Comultiplication on monoids

VOL. 20 NO. 4 (1997) 803-812

COMULTIPLICATION

ON MONOIDS

MARTINARKOWITZ Mathematics

Department

Dartmouth College

Hanover,

NH

03755

E-mailaddress:

[email protected]

MAURIClOGUTIERREZ Mathematics

Department

TuftsUniversity

Medford,

MA

02155

E-mailaddress: [email protected]

(Received

April 14,

1997)

ABSTRACT.

A

comultiplicationonamonoid

S

is ahomomorphismrn:

S

S

S

(the

free

productof

S

with

itself)

whose composition with each projectionisthe identity homomorphism.

We

investigate how the existenceofacomultiplicationonSrestrictsthestructure of

S. We

show thatamonoid which satisfies theinversepropertyand hasacomultiplicationiscancellativeand equidivisible.

Our

mainresultisthatamonoid

S

whichsatisfies theinversepropertyadmitsa

comultiplicationif andonlyif

S

isthe free productofafreemonoidanda free group. Wecall these monoids semi-free andwestudydifferent comultiplicationsonthem.

KEY

WORDS

AND PHRASES:

Monoid, freeproduct,comultiplication,inverseproperty. cancellativemonoid,equidivisibility, semi-free monoid.

1991

AMS

SUBJECT CLASSIFICATION CODES:

Primary:

20M05,

20M10 Secondary:

20M99,

20N99.

1.

INTRODUCTION

Comultiplicationscanbedefinedfor objectsinanycategorywithcoproductsand zero mor-phisms. Given such an object

X,

a comultiplication is a morphism m

X

X

II

X (the

coproduct of

X

with

itself),

suchthat the composition of rnwith eitherprojection

X

t

X

X

isthe identitymorphism. Thegeneral theoryhas beensurveyedin

[1],

andin

[3]

it isspecialized to algebraic systems.

In

[2], [5]

and

[6],

comultiplicationson groups have been studied.

In

this paperweextend some of these results to the categoryof monoids.

We

are interested in the following kindofquestion: givenamonoidwithacomultiplication, whatrestrictionsdoesthis placeonthestructure ofthemonoid? This question has beenanswered for groups by

Kan

who

showedthatagroup admitsacomultiplicationif andonlyifit isfree

[6].

For

monoids,

Bergman

and Hausknecht have shown that the existenceofan associativecomultiplication yieldsa pre-sentationof the monoid by generatorsand relations

[3,

Thin.

20.16].

Here

westudyarbitrary

comultiplicationsonamonoidwhich satisfiesanadditional condition

(the

inverse

property).

We

inverseproperty which may beof independent interest.

In

particular, weshow that ifsuch a monoidhasacomultiplication, thenit iscancellative

(Theorem 3.5)

andequidivisible

(Theorem

4.2).

In

addition,we studythe possiblecomultiplications onsemi-free monoidsin

5.

2.

BASIC FACTS AND DEFINITIONS

A

monoid is a set

S

with an associative, binary operation and an identity element. The operationisusually called multiplication and denoted by juxtaposition and the identity element isdenotedby 1. Thenotionsofsubmonoidandhomomorphismof monoidsareanalogousto thecorresponding notionsin group theory. Thereisalso thenotion ofafree monoiddefined

bythe usual universalproperty.

A

good general referenceonmonoidsis

[4].

Every

freemonoid hasabasisin whicheach elementcanbeexpressedasaword.

Moreover,

it isprovedin

[7,

5.1]

thatafree monoid hasauniquebasiswhichwecall the canonical basis.

If

S

and

T

aremonoids, thenone canformthefree product

S

T

inanalogytothe free productof groups.

A

typical elementofS,

T

is a sltls2t2..,s,t,,where s, E

S

andtj E

T.

A

product ofthefirst kfactors of a,

_<

k

<

2n,

iscalledaninitialsegmentof a. Ifsome

factor,

say s, 1,then the expression t,_l 1t, inacanbereplaced by t,_t,. Ifs2,...

,s,,t,...

,t,_ areall

#

1,thena is saidtobereduced. Ifa

S

T,

then the number of non-trivialfactors

ofa inreduced

form,

is denoted

[a[

and called thelengthofa. If

f"

S

S’

andg"

T

T’

arehomomorphisms, thenahomomorphism

f

g

S

T

S’

T’

isdefinedby

(f

g)(stl

st.)

f(s)g(t).. f(s)g(t).

We

alsohave injectionhomomorphisms

i

S

S

T

and

i2

T

S

T

definedby

i

(s)

s and

i(t)

t and projection homomorphisms p

S

T

S

and p

S

T

T

defined by

pl(a)

l-I

s, and

p(a)

[-I

tj. If

S-

T,

we write

i,(s)

s’

and

is(t)

t"

sothat atypical elementaof

S

S

canbeexpressed

a

st...s,,,

s,,t,

S.

(2.1)

The equalizer

Es

of

S

is the submonoid

{a]ae

S,S,p(a)

p2(a)}

of S,S. Then

PlIEs

P2[Es

andwedenotethishomomorphismbyp"

Es

S.

A

comultiplicationrnon

S

is ahomomorphismrn

S

S S

such thatplm prn id"

S

S.

The comultiplicationm iscalled associative if

(id,m)m

(m,id)m"

S S,S,S.

A

section of pis ahomomorphism #

S

Es

such thatplz id"

S

S.

Clearlycomultiplicationsof

S

correspondtosectionsof p.

In

particular,if

S

admitsacomultiplication,then

S

canbe embeddedin

Es.

Givenamonoid

S,

let

{z8

Is

S}

bea set in one-to-onecorrespondencewith

S.

Thefree group in

S,

[4, 12.1],

is the group

S

with presentation

(z8 ]zzt

z,t, s,t

e S).

Then a

homomorphism

-

S

S

is definedby

t(s)

z. Moreover,

if

G

isagroup and

f

S

G

is ahomomorphismof monoids,then thereisauniquelydefinedgroup homomorphism

f"

S

G

with

ft

f.

Thus,

if

S

admitsacomultiplication m,then

S

admitsacomultiplication with t

(t t)rn.

Finally, by

[4,

Thm.

12.4],

S

embedsinagroup if andonlyift"

S

S

is an embedding.

If

S

is a monoid and a, b

S

withab 1, then bis called aright inverseofa anda is calledaleft inverseofb. Ifa

S

hasaleftinverseandaright inverse, then theyare unique

and equal.

We

then say thataisinvertible with inverse a

-.

The set

Us

ofinvertibleelements

of

S

is asubmonoidof

S

which isagroup.

Definition 2.2.

Let S

beamonoid.

(2)

S

iscancellativeor acancellation monoidif ab ac orba caimpliesb c, for all a,b, cE

S.

In

verifying cancellation,weusuallyestablishoneofthe two implications. The otherisproved analogously.

We

note that ifa monoidsatisfies eitherproperty ofDefinition

2.2,

sodoes every submonoid.

Clearly every cancellation monoid has theinverse property:

For

if ab 1, then

(ba)(ba)

(ba)l,

andso ba 1.

However,

theconverse is not truesince the monoid S 1, a,a with a ahas theinverse propertybut isnot cancellative.

3.

THE

INVERSE PROPERTY AND CANCELLATION

In

thissectionweshowthatamonoid withacomultiplicationwhichhasthe inverseproperty iscancellative.

We

beginwithsomesimple lemmas.

Lemma

3.1. If

S

and

T

have theinversepropertyand a, 6

S

T

areinreduced

form,

then thereexists

a,,

S T

suchthat isinvertible,a

a

-,

and

a

isin reduced form. Furthermore, 1 or isaninitialsegmentof

.

Proof.

Express

aand inreduced form as

s,,

=

z,,,

(3.2)

I=1 :=1

wheres,,x

S

andt,,yj

T

and consider

a

s’ t"x’

".

We

listseveralcases.

Case1:

to

and

x

areboth

-#

1. Then

a

isinreduced formandso we set 1.

Case:

to=l

Thena=

...t soXl)

y

xpy. Let

be thesmallest integer

_>

0 such that either

(i)

so_tx+ 1 or

(ii)

to__,y,+ 1.

We

onlyconsider

(i)

since

(ii)

isanalogous.

We

have

sox

1, to_y 1, t,_ty and

xl

a

st

..t__

Y+

By

the inverse

property

for

S

and

T,

x[ so,

y-

to_,

and

xy

xzy is invertible with

-

t,_"

s’o_z+z

s andweset

x+y+

y.

Case

3:

xo

1. This issimilar to

Case

2, and henceomitted.

Lemma

3.3.

Let S

and

T

be monoids.

(1)

If

S

and

T

have theinverseproperty, then

S

T

has theinverseproperty.

(2)

If

S

and

T

arecacellative, then

S

T

iscancellative.

Proof.

(1)

Suppose

a,

@

S.

T

with

af

1.

By Lemma 3.1,

thereexists

,a,f

ES.

T

such that is invertible, a

a

-,

f

Sf

and

af

is in reduced form. Since

af

af

1, it followsthat either

a

f

or

a

1

.

In

eithercase

fa

1.

(2)

Suppose

af

3f,

where

a,,f

S.

T

are allreduced.

By Lemma 3.1,

there exists

6t,c,,6:,/, e

S

_*

T

such that

6t,6

are invertible, a

:

and

a

and/

arereduced.

By Lemma 3.1,

6

and

6

are either or aninitial

If

[6,[ < I=l,

then

6=

6/

for some invertible 7.

Hence

11

12

{1’2.

Thus

f

7f=-

Therefore

aVf=

af

fl=f=.

Therearenow several casesto considerdepending onwhethera7and

fl=

end withaprimedordouble-primedterm and

fz

begins withaprimed or double-primed term.

For

example, suppose that a7 and

=

end with primed terms

(say

s’,

and

u’;,

respectively)

and begs with aprimed term

(say

z).

Then s,za

uz.

By

cancellation we obtn s,

u

andm a7

fl=.

All othercas aretreated similarly. Thus

a

a?

af

ft.

Corolly 8.4.

Let S

beamonoid.

(1)

ff

S

htheinverse

prope

then

Es

htheinverseproperty, ff

S

isccellative, then

Es

iscanceHative.

(2)

Let

Shavea comtiplication. If

Es

htheinverse

prope

then

S

h theinverse

propey.

If

Es

iscancellative, then

S

iscancellative.

Theorem 3.5. If

S

is amonoid with theinversepropertyand

S

admitsacomultiph’cation,

then

S

and

Es

arecancellative.

Proof.

We

firstshow that if E

Es

and 1, then

[21

> [[.

Suppose

[21

_< [[.

Wewrite as

3’s’

or

7t’

forsome

,,

E

S.S

and s, non-trivialelements of

S.

If

7s’,

then

a

7s’/s’

and so

67

1.

By 2.9.,

7

-

and hence

7s’/-.

Therefore

pa()

pa(/)pa(/-)

1.

Hence

1

p()

pa(7)s(p(/))

-

andsos 1. This contradicts

#

1.

A

similarargument

holds if

7t".

Next

we show that

Es

has the followingweak cancellation property:

a

a or

a

a implies 1 for

a,

Es.

Suppose

c

aand 1. Then

[a[ > [[.

Then

a

a,for all k

>_

1, andwechoose

N

suchthat

[N[

>

2[0[.

Then

a

N_cannotequal

asincetheirlengthsare different. This contradicts 1. The other implicationisprovedsimilarly.

It

nowfollows that

S

has this same weak cancellationproperty: ax a or za a implies z 1. This is because thecomultiplicationmon

S

provides anembedding of

S

into

Es

and the weak cancellation propertyisinheritedbysubmonoids.

Now

weprove that

S

iscancellative.

Suppose

az bx in

S.

Then

a

3,

wherea

re(a),

re(b)

and

re(x).

We

representaand by

(3.2)

and

12I

l,_kV

(3.6)

k=l

whichareall assumed to be reduced. Againweconsidercases.

Case

1: t,

v,

x

areall 1. Clearlya andsoa b.

Case

:

x

1. Then

,(

vqy

..;.

Ift,yx andvqyi 1, then, comparing both sidesofthe above equation from the right, we obtain

sx:

x:.

This implies that

s

1, contradicting the fact that c is reduced. Thus

Vqy 1.

We

continue in this manner and conclude that if k cancellations arerequired to write

(

inreduced

form,

then exactlyk cancellationsareneeded toput

f

intoreducedform.

forsome

a.,,(2

E S,

S.

Thus

a(sx)’(

B(ux,)’(.

Sincethese arereduced,wecancel

2

from both sides and then multiplyon the right by

-1,

getting

a(sx)’$

Applyingp, wehave

a

p2(ot)

p2(Ol2(SkXl)’l

-1)

p2(2(rX/)’

-1)

P2()

b.

Case

3: x

7

1and

t

or

v

1. Thiscaseislike

Case

2,andhenceomitted.

This proves that

S

iscancellative.

By

Corollary 3.4,

Es

iscancellative. 4.

EQUIDIVISIBILITY

We

have needof thefollowingdefinition

[7,

p.

103].

Definition 4.1.

A

monoid

S

isequidivisibleif the equationax byin

S

implies that either there existsac E

S

such thata bcandcx y orthere existsa d

S

such that b ad and

Note

that if

S

iscancellative,then

S

isequidivisible ifax byimplies that either thereexists a c Ssuchthata bcorthereexists ad

S

such that b ad.

Theorem 4.2. /fthe monoid

S

has theinversepropertyand admitsacomultiplication, then

Sisequidivisible.

Proof.

We

assumeax byin

S

andapply rn to obtain

a

3r/,

where a

rn(a), 1

rn(b),

re(x)

andr/=

rn(y).

By

Lemma

3.1,thereareelements

6,

0,

a,

(1, 1,

rh

S

S

such that and 0areinvertible,a

a15

-1,

$,

B

10

-1,

r/=0rh and

al

and

r

h arereduced.

Case

1:

[al[

< [/31[.

Sincea(

r

h isanequalityofreduced expressions,

alA

1

forsome

A

or.

S,S. Then

a(6o-)

a-o

-

3o

-

3.

We

apply p to thisandget ad

b,

where d

Case

:

131

<

[al

1.

Thisissimilar to

Case

1.

Case

3:

[al[

[31 [.

Let

us assumethatalends with

s

(a

similarargument holds whenalends with

t)

andso

1

begins withsome

x.

We

writeal

ls

and

1

xl.’

Since

[al[= [1[,

andrh=wtrh Then

We write/31

=/lu

/31

endswith some

u’

andso rh beginswithsome

w

l(sx)’(l

31(uw)’l

yields

1

BI.

Now

a

p(a)

p(ls’5

-1)

p(,)p(5

-1)

and

()

(;0-’)

p(,)(o-’).

Thusa cu and b cv, whereuand vareinvertible, andso a

b(v-lu).

F

Corollary 4.3.

It"

amonoid

S

adm/ts acomLdtiplication rn and has the inverseproperty, then

S

S

and

Es

areequidivisiNe.

Pro@

First note thatrninducesacomultiplicationon

S

S

givenby

S

,

S

:-

S

,

S

,

S

,

S

’:-2

S

,

S

,

S

,

S,

where

T

S

S

S

S

interchanges the twofactors.

By

3.3, S

S

has the inverseproperty.

By

4.2,

S

S

isequidivisible.

Now

suppose

a(

r

in

Es.

Since

Es

C_

S

S

and

S

S

is equidivisible, there exists q, S,S such that a

B’r

or there existsa S,S such that

aS.

In

the former

case,

pl(3)pl()

p,(a)

p(a)

p()p(/).

Since

Pl()

p(/3)

and

S

is cancellative,

Pl(/)

P(V).

Thus/e

Es.

Theothercaseissimilar. Equidivisibility for

Es

now follows by

3.4.

Proposition 4.4.

Let

S

beamonoid. Then the foBowingareequivalent:

(1)

S admits a comultiplication,

S

has the inverseproperty and the group ofinvertible elements

Us

1.

(2)

S

isafree monoid.

In

thiscase, any comultiplication ofSisassociative.

Proof.

We

firstshow

(1)

implies

(2).

Ifx andwecanwritex uv foru

#

andv

#

1,then uiscalledaleft factorof x.

By

[7,

Cor.

5.1.7],

itsuffices to show that every non-trivial x

S

hasfinitelymanyleft factors.

Suppose

u is aleft factorof x. Then

rn(x)

m(u)rn(v).

There canbenocancellationbetween the last factor of

re(u)

and the firstfactor of

re(v)

because

S

has theinversepropertyand

Us

1. Thus

m(u)rn(v)

isinreduced form. Ifwe write

(

l-I,__l z:v:’,

then forsomel, 0

<_

<

p,

re(u)=

x;y’

z,+l or

s’,

wherer orisaleft factor ofyz+land wheres 1or isaleft factorofxz+l.

In

the first case

-l+

u

prn(u)

,,,=x, andin the secondcase u

p2rn(u)

[l,=

y,. Thus everyleft factoru

of x has the form u

I-I

+,=1x,oru

1,=

y,. Since thereareonly finitelymanyof these, x has

finitely many left factors. This proves

(2).

For

(2)

implies

(1)

itisclear that if

S

is

free,

then

S

has theinversepropertyand

Us

1. If

Y

C_

S

isabasis, thenacomultiplication of

S

isdefinedby

re(y)

y’y"

fory

Y.

This proves

().

If Sisafree monoid withbasis

Y,

then

re(y)

t ..-to

y, and this impliessomes, y,sometj yand all other factorsaretrivial. Therefore

re(y)

y’y"

or

re(y)

5.

SEMIFREE MONOIDS

In

thissection we study comultiplications oncertainmonoids

(called semi-free).

We

shall see in

6

that a large class of monoids with comultiplication

(namely

those with the inverse

property)

aresemi-free.

We

begin withsome preliminaries.

We

recallfrom

[2]

somebasic facts aboutcomultiplicationsongroups.

Let

F

beagroup with

comultiplication n.

We

know that

EF

isafree groupwith basis

a’a",

for alla

-

1 in

F,

andso

F

is afreegroup

[5].

If

X

is abasisfor

F,

thenforevery x

e X,

n(x)

canbeexpressedas areduced wordinfinitelymany of thegenerators

6,(,),

where

6,(x)

F,

and 1,... ,k.

(We

have infactgivenanalgorithmin

[2]

for finding the

$,(x).)

Then the set

A,

{$,(x)[

z

X}

is thequasi-diagonal set ofn andisessential toourstudy of comultiplicationsongroups in

[2].

For

example,if

D

{dl

d

F,

d 1,

n(d)

d’d"}

is thediagonal set of n,then

Do

C

Ao,

andn is associativeif andonlyif

D,

A.

We

next consideranaloguesof these sets for monoids.

Let S

beamonoidwithcomultiplication rn.

We

define thediagonal set

D,

andantidiagonal set

D,

ofmby

D,,

{did

S,

d 1,

re(d)

d’d"}

and

D,

{did

S,

d notaunit,

re(d)

d"d’}.

Definition 5.1.

A

monoid S issemifree ifS

U

M,

where

U

isafree group and

M

isa free monoid.

Hence

if

S

is

semifree,

the group of invertible elements

Us

U

and

M

has a canonical basis

Y

(2).

If

X

is any basis of

U,

we say that

(X, Y)

is abasis for

S.

The cardinality of

X

t2

Y

is called the rank of

S. Clearly S

U

M,

and

M

is the free group with basis

Y.

Ifm isa comultiplication on

S

and isthe induced comultiplication on

,

then

[u

rn]u

U

U

,

U.

Moreover,

for each y E

Y,

rn(y)

IJs’,t’,’

S

,

S

which we assume is reduced. Since

l-I

s,

l-I

t, y,it follows that for precisely one indexj

(resp.,

k)

sj ujyv

and

u,va,s,

U

for all j

(resp.,

t

wiyzi andw,z,t,

U

for all

# k). Moreover,

sl...s_luj vsj+..,

s

1 and similar equations holdforthe

t,,

wi,

z.

Now

weproceedto

computethequasi-diagonalset

A

of

.

Clearly

zh

A,,lu

U

A,

whereA

{,(y)[y

y}.

By

theabove

remarks,

if j

<

kand y

Y,

then

6,(y)

e

U

forall

q

[22-

1, 2k-

1]

and

,

(y)

is of theformu, yv, for all

e

[2j-

1,2k-

1],

whereu,, v,

U.

Ifk

<

j andy

Y,

then

,(y)

e

U

for

all/ [2k-

1,2j-

1]

and

$,(y) =u,y-lv,

forall/e

[2k-

1,2j-

1],

whereu,,v,

e

U.

Thus the expressions for the 6,

(y)

E A which arenotin

U

areallofthe form uyvor all of theform

uy-lv,

whereu,v

U.

If

A

lies inSwesay that isaquasidiagonal element ofrnand denote the set ofquasidiagonalelementsby

A,,.

If$- At3

S

and isnotaunit, thenwesaythat 6- is aquasi-antidiagonalelementofrnand denote the setofsuch elements by

A,.

Thus

A,

istheunionof

A,u,

the units in A and all elements of theform uyvin

A,

and

A%

consistsof all elements of the form

v-yu

-,

where

uy-v

A.

Clearly

D,

C

A,,

and

D,

C_

A,.

We

illustrate all of this with a concrete example.

For

easeof notationwe write for the inverseofagroup elements and for

a’a"

e

Es.

We

usethe algorithm of

[2]

to expressan elementof

Es

asawordinthe

’s.

Example 5.2.

Let S

beasemifree monoidof rank 4 with basis

X

Y,

where

X

{xl,x}

and

Y

{y,

y

},

and defineacomultiplication rnby

Note

that the comultiplication

ml=

isExample

3.7(2)

of

[2].

Thenwe have

A

{xl,

x:,

"., x,x’-, xyx,

x,x2,

2x,x:}

and

{x,,

xly’},

andso

A,,

{x,

x,

,

x,5,

xx},

D,,

{x,},

A:,

{xyx, xx,y}

and

D

{xyx}.

The following theoremisthenproved analogouslyto

[2,

(3.6)

and

(4.5)].

Theorem 5.3. If

S

isafiniteranksemifreemonoid withcomultiplicationrn, then

(1)

Theset

A,

isafinite set of generators of

Us

and

A,

U A is afiniteset of generators of

S.

6.

THE

MAIN THEOREM

In

thissectionweprove the mainresult of the paper

(Theorem 6.3).

Ifaand bareelements ofamonoid

S,

thenaand baresaid to beassociate ifa bu forsomeu E

Us.

Clearlythis is anequivalence relationon

S.

We

denoteby

{a>

theequivalenceclassofaE

S.

Definition 6.1.

Let S

beamonoid anda

S.

Ifthe set

(u)

lu

aleft factor of

a}

isfinite, thenaiscalledfinitely

decomposable.

/f

{ (u>

lu

aleft factor of

a}

hasonedement, thena iscalledindecomposable.

Thefollowingis ageneralization of

[4,

Thm.

9.6].

Lemma 6.2.

A

monoid

S

issemi-free if andonly

(1)

S

iscancellative,

(2)

Us

isafree group,

(3)

S

isequidivisible and

(4)

eachelement of

S

isfinitelydecomposable.

Proof.

Theseareclearlynecessary conditions.

To

showsufficiency,let

Y

be the setofnon-units of

S

which areindecomposable.

As

in

[4,

Thin.

9.6],

Y

generatesafree monoid

M

with

UM

and

Us

M

1.

We

shownowthat

Us

and

Y

generate

S.

Ifaisindecomposable, theneither a

Us

or a

Y.

If not,a aa,where

a

isnotaunit. If

a

and

a

areindecomposable, then wearedone. Otherwisewecontinuethis process and obtainforeach naproducta

a...a

witha...a, notassociatetoa...a,+.

We

claim that

(a>,

(aa>,

(a-..

a,>

aredistinct classes.

For

if a...a, isequivalent to

a...a,

j

>

i, thenby cancellation,

a,+...a

is a unit. Thus a,+,...

,a

are units. This contradicts our previous assumption. Thus for any n, we obtain ndistinct equivalence classes of left factors ofa. This contradicts

(4).

Finally, to show that each elementof

S

canbe uniquelyrepresentedin termsof

Us

and

Y,

see

[4,

Thm.

9.6].

We

nowproveourmaintheorem. Theresultis in the spirit of

Kan’s

workongroups witha comultiplication

[6].

Its

prototype isthe result ofBergman-Hausknecht

[3,

Thm.

20.16]

which classifiesall monoids which admitanassociativecomultiplication.

For

suchmonoids, Theorem

6.3follows from

[3,

Thin.

20.16].

Theorem 6.3. If

S

is amonoidwiththeinverseproperty,then

S

admitsacomultiplicationif andonlyif

S

issemi-free.

Proof.

Clearly if

S

is semi-free,

S

admits a comultiplication

(see 5).

Now

suppose that

S

admitsacomultiplicationm. ThenbyTheorem

3.5, S

iseancellative andbyTheorem4.5,

S

is equidivisible. Also m inducesacomultiplicationon

Us,

and so

Us

isa free group

[6].

Thus it sufficestoshow that

S

hasproperty

(4)

of

Lemma

6.2.

We

do this inasimilarwaytothe proof

of Proposition 4.4.

Suppose

x

S

and uis aleftfactor of x. Then x uvforsomev E

S,

where

#

=d

#

,

=do

()

(/().

Suppo

()

,,,,

dud.

By

Lemma

3.1,

re(u)

a

and

re(v)

-,

where

a,

for a,

,

e

S

S.

Thus u

p,m(u)

is aproduct of the form

(1-1l

x)d

or

(1-1l

y)e,

where

d,

e

e

Us

istheimageofp, orp of$and

Thushereareonly finitely many equivalenceclassesof lef factors ofz. Thiscompleeshe

proof.

Corollary 6.4. If Sisacancellativeand equidivisiblemonoid, then

Es

issemi-free.

Corollary 6.5. If

S

is a commutativemonoid with comultiplication, then

S

,

N

,

the [ree monoidon onegenerator, or

S

,m

Z,

the[ree

group

ononegenerator.

Note

that there areexactly two comultiplicationson

N

(see

the proof of Proposition

4.4)

andthat the comultiplicationson7/,_{have been classifiedin}

[2,

Lem.

6.9].

ACKNOWLEDGEMENT.

We would like to thank J. M. Howie and G. Lallement for valuable clarifications.

Part

ofthisworkwasdonewhilethesecond-named authorwas avisiting scholar at Dartmouth

College.

PEFERENCES

1.

M.

Arkowitz, Comultiplications in algebra and topology, Rend. del

Sem.

Mat.

e Fisicodi Milano

(to

appear).

2.

M.

Arkowitzand

M.

Gutierrez, Comultiplications on

free

groupsandwedges

of

circles,

Trans.

Amer.

Math.

Soc.

(to

appear).

3.

G.

Bergrnan

and

A.

Hausknecht, Co-groups

andCo-ringsinCategories

of

Associative Rings,

Mem.

Amer.

Math.

Soc.,

1996.

4.

A.H.

Clifford and

G.B.

Preston,

The Algebraic Theory

of

Semigroups,vol. and

II,

Amer.

Math.

Soc.,

1967.

5.

B.

Eckmannand

P.

Hilton,

Structure

mapsingroup theory, Fund.Math.50

(1961),

207-221.

6.

D.

Kan,

Monoids and their

duals,

Bol.

Soc.

Math.

Mex.

3

(1958),

52-61.