subset of onse utive integersof

{1, . . . , n + 1}

and

|J| ≥ 2}

(1.7)

We an nowdene:

Denition 1. Let

(g, b)

in

(G × P)

and

u ∈ E

su hthat

u ≺ b

(resp

g ≻ u

). We thenset

g

′

_{= g · u}

and

b

′

_{= u}

−1

_{· b}

(resp.

g

′

_{= g · u}

−1

and

b

′

_{= u · b}

). Now, if

g

′

satisesthefollowing onditions:

1.

g

′

belongs to

G

2.

τ (g

′

_{) = τ (g) ± 1}

,

then

u

is alledan eligible movingpart of

(g, b)

.

Observe that for

|J| ≥ 2

,

∆

J

· δ

−1

J

= ∆

J−1

. So there is at least one eligible moving part of

(g, b)

if

g 6= 1

. For

(1, b)

there is learly at least one eligible moving part unless

b = 1

. So for

(g, b) 6= (1, 1)

, it exists at leastone eligible moving part of

(g, b)

. In this ase we onsider theeligible movingpartmaximalforthelexi ographi orderingindu edbytheordering

σ

1 < σ

2 < . . . < σ

n

on the generators. We all it the movingpart of

(g, b)

anddenoteit

k

. Sin e

k

annotbebothfrom

g

to

b

andfrom

b

to

g

,we an set:

Ψ(g, b) =







(1, 1)

if

(g, b) = (1, 1)

(gk, k

−1

b)

if

k

is from

b

to

g

(gk

−1

_{, kb)}

if

k

is from

g

to

b

Figure3showsexamplesofhow

Ψ

worksonsome ouplesof

G

5 × P

5

(eligible moving partsarerepresentedwithdashedlines).

Lemma 1. The fun tion

Ψ

is an involution from

(G × P)

into itself whose unique xedpointis

(1, 1)

.

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Ψ

( ase4) ( ase3) ( ase1)

( ase2)

Ψ

Figure3: Examples ofhow

Ψ

works. Case numbers refer to lemma2. Ele-mentsof

G

5

arerepresented atthe top and those of

P

5

at thebottom

Proof. Assumethat

Ψ

xes

(g, b)

in

(G ×P)

,itimpliesthatthereisnoeligible movingpartof

(g, b)

. Nowitimpliesthat

g = 1

and thenaneligiblemoving partisanyelement of left

(b)

,hen e

b = 1

.

Now, given a ouple

(g, b)

in

(G × P)

we look at the maximal eligible moving part. From Denition 1,we an see,bydire tinspe tion,that:

Lemma 2. Let

(g, b)

in

(G × P)

. We write

g = ∆

J

1 · . . . · ∆

J

p

. Let

Ψ(g, b) =

(g

′

_{, b}

′

₎

and let

σ

max

bethe biggest element ofleft

(b)

then:

1. If

σ

max

≤ σ

m(J

p

)

,thenthemovingpartis

δ

J

p

and

(g

′

_{, b}

′

_{) = (∆}

J

1 . . . ∆

J

p

−1

,

δ

J

p

· b)

. 2. If

σ

max

isequal to

σ

m(J

p

)+1

but

δ

J

p

+1

⊀

b

, thenthe on lusion of the previous ase still holds.

3. If

σ

max

is equal to

σ

m(J

p

)+1

and

δ

J

p

+1

≺ b

, then the moving part is

δ

J

p

+1

,and

(g

′

_{, b}

′

_{) = (∆}

J

1 . . . ∆

J

p

+1

, (δ

J

p

+1

)

−1

· b)

.

4. If

σ

max

> σ

m(J

p

)+1

, then the moving part is

σ

max

and

(g

′

_{, b}

′

_{) = (g ·}

∆

{σ

max

,σ

max +1

}

, (∆

{σ

max

,σ

max +1

}

)

−1

_{· b)}

.

Examples of thedierent ases ofLemma 2 aregiveninFigure 3.

We ontinue theproof of lemma 1. Let

(g, b)

be an element of

(G × P)

and

(g

′

_{, b}

′

₎

beitsimageby

Ψ

,wewant toprove that

Ψ((g

′

_{, b}

′

_{)) = (g, b)}

. Two

ases have to be onsidered: either the moving partwasfrom

g

to

b

( ases 1 and 2 of lemma 2) or the moving part wasfrom

b

to

g

( ases 3 and 4 of lemma2).

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•

if

| J

p

| = 2

,we get

g

′

_{= ∆}

J

1 . . . ∆

J

_p−1

and

b

′

_{= δ}

J

p

· b

. Applying then ase4 oflemma 2to

(g

′

_{, b}

′

₎

gives

Ψ(g

′

_{, b}

′

_{) = (g, b)}

.

•

if

| J

p

| > 2

, we get

g

′

_{= ∆}

J

1 . . . ∆

J

p

−1

and

b

′

_{= δ}

J

p

· b

. Applying then ase3 oflemma 2to

(g

′

_{, b}

′

₎

gives

Ψ(g

′

_{, b}

′

_{) = (g, b)}

.

We have to be a little more areful for the se ond ase as we have to

study how left

(b

′

₎

behaves. Let

k

be the moving part of

(g, b)

from

b

to

g

, we write

k = δ

{i,...,i+p}

. If

σ

belongs to left

(b

′

₎

then

σ

is not bigger than

σ

i+p

. Indeedifweassumethatthereexists

j > i + p

su hthat

σ

j

∈

left

(b

′

₎

,

using the ommutation relations, we an rewrite

b = k · σ

j

. . . = σ

j

· k . . .

. Applying now lemma2 ontradi ts thefa tthat

k

isthe moving part from

b

to

g

.

So we are in ase 2 or 3. It remains to prove that we are not in ase 3 or

equivalentlythat

δ

{i,...,i+p,i+p+1}

⊀

b

′

. Assumethat

δ

{i,...,i+p,i+p+1}

≺ b

′

then we anrewrite :

b = δ

{i,...,i+p}

· δ

{i,...,i+p,i+p+1}

· b

1 = δ

_{{i,...,i+p+1}}

· δ

_{{i+1,...,i+p}}

· b

1

Sin e

δ

{i,...,i+p+1}

= σ

i+p

. . . σ

i

,itshowsthat

σ

i+p

belongstoleft

(b)

andon e morelemma2 leadsto a ontradi tion with

k

beingthemovingpartfrom

b

to

g

. This on ludesthe proof.

1.5 Proof of Theorem 1 and Corollary 1

For

(g, b) ∈ (G × P)\{(1, 1)}

, let

Ψ(g, b) = (g

′

_{, b}

′

₎

. We see that

gb

and

g

′

_b

′

arein

P

equal and that

τ (g)

diers from

τ (g

′

₎

only by 1. We an translate

thesetwo observations into the following equality (in

Zhh P ii

):

(−1)

τ(g)

gb + (−1)

τ(g

′

)

g

′

b

′

= 0

Besides,a ording to lemma1,

Ψ

is an involution whose unique xed point is

(1, 1)

. Wede omposethesumbelowon

(G × P)

alongtheorbitsof

Ψ

and dedu ethefollowing equalities (in

Zhh P ii

):

(

X

g∈G

(−1)

τ(g)

g) · (

X

b

∈P

b) = 1 +

X

(g,b)∈(G×P)\{(1,1)}

(−1)

τ(g)

gb = 1

(1.8)

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1 +

X

(g,b)∈(G×P)\{(1,1)}

(−1)

τ(g)

gb = 1

Proje ting it into

Z[t]

by identifying

σ

1 , . . . , σ

n

with

t

and noti ing that

|gb|

S

= |g|

S

+ |b|

S

give:

1 +

X

(g,b)∈(G×P)\{(1,1)}

(−1)

τ(g)

t

|gb|

S

_{) = 1}

1 +

X

(g,b)∈(G×P)\{(1,1)}

(−1)

τ(g)

t

|g|

S

_t

|b|

S

_{= 1}

(

X

g∈G

(−1)

τ(g)

t

|g|

S

_{) · (}

X

b

∈P

t

|b|

S

_{) = 1}

This on ludestheproof ofthe orollary.

Now that we dene an involution for braid monoids, we show that our

onstru tion remains valid for alarger lass ofmonoidsinthenext se tion.

2 Generalisation

2.1 Denition of a larger lass of monoids

We follow the work of Bronfman (see [Bro01 ℄) to extend our result to a

larger lassofmonoids. Fromnowonweonly onsidermonoids

M

= h S|R i

(where

S

is a nite generating set in

R

a nite set of relations), satisfying thefollowing properties:

1.

M

ishomogeneous(ie: all relations are

R

arelength-preserving) 2.

M

is left- an ellative, ie: if

a, u, v ∈ M

are su h that

au = av

then

u = v

.

3. Ifasubset

{s

j

|j ∈ J}

of the generatingset

S

hasa ommon multiple, thenthissubset hasa least ommon multiple.

We annoti ethatProperty1impliestheuni ityofaleast ommonmultiple

ifitexists.

Let

J ⊂ S

su h that

J

has a ommmon multiple and hen e a unique least ommon multiple byproperties1 and3. We denotethisleast ommon

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multipleby

W(J)

. Subse tion1.2leadsustodene anewset

G

ofgenerators as:

G = {

_

(J),

for all

J ⊂ S

su hthat

J

hasa ommon multiple

}

Observethatfor

K ⊂ J ⊂ S

su h that

J

hasa ommon multiple,

K

has alsoa ommon multipleand

W(K) ≺ W(J)

(we re allthat

≺

representsthe divisibilityontheleft).

Notation. For

J

and

K

dened asabove,we denote:

_

(K) · δ

_K

J\K

=

_

(J)

For

g =

W({s

i

1 , . . . , s

i

p

}) ∈ G

, we set

τ (g) = p

. This extendsthe deni-tionof

τ

of subse tion1.2.

2.2 Growth fun tion for monoids

Theorem 2. Let

M

be a monoid satisfying Properties 1,2 and 3 of se tion 2.1 and

S = {s

1 , . . . , s

n

}

be the set of generators of

M

. In

Zhh M ii

the following identityholds:

(

X

g∈G

(−1)

τ(g)

g) · (

X

m

∈M

b) = 1

(2.1)

The growthfun tion of

M

isequal to:

F (t) =

X

m∈M

t

|m|

S

₌





X

g∈G(M)

(−1)

τ(g)

t

|g|

S





−1

(2.2)

Proof of Theorem 2. For thepassage from 2.1to 2.2,theproof ofCorollary

1remains validsin e we assumethat

M

is homogeneous.

For the rst part of the theorem, we stay very lose from the proof of

Theorem 1 (see subse tions 1.4 and 1.5for more details). We onstru t an

involution

Ψ

from

(G, M)

to itself. We dene the set

E

of elements whi h willbe allowed tomove:

E = {δ

_J

{k}

,

where

J ⊂ {s

1 , . . . , s

n

}

and

k /

∈ J}

(2.3)

Denition 1 of an eligible moving part remains valid for a ouple

(g, m) ∈

(12)

s

1 < . . . < s

n

on the generators indu es a partial ordering on

E

by the following:

δ

_J

{i}

< δ

{i

_J

′

}

ifandonly if

i < i

′

It is straightforward to see that given any ouple of

G × M \ {(1, 1)}

, there is a unique maximal eligible moving part for this ordering, we all it the

movingpart and denoteit

k

. Wethendene:

Ψ(g, m) =

(

(gk, k

−1

_m)

if

k ≺ m

(gk

−1

, km)

if

g ≻ k

Lemma 3. The fun tion

Ψ

is an involution from

G × M

into itself whose unique xedpointis

(1, 1)

.

Ifthelemmaaboveisproved we anthen on lude theproofofTheorem

2exa tlyinthe same waywe didinsubse tion 1.5.

Proof of lemma 3. Thefa tthat

(1, 1)

istheuniquexedpoint is lear(see proofof lemma1 for details).

Let

(g, m) ∈ G × M

, we set

Ψ((g, m)) = (g

′

_{, m}

′

₎

and we want to show that

Ψ((g

′

, m

′

)) = (g, m)

. Asinthe proofof lemma 1, we distinguish two ases: eitherthe moving partwasfrom

g

to

m

or from

m

to

g

.

We beginwiththe ase ofa movingpart from

m

to

g

,we set:

(

g =

W(J)

m = δ

_J

{i}

· m

1 ,

and

(

g

′

=

W(J) · δ

{i}

_J

=

W(J ∪ {i})

m

′

_{= m}

1 ,

(2.4)

Assume the moving part for

(g

′

_{, m}

′

₎

is from

m

′

to

g

′

, it means that there

exists

l > i

su hthat

m

′

_{= δ}

{l}

J∪{i}

· m

′

1

. Then:

g · m =

_

(J) · δ

{i}

_J

· δ

_J∪{i}

{l}

· m

′

₁

(2.5)

=

_

(J ∪ {i, l}) · m

′

₁

(2.6)

=

_

(J) · δ

{l}

_J

· δ

{i}

_J∪{l}

· m

′

₁

(2.7)

This ontradi tsthefa tthat

δ

{i}

J

isthemovingpartfrom

m

to

g

. Itremains toshowthat

∀j ∈ J, j < i

whi hwillimplythat

δ

{i}

J

isthemaximaleligible moving partfrom

g

′

to

m

′

. Assumethereexists

j ∈ J

su h that

j > i

,then

δ

{j}

_J\{j}

isaneligiblemovingpartfor

(g, m)

biggerthan

δ

{i}

J

,whi h ontradi ts the maximalityof

δ

{i}

(13)

We dealnowwiththe ase ofa moving partfrom

g

to

m

and set:

(

g =

W(J ∪ {l}) = W(J) · δ

_J

{l}

m = m,

and

(

g

′

=

W(J)

m

′

= δ

{l}

_J

· m

(2.8)

It is lear that the moving part for

(g

′

_{, m}

′

₎

is from

m

′

to

g

′

(otherwise it

ontradi ts themaximality of

l

among elements of

J ∪ {l}

). Assume

δ

{l}

J

is not the moving part for

(g

′

_{, m}

′

₎

, it implies that

m

′

_{= δ}

{i}

J

· m

1

with

i > l

. Hen e we an write

gm = g

′

_m

′

₌

W(J ∪ {i}) · m

1

whi h implies

i ≺ gm

. Lastlyfor all

j ∈ J, j ≺

W(J)

and

l ≺ g

hen e :

_

(J ∪ {l} ∪ {i}) ≺ gm

_

(J ∪ {l}) · δ

_J∪{l}

{i}

≺ gm

g · δ

_J∪{l}

{i}

≺ gm

Now we assumed that

M

has the left an ellation property thus

δ

{i}

J∪{l}

≺

m

and

δ

{i}

J∪{l}

is an eligible moving part for

(g, m)

bigger than

δ

{l}

J

. This ontradi ts thefa tthat

δ

{l}

J

isthemovingpartfor

(g, m)

and on ludesthe proof.

2.3 A few examples of monoids

2.3.1 Artin-Tits monoids

Artin-Tits monoids are a generalisation of both braid and tra e monoids.

Given a nite set

S

and a symmetri matrix

M = (m

s,t

)

s,t∈S

su h that

m

s,t

∈ N ∪ {∞}

and

m

s,s

= 1

,theArtin-Tit monoid

M

asso iatedto

S

and

M

hasthe following presentation:

M

= hs ∈ S| sts . . .

| {z }

m

s,t

terms

=

tst . . .

| {z }

m

s,t

terms if

m

s,t

6= ∞i

(2.9)

Now anArtin-Titsmonoidis learlyhomogeneous, he hastheleftand right

an ellation property (see Mi hel, Proposition 2.4 of [Mi 99 ℄) and has the

least- ommonmultipleproperty(seeBrieskorn andSaito,Proposition4.1of

(14)

In[DP99 ℄, Dehornoy and Paris generalise Artin-Titsgroups. Theydene a

right Gaussian monoid asanitely generated monoid

M

su h that:

1. There existsa mapping

ν : M → N

su hthat

ν(a) < ν(ab)

for all

a, b

in

M

,

b 6= 1

.

2.

M

isleft an ellative.

3. All

a, b ∈ M

admit a least ommon multiple.

Su h a monoid is not ne essarily homogeneous. We an neverthelesse

loosen onditions ofTheorem2byonlyassumingthat

M

satisesproperties 2and3ofse tion2.1forageneratingset

S

. Wedenetheset

G

by hoosing for ea h ouple

(a, b) ∈ S

2

one least ommon multiple. The rst part of

Theorem 2remains true but itis no longerpossible to ompute thegrowth

fun tionof

M

.

Now, if

M

is a homogeneous right Gaussian monoid, we are exa tly in the onditions of se tion 2.1. We apply this point to Birman-Ko-Lee braid