Systems of reductions

Lecture Notes in Computer Science Edited by G. Goos and J. Hartmanis

277 B. Benninghofen S. Kemmerich M.M. Richter

Systems of Reductions I

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editorial Board

D. Barstow W. Brauer R Brinch Hansen D. Gries D. Luckham C. Meier A. Pnueli G. Seegm(Jller J. Stoer N. Wirth Authors

Benjamin Benninghofen MBB D-8012 Ottobrunn, FRG Susanne Kemmerich Technische Hochschule Aachen Lehrstuhl f~JrAngewandte Mathematik, insbesondere Informatik Templergraben 64, D-5100 Aachen, FRG Michael M. Richter Universit~t Kaiserslautern, Fachbereich Informatik Postfach 30 49, D-6750 Kaiserslautern, FRG

CR Subject Classification (1987): R2.2, 1.2.3, R4.1, E4.2 ISBN 3-540-18598-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-t8598-4 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned, specificallythe rights of translation,reprinting,re-useof illustrations,recitation, brcadcasting, reproductionon microfilmsor in other ways, and storage in data banks. Duplication of this publicationor parts thereof is only permitted underthe provisionsof the German Copyright Law of September 9, 1965, in its version of June24, 1985, and a copyrightfee must always be paid. Violationsfall underthe prosecutionact of the German Copyright Law. © Spr[nger-VerlagBerlin Heidelberg 1987 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 2145/8140-543210

Introduction

Systems

of r e d u c t i o n s

called)

enjoy

science. tional

this

have

algebra;

also

become

these

and have

common

areas

much

is

the

equality

is the only predicate. with

a

fragment

class

of

tool

anyway An

are

not

In practice, problems

often

computer computa-

very

well

branch

logic".

of p r e d i c a t e

restricted as well as more general.

in

important

"equational

speaking,

concerned

a useful

in common.

background

as they

in t h e o r e t i c a l

are

strictly

is

rules

a growing popularity

They

separated

(or r e w r i t e

This

logic, equality

which

of is,

where logic

are

more

Typical are implications

of the form

Z~p Here

~ is

however,

a

universally

quantified

set

need not to be an equation

of first order predicate

of

equations;

and not

P,

even a formula

logic.

An example of the latter type of problems "Is each algebra in which ~ h o l d s

is:

finite?"

or equivalently "Is the free algebra defined by ~ finite?"

The

answer

one

expects

is an a l g o r i t h m i c single

questions

in equality

one.

but

That

whole

logic to

means,

classes

one

such questions

does

not

of questiQns.

consider These

may

arise in different ways:

i)

is fixed and P varies. An example is the word problem,

where ~ defines the

algebra and P varies over all equations

2)

s = t.

varies and P is fixed. An

example

algebras

is

the

(which

infinity

are

under consideration).

defined

problem by

the

for

a

axiom

class system

of

IV

3)

and P both vary. An example is the uniform word problem for a class of algebras.

Algorithmic mainly

decision

in the

theorem

context

proving.

in p a r t i c u l a r ago,

at

were

computable. abstract

of

more

In

algorithmic

algebra,

time

arise

of abstract

The

in

"the

problems

problems

less

first

tradition

much

half

older.

the

methods

There was still an enormous

interest

computability

problems,

decision

and automated

A

hundred

solutions

of

science

in mathematics,

automatically

and non-constructive

and

computer

data types

Kronecker,

or

the

is

in

to

algebraic

expected

twentieth

became

years

to

century

more dominant.

in principal aspects of but

constructive

me-

thods did not play such a big role in the every-day-work mathematicians.

This has radically changed under

ence of the development of computer science. tive

methods

are

not

only

certain applied situations; relevant

regarded

Among the defining properties one.

The basic

as

for

direction.

groups

1912-14

which

being

useful

in

of equality reflexivity idea of a reduction

to

of a specific for of

course,

search

on Dehn's

which

aspects

one-sided

than some

full

use.The

idea

and

fifty

years.

of the

power

algorithm was of

Dehn

and

Dehn's algorithm is the application

system of reductions more

M.

word problem

and which was one of the first decision problems In fact,

by

is to give

of equals by equals

is as old as the

considered

is the

in

studied

was

idea

Today construc-

A.Thue

be formulated.

looses,

This

the influ-

insights.

up symmetry and allow only replacements in one

of

their main purpose is to provide

structural and combinatorial

only trivial

be

Giving

are

regaining

up

with the

The

of

re-

question

preserved part

were

symmetry

of equality.

concerned

equality of

its properties

the

by

its power

p r o v i d e d by symmetry by systematically adding new reductions came

up m u c h

later.

This

leads

to

concepts

of

systems

of

reductions like the Church-Rosser and the weak Church-Rosser property

(which

are

known

under

finite t e r m i n a t i o n p r o p e r t y ;

various

names)

and

in c o n n e c t i o n w i t h the

the

latter

t h e r e is g r o w i n g interest in p a r t i a l and total w e l l - f o u n d e d orderings on the terms.

The

most

ensures

useful

that

property

is

certainly

e a c h t e r m t reduces

i r r e d u c i b l e t e r m t#;

to

completeness,

it

a uniquely determined

t # is the canonical form of t. The aim

of the completion a l g o r i t h m is to enlarge a given system of reductions system

in

order

to

(if it exists)

obtain

a

complete

one.

A

complete

can be regarded as a link b e t w e e n the

finite system of equations and the algebra defined by these equations which is a set-theoretic,

There one

are two main studies

criteria

the

which

to

terminate

systems.

lines of research here. completion

ensure

a l g o r i t h m in many

its

algorithm

termination.

leads

to

cases these

the

and As

searches the

investigation

can be

object.

On the one hand

(one is tempted to say "most")

this

In many

often infinite,

for

completion cases fails of

infinite

finitely d e s c r i b e d and

are as useful as finite systems.

The other type

of investigations

is concerned with the use

of complete systems. A complete system certainly provides an answer

to

the

word

problem

but

unravels

much

structure of the algebra under investigation.

more

of

the

This turns out

to be most apparent in the case of groups.

Most

of

the

years after partially

material

in

these

notes

obtained

in

1978 at the Technische H o c h s c h u l e Aachen;

contained

in

the

dissertations

Patrick Horster and Susanne Kemmerich; wrote

was

of

Hans

the

it is

BOcken,

Patrick Horster also

section IV.2.Very useful for computer experiments was

Vl

an i m p l e m e n t a t i o n semigroups

as w e l l

implementation Cyber

175.

Most

that

by

part

the

interest

Klaus of

Dittrich

these

material

is

of the a u t h o r s

applications

however.

these

forward-backward

notes part

algorithm.

This

in

on

Pascal

was

of

his

are

is in g e n e r a l

are

in

T h e r e are s e v e r a l familiar

feeling

for d i f f i c u l t i e s

general

universal

group

possibilities

reasons

structures and

algebras.

for

and

importance

Another

interesting

and is also u s e f u l

order

general

by

Habilita-

to

reader with

the

for this. one

has

r e a s o n is that

is

is

understanding

somewhat

included.

meant

of

and

One

is

a better

of r e s u l t s

connections

volume

an at least

completeness

semigroup

than

in

in t h e s e

This gives comparisons

for a fruitful competition.

make

material

principles.

and

areas c o m p u t a t i o n a l m e t h o d s are well e s t a b l i s h e d .

In

written

a

at the U n i v e r s i t y of K a i s e r s l a u t e r n .

concrete

theory,

done

last

Otto;

tionsschrift

as the

was

The

Friedrich

The m a i n

of the c o m p l e t i o n a l g o r i t h m for g r o u p s and

the

The

almost in

i d e a was

complete

the

rest

self-contained

sense

of t h e

much

to p r o v i d e

introduction.

that

suffices

material.

It

the Here

for

is

an

natural

that m a n y aspects h a d to be left out.

The

authors

remarks.

have

O n the

independently situation

was

also

been

one hand,

many

by d i f f e r e n t not

so

very

reluctant

results

authors.

clear

to

us

with

historical

s e e m to b e

On the that

obtained

o t h e r hand,

we

dared

to

the make

s t a t e m e n t s on p r i o r i t y questions.

There

are

several people

w h o m the

u s e f u l h e l p and d i s c u s s i o n s students Dittrich,

of

Aachen

influenced

are i n d e b t e d

over the years.

would

name

Hans

P e t r a Z i m m e r m a n n and Tom Beske.

w a n t s to m e n t i o n early

we

authors

A m o n g the former BQcken,

Woody

Bledsoe.

Klaus

One of the a u t h o r s

D a l l a s L a n k f o r d in p a r t i c u l a r ; by

for

Later

he was also on

useful

VJl

discussions Buchberger,

took Richard

place

with

Mike

Ballantyne,

G~bel,

Deepak

Kapur,

Wolfgang

Bruno

K~chlin

and J. N e u b ~ s e r .

All

authors

are

indebted

to

Mr.

v.

N~kel

for the e x c e l l e n t

preparation

Last

not

our

thanks

are

Forschungsgemeinschaft

without

their

this

least

research

Hehl

and

Mrs.

Scarlet

of the m a n u s c r i p t .

due

to

the

support

over

w o u l d not have b e e n possible.

Deutsche the

years

I.

General

Concepts

Algebra

1

Terms and Substitutions

1

I.l

Algebras,

1.2.

Some Concepts Languages

II.

from Universal

of the Theory of Formal

and Automata

14

1.3.

Decidability

24

Finite

Sets of Reductions

26

II.l.

First Concepts

26

II.2.

The Completion Algorithm

38

II.3.

The Ground Case

48

II.4.

First Analysis

II.5.

The Special Word Problem

of the Completion Algorithm

Small Cancellation II.5.1.

for Groups

and

Theory

82

Superpos-Deduction-Chains Criteria

and

for the Solvability

of

the Word Problem II.5.2.

84

The Small Cancellation

Conditions

and the Condition K 11.6.

Relations

III.

Infinite III.l

96

between the Completion

and the Todd-Coxeter

Procedure

Algorithm

106

Sets of Reductions

Regular III.l.l.

Infinite III.l.2.

116

Systems Regular

116 Systems as Special Systems

Applications

116

of Regular Reduction

Systems to Sets of Words III.l.3.

The Undecidability

121

of the Church-

Rosser Property III.l.4.

A Possible

III.2. Forward-Backward III.3. The Church-Rosser Backward

Systems

69

137

Church-Rosser

Test

Systems Property

140 146

of Forward153

IV.

Automata and Reductions

170

IV.l.

General Aspects

170

IV.2.

The Complexity of Reductions Algorithms

181

IV.3.

The Cycle Structure and the Growth Function

190

IV.4.

Effective Aspects of Gromov's Theorem

202

IV.5.

A Relation between the Growth Function and the Completion Algorithm

V.

Deciding Algebraic Properties of Finitely

210

218

Presented Monoids by Friedrich Otto V.I.

Monoid Presentations and Tietze Transformations

V.2.

218

Markov Properties of Finitely Presented Monoids

221

V.3.

Automata for Reduction Systems

228

V.4.

Deciding Algebraic Properties of Monoids Presented by Finite Complete Reduction Systems

V.5.

234

Deciding Algebraic Properties of Monoids Presented by Finite Monadic Complete Reduction Systems

243

References

256

Subject Index

264

List of Symbols and Abbreviations

265

I.

General

1.1.

Concepts

Algebras,

Although algebra

we we

from_Universal

Terms

and Substitutions

assume recall

concerned

with

Algebra

familiarity some

of

the

finitar Z

with

the

basic

definitions.

alsebras,

We

i.e.

concepts will

only

algebras

of be

of

the

ni ~

0 a

form A

where

A fi:

the

Ani

natural

number.

it w i t h

an e l e m e n t

signature

fix

it t h r o u g h o u t

of A

from

i.e.

I i ~ i ~ m>

are

ni-ary

we

from

A, t h e c a r r i e r

of

A ~ B

call

f~

operations,

ni=0

most

h:

B,

A

is the

Homomorphisms to

~

If

The

A

= < A , f ~±

sequence the

a constant

and

identify

of A°

~ = ~(A)

= ;

we

discussion.

are

structure

h ( f ~ ( a I ..... a n i ) )

=

preserving

mappings

f ~ ( h ( a I) ..... h ( a n i ) )

holds. Isomorphisms

are

homomorphisms A s B means

that

relation

for

R ( a k , b k) The

homomorphisms

that A and B are

A congruence such

i-i

and

endomorphisms

are

A ~ A.

congruence

isomorphic.

for A is an e q u i v a l e n c e

relation

RcA 2

l~k~n i implies class

R { f ~ ( a I ..... a n i ) , f ~ ( b I ..... b n i ) ) .

of a ~ A m o d u l o

R is d e n o t e d

by

lair or

2 simply

by

[a];

the q u o t i e n t

A/R

the c a n o n i c a l Defining

of A m o d u l o

= ( [a] I a ~ A};

mapping

7:

the o p e r a t i o n s

A ~ A/R

the q u o t i e n t

The c o n g r u e n c e

(i)

a l g e b r a A/R.

= h(b));

The

interrelations canonical

kernel (ii)

it is a l s o

h: A ~ B

called

If

h:

A ~

The

R h,

intersection

ence

relation.

a smallest

This

is c a l l e d

R ~

the kernel

implies each

R

is

congruence

is a h o m o m o r p h i s m

a

surjective

of c o n g r u e n c e

Therefore

the

A 2 there relation

operator sense

each

relation

congruence

the

with

is a g a i n

relation

generated

a congru-

R c A2

s.t. R [ by

R;

there



the

c A 2.

operator

operator.

relation

In s o m e

A/R

homomorphism

relations

for

congruence

closure

A ~

then A / R h s B.

R(h(a),h(b)) c

are:

is R.

B

is a c l o s u r e

congruence

between these concepts

mapping

of w h i c h

kernel

A

[f~(a I) .....f ~ ( a n i ) ]

h.

The b a s i c

is

=

r e l a t i o n i n d u c e d by a h o m o m o r p h i s m

is R h = ((a,b) I h(a) of

is g i v e n by ~(a) = [a].

on A / R by

f ~ / R ( [ a l ] ..... [ani])

one o b t a i n s

R is

R

for

each

is R

is

the c

fu_!l!z i n v a r i a n t endomorphism smallest

(R~;

the

h.

fully

operator

if

R(a,b)

Again,

for

invariant R ~ (Rm

is

a

too.

the m a i n

g a t i o n of this o p e r a t o r

topic

of t h e s e

notes

f r o m the a l g o r i t h m i c

is the

investi-

p o i n t of view.

For a set

X an a l g e b r a

free a l g e b r a

for

generated ~

every

nature

free

extension

equivalent

If

(ii>

all

l~i~m>

mapping

(of the

h:

same

sig-

X ~ B there

is a

of h to a h o m o m o r p h i s m

possibility

k,

of

h:

describing

A

an

~

B.

absolutely

{iii)

A

the

{0)°

In t h i s

algebra

isomorphism, A(X)

various

the

which

An absolutely

free

an alphabet

For

absolutely

tk=s k

all

operations

f~

and

all



of

conditions

algebra

the

generated

motivates

by

Xo

X u {fi~

free

of A

t h e n B = A.

(i) -

the

natural

by the (iii)

numbers

singleton

set

are

the

alternative

just

name

Peano

b y X is u n i q u e

up to

free algebra.

generated

It

of c o n s t r u c t i n g

in X.

If

i=j,

h e n c e w e can r e f e r to the a b s o l u t e l y

not

induction

then

i.e. if B is a s u b a l g e b r a

of a b s o l u t e l y

generated ways

b y X,

operation

case

instead

for

algebra

successor

axioms

X

X ¢ B,

is t h e

with

f ~ ( s I ..... Snj),

t k-

such t h a t

Peano

~

is g e n e r a t e d

example

=

l~k~n i,

f ~ ( t I .....tni) arguments

over

every

f ~ ( t I .....tni)

for

bra

=

~round

always

there

are

it

easy

Again

K over

pair

not

is

exist.

however to

see

isomorphic.

of

terms

e~uation

is

and

will

an

be

equation

terms.

h:

= h(t);

for

will

theories

algebras.

ordered

A homomorphism h(s)

free

an

s

algebras

T ~ A

s ~ t

is

satisfies

satisfies

true

an

in A

s ~ t. T h i s

equation

iff

last

every

s ~ t

iff

homomorphism

situation

is d e n o t e d

by

A I~ s ~ t.

The

models

for A

which

set

A I~ s ~

semantic

equations tion

for

t

E

of e q u a t i o n s

holds

theory

are

K

variet[)

for

all

are those

s ~ t e E

algebras

(denoted

by

where tant

Hence

of

A i~ E i m p l i e s

equations

are

those

A I~ s ~ t; t h e

nota-

of

equations

a class

closed

K of

under

algebras

consequences.

iff

K

definable

is d e f i n e d

by

some

E;

in t h i s

by

quotient

theory

A

= { A I A ~ E).

Z

the

(or case

a

the

K(E).

congruence TE

relation

= T/<E >

T / < E > is

the

is

generated

generated

canonical

by

mapping.

It

by XE is

E in T = ~E(X) impor-

notice:

varieties

a class

E of

e~uationall

E is t h e

T ~

set

is

denoted

~E:

which

a

algebras

~

to

of

E ~ s ~ t.

sets

iff K

is

<E>

is

defines

class

then

s ~ t for

E

class

consequences

this

Theories

If

a

t~ E).

The

A

of

of

T E is

free

for

have

free

algebras;

algebras

has

free

K(E)

over

X E.

moreover

algebras

iff

it

we

mention

that

is a v a r i e t y .

In

the

if

E contains

K(E)

same

are

way only

the

in g e n e r a l In f a c t

WE

ground

same

more

{s

as

Because

the

word

we

~

= <E>

can

refer

algebras

to are

also

assume

added The

the

initial of

has

gruence "below"

algebra and

the

this

these

up

to

algebra

of

data

data

type,

which

contains

<E>

defined

relation

in

is as

follows.

only

D c A x A we put

D

are

invariant

endomorphism

isomorphisms

K(E).

Such

types,

E

Here

we

we

initial

is

called

recall

principle

{s

-- t

as

{

of e q u a l i t y

In

fact,

in t h e

<E>

con-

invariant

con-

E:

t}.

fully

of

arbitrary

cases

R(D)

A

~E~

s>l

<s,

from

algebras

= T and

A

A)

E

D and

h is a n

t>

of A}

(iii)

S(D)

= D u {

~ D}

from = D u

A

= W.

endomorphism

T(D)

a

that to

contained

construction

in t h e

t> I t

E.

of

I E I~ s

It w o r k s

interested

u

all

generated

above

For

(v)

in

case.

information

=

we

(ii)

fully

non-trivial

abstract

WE

but

(i)

no

initial

semantic

been

are

in T E a n d

true

E.

unique

signature

<E>

(Em

are

abstract

relation

just

in K(E)

specification.

K(E)

congruence tains

the

a fixed

to

logic

of

algebra

equations

is t h e

by

ground

called

The

true

=: ~

has

algebras the

initial

<E>.

generated

in t h e

W E as

specification we

just

algebra

initial

an

equations

~ t I T E ~ s ~ t) relation

Because

is

equations.

the

than

congruence

have

= W/<E >

( ( s , t ) I 3 t I ..... t n,

D for tl=s,

l~k~n i,

l

resp.

the

and group

In t h e form the

case

ui

groups

= si.(ti )-I

words

The

of

u i,

the

~

e,

of

F G ( a I ..... a m ) / N

generated

by

the

elements

of

the

contains

form

tot; These

a

denoted

by

such x.

0(x)

= s

be

For

word

N

u

is

<S> the

is

given

then

normal

we

by

the

subgroup

group

identify

in

words

the

or

e

or

of

a genera-i b i • bi+ 1 •

they

of

identity

such

b i is

simply

carrier

each

either

addition

b l . b 2 ... bk;

the

words;

is

and

b i • e and

have

as

classes

which

where

endomorphisms

0(x)

= x for

substitution

for

which

in

a term

way.

0:

all

is

s • x,

this

is

are

usually

words.

FM

resp.

element

T ~ T

except

thus

e

FG

with

is

also

The

of t h e t e r m

finitely

determined

x a variable, set

a substitution

of

many

by

and

alge-

all

will

variavalues

in g e n e r a l

substitutions

is

closed

always

V 2 are

two

02(V 2)

the

variables

The

notion

the

a

two

in of

of

variable

of

renaming

= 6;

replacement

0 the

term

0(t)

is c a l l e d

an

one-one

then

is

t.

a renaming

If V 1 a n d always

t and

of

0(x)

need

is

composition.

called

n

resp.

are

that A

instance If

in t h e

i.

defined

under

presentation

congruence

written

these

Substitutions

bles

are

case

will

of

written

l~i~n.

unique

are

we

set

bra

FM

semigrouR

usual

the

u i,

group

terms

called As

the

be

modulo

where

(bl*(b2.(...)...)

in

the

F G ( a I ..... a m )

quotient

class

hence

can

ISd_~n.

quotient

The

equations

0

is

0

variables. finite

sets

of v a r i a b l e s

substitutions

this

case

V 1 and

V 2.

substitution

of c e r t a i n

following

and

we

say

does

subterms

terminology:

81 a n d that

not

82

such

01

and

cover

of a t e r m

then

there

that 02

the

by o t h e r

are

81(V I)

separate

concept terms.

of We

i.

Definition:

(i)

(ii)

A

place

is

numbers;

the

The

a

empty

subterm

defined

tupie

of

l

=

place

a

term

(i I ..... i n )

( ) is a l s o t at

place

))

inductively

= sbt(ti,X) and

replacement

in a t e r m

of

t by

the

subterm

a term

for

undefined occuring

r is i n d u c t i v e l y

r e p ( t , ( ),r)

l is

finitei Z

presented. Notation:

X+/

=: S G w ( Z , R )

E * / I R > =: M w ( Z , R ) By

SGw(Z)

resp.

6.

resp.

the

free

Mw(Z)

we

monoid

also

denote

generated

by

the

free

semigroup

Z.

Definition:

Let

Z

(0:

Z ~

every

=

{a I .....a k} M\{0)

be

a

a

u

alphabet

function

( over

X

= UlU2...u n and

l i,

in

hence

x1

x i and

y contains

have

that

xi~

xy

we

variables.

by

some

to

that

(d) t h a t

z; we

because see

some

that

other

xk

a contradiction.

the

works

special

reductions

generates

the our

minimu~

because weight

considerations

of

from

which

complete

of

xm

does

is

idempotent

not

semi-

system.

arbitrary because x for

with

infinite

by t h e

otherwise

every

n ~ m

also

ordering.

and

variable violate

compatible arguments

with from

Any the xi FTP. the

above.~

case. u

= yaw ~(a)

an

replaces

system

ground

u

generated

reducing

xn ~

semigroups

system.

case

length

= < a I ..... anl

that

with

KB-orderings

of i d e m p o t e n t

a finite

hence

i.e.

all

is a l w a y s

finitely

be

to t h e

for

this

general

to

and

such

reduced

occurrences

complete

complete

weights

to

hence

theory

any

once,

is

of

and

KB-orderings

reduction

exactly

z has

Combining

a

G

sn

sees

renamings

x starts

means

substitution

Suppose

to of

variables

the

has

turn

be

The

a finite

KB-orderings

We

have

argument

admits

Therefore

one

same

be o m i t t e d ,

group

give

First

remaining

general

consider

in

x I has

specific

special

one.

s n.

Each

reduction

a typical

(a) a n d

for

have (2)

the

• (z);

from

this

(i)

cases

and

This

cannot

we

xz

two

that

above

~

of

is

subword

Therefore

not

which

a

xyz

xyz.

there

that

form

first

to

all

involved

e.g.

the

proof

~

e> and

> ~(v)

where a

is

+ ~(w).

some not

a

= a i occurs

in vw.

Then

the

We

choose

completion

80 R ® = {a ~ v - l w -1, a -1 ~ wv}.

We

observe

here

"Freiheitssatz" that

for one

defining

of

relation

there

Magnus

relator

word denoting Notice

that

groups

occurs

the

identity

connection

e.g.

each

generator

in

[Ly-Sch]).

each

to This

occurring

non-trivial

the says

in the reduced

element. c h o i c e of w e i g h t s

may

lead to an

limit s y t e m R m.

F i n a l l y w e will give an e x a m p l e

w h e r e the l i m i t

inifinte

for all K B - o r d e r i n g s .

Consider

the so c a l l e d G r e e n d l i n g e r

G = < A, B, C

For each

a

(cf.

also

that a less skillful

infinite

is

weight

I

function

group

ABC ~ CBA

~ABC

E { ~

s y s t e m R W is

>

I~ is a w e i g h t

function

s.t. ~ ( A ) < ~ ( B ) < ~ ( C ) } the c o m p l e t i o n

algorithm

GCA generates:

R W = {CBA ~ ABC, C-IB-IA-I

A - I c B ~ BCA -I,

~ A - I B - I c -I, B - I c - I A ~ A C - I B -I,

c-lAB ~ BAC -I, A ( B C ) n A -I ~

is

easily

functions

seen

WACB'

that

~BAC'

the

~BCA'

B ( C A - I ) n B -I ~

(A-Ic) n,

(CB) n, B ( A C - I ) n B -I ~

C( A - I B - I ) n C - I

It

B - I A - I c ~ CA-IB -I

~

(B-IA-I)n

limit ~CAB'

systems

I

nE~

for

W C B A are also

(C-IA) n, }.

all

weight

infinite.

81 Additional

For

remarks:

practical

purposes

it

is

important

to

have

algorithms

w h i c h c o m p u t e the c o m p l e t e s y s t e m R ® (if it is finite) fast. An i n t e r e s t i n g w a y to avoid the i n v e s t i g a t i o n of u n n e c e s s a r y critical

pairs

is d e s c r i b e d

however,

that such m e t h o d s

of the a l g o r i t h m

in

[K~ 86].

do not

It should be added,

always

improve

the

speed

: One a l w a y s has to test w h e t h e r the m e t h o d

a p p l i e s and these tests also c o n s u m e s o m e t i m e and space. Other

heuristics

and

strategies

are e.g. f i r s t to o v e r l a p

short w o r d s and s i m i l a r devices; each i m p l e m e n t a t i o n usually contains

such

methods.

n e g a t i v e aspects; nearly unknown.

T h e y all

theoretical

have

results

positive in this

as

well

direction

as are

82

The

II.5.

Special

Word

Cancellation

The oldest problem It

for

allows

relator power

of

~u I >

success

algorithm

on

important

of

Dehn's

that

E S

In t h i s

v

if

is

u v -I

the

and

Small

s o l v e s the w o r d

Dehn's

= e

length

found

Group

in this

function

I...I. T h e

over

in the b o o k

Theory" context

which

algorithm.

is a d e f i n i n g

investigated

c a n be

conditions

are

several by L y n d o n

([Ly-Sch]). is the c o n c e p t

sufficient

for

of the

algorithm. conditions

Sc(I

A ~

we

already

lwl

met

are

the

so

~+.

u I-l) * s a t i s f i e s

implies

section

Groups

which

groups

was

notion

C'(A)-conditions,

Remember

of

"Combinatorial

cancellation

wu,wv

u by

Iv I f o r

cancellation

called

classes

a good overview

The most

Small

popular algorithm

replace

this

Schupp

small

most

to

for

Theory

certain

and

decades; and

and

Problem

the

C'(E)-condition

iff

< A • lwvl.

we need some

more

conditions,

the

so c a l l e d

T(p)-conditions.

I. D e f i n i t i o n :

Let S

S ¢ (I u I-l) * and p E N

satifies

w I ....W n ~ S for

iSi~n,

the

T(p)-conditi0n,

where

iff

w i is not t h e

there

w 2 w 3 ..... W n W 1

be given.

is a t

which

least

for

literally

all

n,

3~n 0 then each

is

respect

with

with

respect

En(W2):=

put

to R G

respect

respect

to R G

to R G

to R G

sw2t;

in

w 2.

= Gw(Z/R)

be

a finitely

presented

group.

Assume

and

Then chain

IkU k ~ r k ~ Sym(R)

for

kE{0,1,2),

uil i ~ r i e S y m ( R )

for

iE{4,5}

uol 3 ~ r 3 ~ Sym(R).

we of

have

for

length

instance 4 for

the

following

(Wl,W 2) =

superpos-deduction-

( r o l 3 , 1 o r 3)

i

0

I

k,-,

,,~

I

~'~

N

N

I

~

~

k-,,

I-~

I

II

~

o

~

v

~

(-~

~

~

0

~

I~

I

~

~

~

I'J

,I~

~

I~

0

II

F"~

i

I

N

< o

II

~

~

o

II

II

N

0F,,.'

F~

N I- ~

0

o

0

II

0

0

o

II

I"

NI I

88

Next

we

dition

will for

obtain

the

a first

solvability

(more of

technical)

the

word

sufficient

con-

problem.

6. T h e o r e m :

Suppose

G

Sym(S) For

=

B

(R4)

u


~-~

words

(R2')

~-~ Bb I

instruction

contain

and

•

introduce



set

a o = u ~ A u Q.

Z := A u Q u {B,,bl,b 2} a n d d e n o t e

(RI')

R will

= {qo .....qh } t h e

symbols.

We

For

Q

initial

= {a I .....a m } its

,bl,b 2 n e w

with

I ' b2

B...B,

induction

desired

a j n>B...B,

on

V;

hence

u = ~(0,n)

and

form.

[]

8. Lemma: Assume

11 = < a i l . . . a i k ' q~ a i k , + l . . . a i k

>,

12 = < a j l ' ' ° a J n, qn a j n , + l ' ' ' a j n > ' Then

wI

~ii--12--~ w 2

Proof:

Consider

By L e m m a

7 we have

for

w i : B...BIIB i~ =

n = a.l

~s(a-l)

bl.nP

+ b,

~

= (bl.l-P).(in)P.

O0,

/ ~x(m))

therefore

t ~ c

< c.

[]

If U h a s shows

an

that

infinite the degree

index

in G t h e n

strictly

the

decreases.

next

proposition

195

~° P r o p o s i t i o n :

Suppose

G

infinite.

Proof: and that

We

is

polynomial

Then

take

U has

finite

representatives

r~R

ly(r) is m i n i m a l

T h e n we h a v e

Because

from

the

sets left

~ n}

Un

:= {g E U~

K n}

Rn

:= G n N R.

is

u

infinite

[G:U]

is

cosets

Ix(u)

we

get

modulo

G

U such

w e p u t for nEN:

card(R k

l~k~n)

u R0.U n

\ Rk_ I) > i;

from

n ~ ~xlk). k=0

>

then

Xx(n)

holds.

and

x c y for U resp.

{(R k \ R k _ l ) . U n _ k l

Yxln)

deg(~x)=m,

Yx(n)w ~ c.(~)

generating

:= {g e G I ly(g)

D

d+l

d e g r e e ~ d.

in rU. F u r t h e r m o r e

this we obtain

If

degree

Gn

Gn

[G:U]

with

This

~ c-n m

for

some

c>0,

therefore

gives n

~y(n)

and

the

right

degree m+l

hand

>

c • • i=0

term

which proves

is

(~);

essentially

a polynomial

of

the a s s e r t i o n .

[]

T h e d e f i n i t i o n of the g r o w t h of

length.

ordering

Alternative]y ~

and

define

function

replacing

concepts

introduced

function

one in

length above.

an by

can

r e f e r r e d to the n o t i o n

refer

analogue weight

and

to

some

other

KB-

way

the

~-growth

all

the

remaining

196

The

fact

little

that

is

all

KB-orderings

expressed

underlying

in

the

differ

next

asymptotically

proposition.

very

Z denotes

the

belonging

to

alphabet°

8. Proposition:

Suppose the

X1 and

same

semigroup

and 42.

(ii)

put

Then

deg(¥ I)

Suppose

e I and

the

~2"

words

iff

= deg(¥2)

weight

iff

Now

if ¥I is b o u n d e d

hold

¥2 is b o u n d e d

and

in this

s.

belonging

> e(n)

e I (u)

=

k.602(u)

~2 (n)


k'

( (n-r)/(m-k)

- I

)k

a.(n-b) k

for

some

a>0,

b>0

for

some

d>0.

> d.n k

Therefore

we

sufficiently

Now

we

two

words.

large.

assume

such that of

get

two

that

¥(n) Hence

let

D.n t

deg(¥)

with

= c(F)

F is e x p o n e n t i a l .

vertices

cycles;

~

start

w I and

t as

above

n

holds.

We choose

a node

at q w h i c h

are

w 2 be

corresponding

the

and

initial

q in F

segments cycle

201 For the

free

SGcL(F). has

semigroup

Therefore

exponential

Because

the

exhaustive

SG g e n e r a t e d

a subset

of

L(F)

by

w I and

and

hence

w 2 we have L(F)

itself

growth.

cases

of

T being

and e x c l u s i v e

polynomial

or

the p r o p o s i t i o n

exponential

are

is proved.

[]

Additional

The

general

cycle tion The

a

connection

structure 13 w a s

of

first

generalization

is of nate

remarks:

interest for

group

growth

the

function.

the g r o w t h

graph in

slower

completion In c h a p t e r than

as

[Gil

of the o r d i n a r y

if the

a

word

observed

KB-orderings. with

between

functions

described 79];

growth

ordinarily

we

the

in

Proposi-

see also

[Ho 82].

f u n c t i o n to a ¥(R)

algorithm IV.4.

and

does

not

termi-

will

investigate

growing

generalized

202

E f f e c t i v e A s p e c t s of G r o m o v ' s Theo,rem.,

IV.4.

It is a n a t u r a l with a first

q u e s t i o n to c h a r a c t e r i z e

polynomial

the c l a s s of g r o u p s

g r o w t h f u n c t i o n by a l g e b r a i c methods.

s t e p in t h i s d i r e c t i o n

w a s d o n e by J. A. W o l f

in

The [Wo].

i. T h e o r e m :

A

finitely

growth

generated

nilpotent

group

has

a polynomial

function.

This

implies

that also finite extensions

grow

polynomially.

also

the

converse

the c e l e b r a t e d

J. M i l n o r in true.

of n i l p o t e n t g r o u p s

conjectured

The

proof

of

in

[Mi 68]

this

that

conjecture

is

r e s u l t of M. G r o m o v :

2. T h e o r e m :

Let G be a f i n i t e l y g e n e r a t e d mial

growth

finite

The

proof

function

iff

G

group. has

a

T h e n G has a p o l y n o -

nilpotent

subgroup

of

index.

uses

deep

analytic

methods

(e.g.

Hilbert's

5 th

problem). There

is a s t r o n g

in p a r t i c u l a r under

in

additional

interest

in s i m p l i f i c a t i o n s

removing

its

assumptions).

of t h i s proof,

nonconstructive Also

one

wants

parts, to

find

(even the

203

nilpotent to

subgroup

exhibit

the

the

growth

proved

by

"explicitly",

significance

function H.

Bass

for

in

of

the

to

determine

its

index

and

the

numerical

magnitudes

of

group.

An

important

step

was

[Ba 75]:

3. T h e o r e m :

Suppose

and

the

rk

nilpotent

group

has

the

lower

G

~

D rn(G)

= {e}

is

= FI(G)

the

rk(G)/rk+l(G).

rank

Then

degree

elementary

been

of

G has

the

free

a polynomial

series

abelian

group

in

growth

function

of

n-i d =

An

...

central

given

~ k=l

proof

k.r k .

for

in

[vdD-Wi]°

an

infinite

the

case

of

linear

growth

has

also

4. T h e o r e m :

If

G

is

function then

G

For

the

the

reader

be

The

~ has

group

recalled

e.g.

some

that

X(n)

subgroup

theoretic to

here,

commutator

groups

such

finitely

[Ka

generated - X(n-l)

U

~ Z

concepts

- Me];

some

group

with

growth

~ n for

some

n ~ ~,

satisfying

mentioned of

the

[G:U]

above

main

~ n4/2.

we

refer

notions

will

and

sub-

however.

of

U,V c_ G t h e

a,b

E G

notation

is

[a,b] [U,V]

= a-lb-lab will

mean

the

for

group

gene-

204

rated of

by

G

all

[a,b]

with

Fk+I(G)

potent

iff

will

word

graph

system

in

Fn(G)

The

with

lower

central

series

...

of

some

G/Fk+I(G)

and

G

is

nil

with

the

n e ~.

of

Gromov's

theorem

group.

section

ordering

the

for

aspects

respect

We c o n s i d e r

D

center

= {i}

the

this

term

D F2(G)

the

connect of

Throughout

G

on to

linear

is

the

finitely words

presented,

and

R~

"4"

= R~(~)

the

some limit

"4"case.

Proposition:

If

R~

is

linear

left

regular

the

word

the

(ii)

and

Proof:

(i)

subgroup

U c G of

G

cyclic

of and

Let

finite

graph

cycles

[G:U]

¢

and

the

R~-growth

function

is

then

(i)

U

bEV.

[Fk(G),G].

is

now

partial

= FI(G)

=

Fk(G)/Fk+I(G)

5.

a~U,

is: G

We

with

~

represent

the

label finite

index

generated

R ~ has

at

path

number

a cycle index. which

by u k for

most

a subgroup

(p(F)/2)-lu i, where

p(r)

u

F of

of

in F; Then

is

k.

is

cycles;

U ~ Z of

index

a cycle

word

F.

u generates

there

normal

some

u

two

is in

some G;

U

a cyclic subgroup is

again

205

Now

suppose

vnEu

for in

1>0

l

of G c o r r e s p o n d

where

c =

system

to

[a,b].

R such

that

N = Irr(R).

I. D e f i n i t i o n : R

= (ca ~

ac,

c a -I ~ a - l c ,

c-la

cb ~ bc,

cb -I ~ b-lc,

c - l b ~ bc -I,

ba ~ a b c -I,

b - l a ~ ab-lc,

~ a c -I

•

c - l a -1 ~ a - l c -I

r

c - l b -I ~ b - l c -I,

ba -I ~ a-lbc, b-la-i

~

a-lb-lc-l).

211

F r o m this d e f i n i t i o n Irr(R)

= N

holds.

it f o l l o w s

Also,

R has

by i m m e d i a t e FTP and

UTP;

i n s p e c t i o n that hence

R

is c o m -

plete. It is s t r a i g h t f o r w a r d ordering. partial

R

is,

term

that

R

however,

ordering

is not c o m p a t i b l e

compatible

with

w i t h any KB-

the

following

"E":

2. D e f i n i t i o n :

For

u = Xl...Xn,

x i E {a,a-l,b,b-l,c,c-l},

we put

Kx(u)

:= c a r d ( { i I x i E {x,x-l)}),

V(u)

:= c a r d ( f ( i , j ) I i<j,

xiE{b,b-l),

xjE{a,a-l}});

V'(u)

:= c a r d ( f ( i , j ) {

xiEfc,c-l),

xjEfb,b-l}})

and f i n a l l y

u E v

i<j,

x E {a,b,c};

iff

(Ka(U),Kb(U),V(u),Kc0

= 0).

4. L e m m a :

Assume

r ~

2(~+B)

satisfying Then

u has

Proof: This

We

obtain Next

Suppose K~(u)

u'~u,

M(r)

[a,b]. u

show this

an a

each

of

u'

that is can

~

1

[a,b]

the

assume

= u(mod

contradiction.

s G)

with case, > 0. which

i-minimal

Ka(U) for

and

hence

[a,b]M(r)(mod

not

the

= aSb-ta-Sbt

of t h e

n starts

is

~ r and

a , a - l , b , b -I

Because =

~(u)

u

= ab-la-lb

permutation

> 0 we

yields

u

shows

also

we

form

w

that

is a c y c l i c word

the

suppose

= M(r),

consider

example

It f o l l o w s

tor

V(u)

and

get

V(u)

= Kb(U) some

= 0.

s>0,

~(u)~r,

t*0.

V(w)=l.

~ i.

occurs

in u.

M(r)-th

power

is

the

in

word

Therefore

u

of t h e c o m m u t a center

of

G

of

a.

we

G). some

positive

i.e. Then

u some

starts

power

= vaSz. cyclic with

aS;

Because

of

permutation we

obtain

214

A

similar

If

u

argument

= a S v a k,

Therefore

shows

then

u

has

the

u now

that

n ends

= as+kv(mod

with

G) a n d

some

as+kv

power

We

have

to

form

which

have

With

p = ti

the

u

For We

m=p+q get

There

= a s l b t l a s2

show

k=2.

and

q

(i)

we

are

t i and

tj

for

= tj

we

get

with

consider

v

w I = UlbmVu2

~(w2)~r

and

by

= av I = v2a.

and

a direct

w 2 = UlVbmu2 • computation

- V(u)

=

V(bmv)

- v(bPvbq),

V ( w 2)

- V(u)

=

V ( v b m)

- v(bPvbq),

V ( w I)

- V(u)

= m. K a ( V )

- P'Ka(V)

=

V ( w 2)

- V(u)

= 0.Ka(V)

- P. K a ( V )

= -P.Ka(V).

three

cases:

q. K a ( V ) = 0

gives

either

which

bmv

V(u) ~

= V ( w 2)

and

wI i u

vb m ~ bPvb q

and

w2 ~ u

(ii)

q°Ka(V)>0

yields

(iii)

q. K a ( V ) < 0

implies

k=2

= V ( w I)

q. K a ( V ) ,

bPvb q

contradicts

again

i<j

sign.

or

immediately

there

k

Z s i = ~ t i = 0. i=l i=l

V ( w I)

Th~s

Therefore

with

= ulbPvbqu 2

~(Wl)Kr,

are

o .o b t k

Otherwise

same

b:

~ u.

k U

of

the

minimality

V(Wl)>V(w)

P. K a ( V ) < 0

of

which

and

= M(r).

ho]ds

u.

is

impossible.

V ( w 2)

> V(u);

a contradiction.

is

from

true; Ka(U)

the

rest

= Kb(U)

of

= 0.

the

assertion

follows

215

5.

Lemma:

(i)

(ii)

M(4~Bn)

= ~Sn 2

If

the

u

is

~(u)

Proof:

~

(i) B y

V(u)

=

From

~(u)

4~Sn

Lemma

= 2~s

f(x)

This

and

n>0.

word

satisfying

Ka(U)=Kb(U)=0,

4

then

u

= a -8 n b

u

= aSb-ta-Sb t

= max

~ 4~B

(s.t I s,t

its

maximum

gives

the

=

V(u) u has

= M(r),

the

form

-~na-S n b - ~ n .

holds

on

at

follows

E R,

the

x = Sn

solution

~Sn 2

this

it

E ~,

(x.y I x , y

= ~/s.x.(2Sn-x)

M(4~Sn)

From

i-minimal

+ 2Bt

max

f has

all

which

gives

s.t.

M(4~Sn)

If

for

x

~s

+ St

x,y~0,

real

~ 2~8n)

~x

+ SY

= 2~n)

interval

and =

that

M'

Sn,

y

= M ~.

[0,2Sn]

= f(Sn)

= ~Sn 2

=

~,

~n

e

then holds.

therefore

holds.

(ii)

follows

directly.

[]

Now

we

come

to

the

Proposition

3:

Proof

of

Then

some

I r r ( R ®) class

finite

=

of

back

L(A).

words

Suppose

automation A

pumping

discussed

A

I r r ( R ~) will

lemma in

Lemma

is

accept

argument 5

regular.

will

Irr(R~), applied

lead

to

i.e.

to

a

the

contra-

diction: Let

Q(z~u)

denote

let

zo

the

be

initial

ciently

large

for

positive

all

the

such

state

reached

state

that

integers

the

of

A.

from We

pumping

state

z by

input

choose

s I ..... s 4

lemma

guarantees

m I ..... m 4 t h e

equality

u;

suffithat

216

Q(zo,alslml

b-s2m2

a-S3m3

=

Taking

mI =

b-S2

the

a-S3

special

bS4)

~.n

~0n

~ , sI

m2 = ~ , s2

m3 = ~ , s3

Lemma

5; t h i s

Taking

word

t

u

such

= a ~n

b -~n

shows

that

that

x

a -~n

b ~n

z I is an

>

y Sy2

- - , Sl

m2 = ~ , s2

m3 = - - , s3

word

w

=

= Kb(W)

= 0

m4

a sy2

= s4

I r r ( R ~)

because

of

state.

= s2.s 4

~y2

another

of

accepting

= t.sl.s 2

holds.

zI)

~0n

e

Sy2

to

(=

choice

S'n

a

leads

Q(zo,aSl

n = Sl. S 2 . S 3 . S 4

gives

mI =

bs4m4)

and

choosing ~y2

m4 s4

b -~x2

a -Sy2

b ~x2

with

Zl=Q(Zo,W). We the

have

Ka(W)

minimal

hence

w

~

word

and

representing

Irr(R~);

w

V(w)

= ~ S ( x y ) 2. B y

is

Lemma

a S X Y b - ~ X Y a - ~ X Y b ~xy

5

• w;

a contradiction. []

The

essential

discussion orderings finitely tions

the

work,

The

part

of

this

of

the

two

"E"

and

"4".

In

many

new

arbitrary simple only

sophisticated

use

proof

growth

of

not

discuss

here.

The

use

nonstandard

functions Gromov's

of

was

first

theorem

model

the cycles

axis,

connected

pumping

in t h e

in word

in

for

in

the

[vdD-Wi]. become

infinitesimal

this

the

The

where

does

case which

of

simplified In

the

as a b b r e v i a -

context

essentially

paths.

with

argument

analysis

the

implicit

situation

occur

lemma

have

an

associated

can

nonstandard

models

in

general

constants

introduced

given

by

more

authors

of

consists

functions

the

the

the

methods

proof

not

use

some

we

will

growth

proof

of

nonstandard a real

half-

assumption

that

217

R ~ is regular then finally leads to a topological bility

between

~3 (arising

from

incompata-

"K") and ~4 ( a r i s i n g

from

V. D e c i d i n g

Algebraic

Properties

of F i n i t e l y

Presented

Monoids

F. Otto F a c h b e r e i c h Informatik Universit~t Kaiserslautern P o s t f a c h 3049 6750 K a i s e r s l a u t e r n

In this chapter ing a l g e b r a i c turn out, general.

we are going

properties

However,

general,

of finitely

all the p r o b l e m s

word p r o b l e m

we are

we have already

and the finiteness

become

decidable

through

tems.

So a f t e r e s t a b l i s h i n g we will

presented

sight

which

Presentations

they are r e s t r i c t e d involving

I. Then

i~ addition

is d e f i n e d

a monoid

[w] R (w 6 E*) (u,v c E*), monoid

MR,

R. As we have

the elements

and the

operation

[e] R serving as the

of the free m o n o i d

for f i n i t e l y

restricted

classes

gain a d d i t i o n a l

and by R a r e d u c t i o n

relation

on Z* c o n t a i n i n g

mentioned

in-

Transformations

to the r e d u c t i o n

gruence

sys-

systems.

~+

defines

in

that are

reduction

problems

certain

of these

on Z*, w h i c h

in

as the

to m o n o i d s

results

decision

through

and Tietze

it will

problems

complete

In this way we will

power

As

concern-

that are also u n d e c i d a b l e

Again we denote by Z a finite alphabet,

ecs/ivalence r e l a t i o n

monoids.

seen that certain

are given

systems.

problems

in are u n d e c i d a b l e

the u n d e c i d a b i l i t y

into the c o m p u t a t i o n a l

Monoid

presented

problem

presentations

reduction

decision

interested

show h o w to solve these

monoids

of complete

V.I.

finite

when

given

above,

to i n v e s t i g a t e

~,

R also

induces

this

identity.

Thus,

con-

congruence

are the c o n g r u e n c e

of which is given by[u] R o

E* g e n e r a t e d

an

as the smallest

seen already,

of which

system on

classes

[V]R = [uv]R

M R is the factor

by Z m o d u l o

the c o n g r u e n c e

7 Whenever ordered

pair

a monoid (E;R)

M happens

is c a l l e d a

ing the set of g e n e r a t o r s of this presentation.

to be isomorphic (monoid)

and R b e i n g

that are f i n i t e l y presented,

monoids

that are given

both

the

sets ~ and

the

through

R are

i.e.,

(M T

MR),

of M with

the E be-

relations

only be d e a l i n g

with

we will only be c o n s i d e r i n g

presentations

finite.

MR

set of d e f i n i n g

In the f o l l o w i n g we will

monoids

to

presentation

of the form

(E;R),

where

219

Let oroblem

(Z;R)

be a finite

for this p r e s e n t a t i o n

INSTANCE:

Two words

QUESTION:

v hold

So the w o r d o r o b l e m word p r o b l e m

there

([Da 58]). scribe

{b,c}, fined

systems

~(a)

finite p r e s e n t a t i o n s

ed ? In o r d e r

this o b s e r v a t i o n

to answer the n o t i o n

1.2. Let

Then p r e s e n t a t i o n application

(3

if

= E, and

may de-

shows.

Then

Z2 =

the m a p p i n g

an i s o m o r p h i s m

(ZI;R I) and

raises

(E2;R2)

~ de-

from the

are dif-

the f o l l o w i n g

question:

depend on the a c t u a l l y

this question, of e l e m e n t a r y

(Z;R) and

(E';R')

(Z';R')

but also Tietze

Does

chosen

of the m o n o i d p r e s e n t for future

reference,

transformation

be two finite

is said to be o b t a i n a b l e

(Z';R')

Tietze

satisfies

R' = R U {(u,v)},

transformation

for finite

condition

presentations. from

(Z;R)

of type

by an

i for some

(i) given below.

w h e r e u , v 6 Z* satisfy

v.

,

R' = R-{(u,v) }, where

Z' = E U {a} for

word p r o b l e m

([Ti 08]).

{ R, but u ~ +

Z' = Z, and

and hence,

presentations

example

or is it a p r o p e r t y

of an e l e m e n t a r y

i 6 {1,2,3,4},

(u,v)

finite

there exist

same monoid.

of the w o r d p r o b l e m

presentations

Definition

(2

w o r d problem,

= b induces i.e.,

of the

finite m o n o i d p r e s e n t a t i o n ,

we i n t r o d u c e

does

is n o t h i n ~ but the

~I = { ( a 3 ' e ) ' ( b 2 ' e ) ' ( a b ' b a ) } '

MR2,

the m o n o i d

the d e c i d a b i l i t y

i.e.,

with u n d e c i d a b l e

different

= cb and ~(b)

ferent

Obviously,

of M,

(Z;R)

R 2 = {(cbcbcb,e), (b2,e),(bcb,c)}.

onto

the word

problem:

R. As is w e l l - k n o w n

as the f o l l o w i n g

MRI

(I) Z'

system

presentations

1.1. Let E I = {a,b},

through

decision

same e l e m e n t

with u n d e c i d a b l e

monoid

monoid

the

for the p r e s e n t a t i o n

On the other hand,

and

M. Then

?

finite m o n o i d

the same monoid,

Examole

is the f o l l o w i n g

for the r e d u c t i o n

reduction exist

of a m o n o i d

u,v 6 E*.

Do u and v r e p r e s e n t u ~+

finite

presentation

some letter

(u,v)

6 R satisfies

a { E, and

u ~

R' = R U {(u,a) } for

v. some

word u 6 Z*. (4

There

exist a letter

(u,a)

6 R. Let ~: Z* ~

by ~(a) E'

The

= u and ~(b)

= Z-{a},

and

following

transformations

a 6 E and a word (Z-{a})*

u 6

denote

!emma

shows

to a finite

b 6 Z-{a}.

induced

Then

£ R-{(u,a)}}.

that by a p p l y i n g

presentation

such that

the h o m o m o r p h i s m

= b for all letters

R' = {(~(l),~(r)) I (/,r)

(Z-{a})*

(E;R)

elementary of a m o n o i d

Tietze M we only

220 get further p r e s e n t a t i o n s of M. The proof of this lemma is straiqhtforward, and t h e r e f o r e Lemma

1.3. Let

(E';R')

(E;R)

it is left to the reader.

and

is obtainable

Tietze transformation. monoid,

i.e.,

(E';~') be two finite p r e s e n t a t i o n s

from

such that

(Z;R) by an a p p l i c a t i o n of an e l e m e n t a r y

Then these two p r e s e n t a t i o n s define the same

MR''

~{R

Observe that the e l e m e n t a r y Tietze t r a n s f o r m a t i o n s of type 2 are inverses of each other,

I and

that the inverse of an e l e m e n t a r y Tietze

t r a n s f o r m a t i o n of type 3 is one of type 4, and that the effect of an e l e m e n t a r y Tietze t r a n s f o r m a t i o n of type 4 on a finite monoid presentation can be r e v e r s e d by a finite number of a p p l i c a t i o n s of elementary Tietze t r a n s f o r m a t i o n s of types

I to 3. We will use this o b s e r v a -

tion in the proof of the f o l l o w i n g theorem,

which describes the basic

reason for c o n s i d e r i n g Tietze transformations. T h e o r e m 1.4. Let same monoid.

(E;R)

and

(E';R') be two finite p r e s e n t a t i o n s of the

Then there exists a finite sequence of e l e m e n t a r y Tietze

t r a n s f o r m a t i o n s that transforms

(E;R)

into

(E';R') .

Proof. W i t h o u t loss of g e n e r a l i t y we may assume that the sets E and X' of g e n e r a t o r s are disjoint. monoid,

we have

M R ~ MR,.

Since

Thus,

(E;R)

and

(Z';R') define the same

for each a 6 E, there exists a word

u a 6 E'* such that a and u a describe the same element of this monoid. Also, for each b 6 E ~, there exists a word v b 6 E* such that b and v b describe the same element. Using these words the p r e s e n t a t i o n

(E;R)

is

t r a n s f o r m e d by a finite sequence of e l e m e n t a r y Tietze t r a n s f o r m a t i o n s as follows: (a)

(E;R) ~

(E U E';R U {(Vb,b)Ib 6 E'}) by

IE~i e l e m e n t a r y Tietze

t r a n s f o r m a t i o n s of type 3. (b)

Let R ° = R U {(Vb,b) ib 6 E'}, and let g denote the i s o m o r p h i s m

from MR~

onto M R that is induced by m a p p i n g b onto,v b for all b 6 E'

Then for all

(/,r)

6 R ~, g(1)

~+

(E U E';R o) ~

(E U E';R ° U R') by

tions of type

1.

(c)

implying i + ~

r. Thus, o IR' I e l e m e n t a r y Tietze t r a n s f o r m a -

Since for each a 6 E, a, u a, and g(u a) all define the same ele-

ment of the monoid

MR,

we have a +~+ g(u a) ~

(Z U E';R o U R') ~ (X U Z~;R U R by

g(r)

u a. Thus, we obtain

U {(Vb,b)Ib E Z ~} U {{Ua,a)!a E E})

IEl e l e m e n t a r y Tietze transformations of type I. Let E" = E U E'

and R" = R U R ~ U {(Vb,b) Ib E E'} u {(Ua,a) la E E ] . been t r a n s f o r m e d into Tietze transformations.

Then

(E;R)

has

(E";R") by a finite sequence of e l e m e n t a r y

221

(d)

In an a n a l o g o u s

so by the remark into

(E';R')

tions.

manner

proceeding

by a finite

the vb

a uniform

process

same m o n o i d (b 6 E')

the t h e o r e m

sequence

into each other,

to the

since

in general.

isomorphism

(Z";R"),

Tietze

transforma-

Two finite

Do these p r e s e n t a t i o n s

presentations

On the other hand, is o b t a i n e d

plication

(E;R)

and

describe

of

(a £ E) and it does not give

(E';R').

the same monoid,

it can be seen easily,

for

(E';RI),

(E;~)

Tietze also

conclude

the f o l l o w i n g

Corollary

1.5. Let

(E;R)

same monoid.

that

i.e.,

induces

Then

and

for

the d e c i d a b i l i t y

finite p r e s e n t a t i o n s .

(E~;R ') be two finite

(E';R')

for

(E;R)

1.4 we can

im-

of the w o r d p r o b l e m

In particular,

there

of

if and

is an invariant

presented

of

the d e c i d a b i l i t y

for a f i n i t e l y

finitely

presentations

is d e c i d a b l e

is decidable.

we can speak a b o u t

undecidability

exist

to

to the word pro-

from T h e o r e m

of the word p r o b l e m

Hence,

ap-

result.

the word p r o b l e m

if the word p r o b l e m

by a single

then a solution

a solution

Hence,

if a p r e s e n t a t i o n

(E;R)

transformation,

and vice versa.

mediately

Thus,

ua

In particular,

from a finite p r e s e n t a t i o n

of an e l e m e n t a r y

the word p r o b l e m for

the words

does not

presentations

M R ~ MR, hold ?

does

(E';R')

construction

two finite

problem:

INSTANCE:

only

into

can be t r a n s f o r m e d

of e l e m e n t a r y

out that the above

QUESTION:

the

(Z";R")

for t r a n s f o r m i n g

are not known

a so l u t i o n

blem

can be t r a n s f o r m e d

[]

It should be p o i n t e d yield

(E~;R')

presented

monoids

or

monoid

M.

with u n d e c i d -

able w o r d problem.

V.2.

Markov

Properties

Given a m o n o i d

M through

like to d e t e r m i n e specifically INSTANCE: I. QUESTION:

of E i n i t e l y

Presented

some finite

presentation,

some of the algebraic

interested

in the f o l l o w i n g

A finite p r e s e n t a t i o n Is the m o n o i d trivial,

i.e.,

one w o u l d

properties decision

often

of M. Here, problems:

(E;R) .

M R given t h r o u g h does

Monoids

this p r e s e n t a t i o n

M R ~ {e} hold

?

we are

222

2. QUESTION:

Is the m o n o i d

M R finite

3. QUESTION:

Is the m o n o i d

MR commutative

4. QUESTION:

Is the m o n o i d

M R cancellative

5. QUESTION:

Is M R a free m o n o i d

6. QUESTION:

Is the m o n o i d

? ? ?

?

M R a group ?

7. QUESTION:

Does

the m o n o i d

M R contain

any n o n - t r i v i a l

idempotents?

8. QUESTION:

Does the m o n o i d

M R contain

any n o n - t r i v i a l

elements

finite 9. QUESTION:

Does

In this ([Ma 51],

c.f.,

the d e c i s i o n will

then

izations

we will

e.g.,

problems

leave

are p r e s e n t e d

M R a free g r o u p ? learn about

[Mo 52]),

2.1.

ever y m o n o i d

complete

(a) A p r o p e r t y

that

(b) A p r o p e r t y

P of f i n i t e l y

if it satisfies

systems

(O)

P is invariant. There

exists

a finitely

property

P, and w h i c h

finitely

presented

There

exists

(c) Finally, hereditary

This

for monoids

or certain

that

special-

a property

presented

Whenever

above

monoid

submonoids

a finitely

if

property

P it-

is a M a r k o v property,

M I which

does not have

to a s u b m o n o i d

having property

P of finitely

of any

P.

monoid M 2 having property presented

presented

monoids

P.

is c a l l e d

m o n o i d M has P, all

of M also have P. property

of finitely

presented

or not P is a M a r k o v property,

can be r e l a x e d

There exists

monoid

isomorphic

presented

a finitely

of whether

invariant

possessing

monoids

presented

P is an h e r e d i t a r y

for c h e c k i n g

(I) g i v e n

is c a l l e d

three conditions:

is not

a finitely

if w h e n e v e r

to a m o n o i d

presented

the f o l l o w i n g

(I)

(1')

of M a r k o v

in general.

these p r o b l e m s

reduction

P of m o n o i d s

is isomorphic

this property.

then

are u n d e c i d a b l e

us the task of solving

self p o s s e s s e s

fini t e l y

result

w h i c h can be used to show that all

stated above

by finite

a fundamental

thereof.

Definition

(2)

M R contain an element of infinite

?

Is the m o n o i d section

of

?

the m o n o i d

order 10. QUESTION:

order

to the f o l l o w i n g

presented

monoids,

condition

condition:

m o n o i d M I not h a v i n g p r o p e r t y

P. Before exam p l e s

stating

and p r o v i n g

of M a r k o v properties.

Markov's

r e s u l t we want

to give

some

223

Definition Then

2.2.

In w h a t

the properties

(a) PI(M)

~

follows

PI,P2,.°.,PIo

M is t r i v i a l ,

(b) P2(M)

.: > M is f i n i t e ;

(c) P3(M)


M is c o m m u t a t i v e ,

m I o m 2 = m 2 o m I, w h e r e ~ (d) P4(M)

let M be a f i n i t e l y

= ml o

m3

implies

monoid.

as f o l l o w s :

M Z {e};

i.e.,

f o r all m l , m 2 6 M, we h a v e

denotes

.~ ~- M is c a n c e l l a t i v e ,

ml ~ m2

defined

presented

the o p e r a t i o n

i.e.,

of M;

f o r all m l , m 2 , m 3 C M,

m 2 = m3,

and m I o m 3 = m 2 o m 3 implies

m I = m2; (e) Ps(M)

~

M is a f r e e m o n o i d ,

alphabet (f) P6(M) m'

6 M such ~

there (h) P8(M)

M ~ E* f o r

some

i.e.,

t h a t m o m'

= e M,

~. M d o e s n o t c o n t a i n

for a l l where

m 6 M,

there

eM denotes

a non-trivial

the

m # e M a n d m o m = m;


I k+n n i and n > O satisfying m = m , where m s t a n d s for m o m o . . . 0

m

(i-times); (i) P9(M)


make

restricted." So recall

right(R)

modulo

recall

R-{(/,r)}.

system

Systems

R on Z, there

tion~

given

determined duction whenever

that

exists

for each

this

6 R. In what

Thus,

R on E is c a l l e d m o n a d i c c E U {e}, where it is c a l l e d

seems a p p r o p r i a t e .

special

I nor r can be r e d u c e d

finite

complete

"reduced"

complete

reduction to R w i t h

systems the

system

are c a l l e d equivalent,

same congruence. system

d e a l i n g with

our a t t e n t i o n

by

reduction

and e q u i v a l e n t

equivalent

when

we can r e s t r i c t

follows

that are even more

R on Z is c a l l e d non-

finite

defining

in p o l y -

= {e}.

system

two r e d u c t i o n

same a l p h a b e t

effectively.

system

6 R, n e i t h e r

a unique

reduc-

length-reducing

of this type are also c a l l e d

R the n o n - r e d u n d a n t

systems

systems

6 R}, and that

R' is n o n - r e d u n d a n t

IRR(R v) = Irr(R) . Here they are on the

(/~r)

?

system R on Z is c a l l e d

(/,r)

right(R)

right(R)

that a r e d u c t i o n

It is known

R' on ~ such that

(/,r)

with

if for each rule

some authors.

rule

and satisfies

6 E*:

if only

v hold

finite

can be a n s w e r e d

a reduction

that a r e d u c t i o n

= {r 6 E*I~l

Finally

hand,

use of r e d u c t i o n

if it is l e n g t h - r e d u c i n g

redundant

Here

does u + ~

or not a given

this q u e s t i o n

IrJ for each

if it is l e n g t h - r e d u c i n g

R, i.e.,

whether

On the other then

time ([Ka-Kr-McN-Na]).

length-reducing

modulo

In a d d i -

R' of R can be

finite

complete

to n o n - r e d u n d a n t

reones

if

230

In the f o l l o w i n g ognizing

certain

we describe

languages

The first one has already only m e n t i o n e d Theorem

3.1.

following

here

associated

There e x i s t s

OUTPUT:

A deterministic Irr(R)

next c o n s t r u c t i o n

finite

Theorem

reduction

systems.

and t h e r e f o r e

it is

construction

that

solves

the

3.1

system

3.2.

following

R on Z.

state a c c e p t o r words modulo imolies

set of irreducible deals with

~u 6 S: u ~ v} of a regular reduction

system

of i r r e d u c i b l e

R on Z, the

the

recognizing

the

set

R.

that for each finite words

is a regular

set of d e s c e n d a n t s

set S ~ Z* with r e s p e c t

reduction

language.

L(S,R)

The

= {v 6 Z*I

to a finite monadic

R on Z.

There

exists

an e f f e c t i v e

construction

that

solves

the

task: A finite m o n a d i c

OUTPUT:

before,

rec-

of completeness.

an e f f e c t i v e

reduction

In particular,

INPUT:

with finite

of a u t o m a t a

task: A finite

Theorem

constructions

been p r e s e n t e d

for reasons

INPUT:

system

three

reduction

acceptor

A with m states

A finite

state

acceptor

system

R on Z, and a finite

recognizing

the

subset

A* w i t h m states

state

S of Z*.

that r e c o g n i z e s

the

set L(S,R). Proof.

Let

R be a finite

r 6 Z U {e}, let D(r) all l e f t - h a n d

reduction

= {l 6 Z*l{l,r)

system on Z. For

E R},

i.e.,

D(r)

sides of rules of R that have r i g h t - h a n d

let S be a regular ministic

monadic

finite

subset

of Z* that

state acceptorr A =

is r e c o g n i z e d

(Q,Z,6,qo,F),

is the

by the n o n d e t e r -

where

Q =

{qo,ql ..... qm_1 } is the finite se__~to~f >tate______~s, 6: Q×Z ~ P(Q) transition of final is the

function,

states

To obtain

an a c c e p t o r transitions

is as follows: some

subsets

suppose

states qi,qj

from qi to qj with transition

P(Q)

denotes

state,

the power

as usual

([Ho-U1]).

for L(S,R)

we m o d i f y

if possible.

that

for

The

set of Q,

function

the a c c e p t e r

a 6 Z,

i.e.,

set P(Q)

6: Q×Z ~ P(Q)

idea of adding

some letter

is the

and F ~ Q is the

A by ad-

transitions

some w o r d I £ D(a),

6 Q, qj 6 6(qi,l) . Then we must add a t r a n s i t i o n

label a,

is to capture

from qi to qj with

initial

of Q. The t r a n s i t i o n

to Q×I* ~ P(Q)

ding c e r t a i n

and

of A. Here

set of all

is e x t e n d e d

qo 6 Q is the

set of

side r. Further,

if qj ¢ 6(qi,a).

the notion

th&t

label a is e q u i v a l e n t

The

since

intent

of adding

this

I ~ a, a t r a n s i t i o n

to a sequence

of t r a n s i t i o n s

231 from qi to qj with label I. £ £ D(e),

Further,

and some states qi,qj

suppose that for some word

E Q, qj E 6(qi,/).

and each qk 6 Q, if qk E 6(qj,a),

qi to qk with label a, if qk { 6(qi'a)" transition

Then for each a 6 Z

then we must add a transition

The intent of adding this

is to capture the notion that since la ~ a, a transition

from qi to qk with label a is equivalent from qi to qk with label la. qj is a final state,

to a sequence of transitions

In addition,

if this situation occurs and

then qi also becomes

a final state.

This whole

process must now be iterated until no further transitions and no additional

final states can be introduced.

IEI'm 2 iterations

suffice.

It is clear that this basic c o n s t r u c t i o n for

a subset of L(S,R).

transitions

On the other hand,

is not difficult

will lead to an accemtor

since the process of adding

to show that the resulting

construction

the set L(S,R).

outlined above

can be added

This means that

is iterated until no further transitions

actually recognizes

P:

from

finite

can be added,

it

state acceptor

Below a formal description

of the

is given.

begin INPUT:

A finite monadic

reduction

system R on E, and a finite

state acceptor A = (Q,E,6,qo,F); (1)

z + I;

(2)

while

[3)

begin for all qi,qj

z
2, or : INTR(a)

idempotents

systems

= {w} a n d a w ~ e .

or

as follows. on Z. T h e n if one

of

248

Proof.

If there exists a w o r d u 6 Irr(R)-{e}

word u d e s c r i b e s 5.11

a non-trivial

(ii) implies

if INTR(a)

idempotent

So a s s u m e i.e.,

that M R , c o n t a i n s

= {w} with a w ~ e

non-trivial

ayx~e,

Thus,

either

Thus,

are w o r d s x,y

is s u f f i c i e n t Define taking

operation (i')

idempotent.

a non-trivial

describing

to Lemma

this

By Lemma Finally, a

idempotent,

a non-trivial

5.9 u satisfies

condition

a 6 Z such that u = xay,

and awa ~ a i m p l y i n g

(ii)

is s a t i s f i e d

system

with a, or c o n d i t i o n

whether

a monoid

R on Z c o n t a in s

to check c o n d i t i o n s

(i) to

CYCLE on the

= {yxlxy

descendant

that w 6 INTR(a).

6 L},

M given by a monadic

a non-trivial (iii)

idempotent,

of T h e o r e m

set P(I*)

on Z by

i.e.,

CYCLE(L)

is the language

of w o r d s

of L. By u s i n g

condition

(i) of T h e o r e m

it

5.12.

of languages

all the cyclic p e r m u t a t i o n s we can express

(iii)

u

to d e t e r m i n e

an o p e r a t i o n

CYCLE(L)

5.9.

a. Take w 6 Z* to be the i r r e d u c i b l e

condition

reduction

contains

M R contains

6 Z* and a letter

w i t h a and w.

in o r d e r

complete

that

R. Then a w ~ e ,

is satisfied

a non-trivial

a word u 6 Irr(R)-{e}

and ayxa ~

of yx m o d u l o

(i), then

for some a 6 Z, then u = aw d e s c r i b e s

of M R . Then a c c o r d i n g

(i), or there

satisfying

of M R by Lemma

of M R.

conversely,

there exists

idempotent

idempotent

that this

5.12 as follows:

e 6 L(CYCLE(Irr(R)-{e}),R). Given

a finite

effectively

set Irr(R)-{e}

theorey

acce p t o r

A 2 for the

a finite

(Theorem 3.2). Lemma

5.13.

complete

a finite

(Theorem 3.1).

auto m a t a

derive

monadic

construct

(c.f.,

[Ho-UI]),

we have

The f o l l o w i n g

INSTANCE:

A finite Does

monadic

there

the

from

a finite

state

from w h i c h we finally

A 3 for the

set L ( C Y C L E ( I r r ( R ) - { e } ) , R )

the f o l l o w i n g

problem

techniques

we then o b t a i n

set C Y C L E ( I r r ( R ) - { e } ) ,

Thus,

system R on £, we can A 1 for r e c o g n i z i n g

By u s i n g w e l l - k n o w n

e.g.,

state a c c e p t o r

QUESTION:

reduction

state a cc e p t o r

result.

is decidable:

complete

reduction

system

exist a word u 6 Irr(R)-{e}

R on Z.

that can be f a c t o r e d

as u = xy w i t h yx ~ e ? Lemma Lemma Then

5.13

5.14.

Let

shows that c o n d i t i o n R be a finite

(i) of T h e o r e m

monadic

complete

for each a 6 Z, one can e f f e c t i v e l y

pushdown INTR(a)

automaton

M a that r e c o g n i z e s

is a d e t e r m i n i s t i c

reduction

construct

the

context-free

5.12

is decidable. system on I.

a deterministic

set INTR(a) ° In particular,

language

for each a 6 Z.

249 Proof.

Given

letter

a 6 Z, one can e f f e c t i v e l y

automaton ceptor

a finite m o n a d i c

(dpda)

complete

M I recognizing

M 2 recognizing

the

set

with awa + ~

6 Irr(R)

niques

for d e a l i n g

nizing

the

Given

compute

with awa ~ a} with

of the

dpda's

complete

reduction

Theorem

5.15

INSTANCE:

of T h e o r e m

([Ot 85a]).

the

set

3.!

pushdown

state ac-

and 3.3).

From

[a] R N a - I r r ( R ) . a

In the p r o o f s solving

comp l e t e

algorithm

able any more. above p r o b l e m involving

we have

from M 3.

M we can d e t e r m i n e

of L. Hence,

the

we can

conditions

for finite monadic

the f o l l o w i n g

problem

(Z;R),

result.

is decidable:

where

R is a m o n a d i c

complete

by

used

problem.

Since

any non-

set Irr(R),

solving

systems

whether

there

remains

open of w h e t h e r

complete

exists

a

exponena more

length-re-

that are not monadic,

the above a l g o r i t h m

can be solved when b e i n g c o n s i d e r e d length-reducing

in general

If finite

are c o n s i d e r e d

to develop

monadic

at first derives

it n e e d s

this problem.

is d e v e l o p e d

given a finite

R on Z, this a l g o r i t h m

So the q u e s t i o n

contain

5.14 an a l g o r i t h m

It is not yet k n o w n for

(Z;R)

?

5.13 and

for the

reduction

finite

a dpda M 4 recog-

by M, and if L is finite,

MR presented

decision

system

time and space.

complete

Thus,

idempotents

the above

the t e c h n i q u e

automaton

tech-

system on ~.

state a c c e p t o r

efficient

[Ha 78])

be c o n s t r u c t e d

The f o l l o w i n g

of Lemmas

reduction

e.g.,

=

By u s i n g w e l l - k n o w n

5.12 are also d e c i d a b l e

the m o n o i d

trivial

is not applicor not the

for p r e s e n t a t i o n s

reduction

systems

that are

not monadic. N o w we turn to the p r o b l e m given

=

we have

[a]R N a. Irr(R).a

all the e l e m e n t s

presentation

reduction Does

pushdown

systems.

A finite

QUESTION:

(cf.,

set L r e c o g n i z e d

a list c o n t a i n i n g (iii)

then

(Theorems

= a'INTR(a).a.

can e f f e c t i v e l y

a deterministic

(ii) and

ducing

a deterministic

R is complete,

if awa ~ a. Thus,

set INTR(a)

cardinality

tial

R on Z and a

[a] R and a finite

recognizes Since

system

w

{awalw

finite

set

a. Irr(R)'a

a}.

w

awa +~+ a if and only

for

construct

the

M I and M 2 we get a dpda M 3 , t h a t {awalw £ Irr(R)

reduction

through

non-trivial Definition

a finite

elements 5.16.

w 6 z* d e s c r i b e s

monadic

of finite

of d e c i d i n g complete

or not a m o n o i d

system c o n t a i n s

order.

Let M be a m o n o i d an e l e m e n t

of w h e t h e r

reduction

presented

of finite

by

(X;R) . Then a w o r d

order of M,

if there exist

any

250 integers

n > O and k >

I such that w n+k +

w n.

If,

in addition,

W

w ~

e,

then

w is

said

to

describe

a non-trivial

element

of

finite

orde r of M. As we saw in Section not a f i n i t e l y finite

order.

acterization for w h i c h

the m o n o i d

this

decidable

whether

given

through

monadic

presented

If M R c o n t a i n s

a non-trivial

elements

finite words

group.

Therefore,

5.77.

before

(X;R)

derive

elements

us with the n e c e s s a r y

Let R be a finite

the

by

(X;R)

following

two

(i)

The m o n o i d

(ii)

There

non-trivial Proof.

statements

Obviously

it suffices

w 6 X* be a shortest order

of M R . If

of finite

Here

system on X. Acwhether

or

idempotent. is a

M R m a y have n o n -

the m o n o i d

idem-

M R is a

information

order

on

of the m o n o i d

idempotents.

The

MR

following

information.

6 left(R)}.

complete

reduction

If the m o n o i d

any n o n - t r i v i a l

MR

idempotents,

then

are equivalent: a non-trivial

element

of finite order.

lwl < ~ such that w d e s c r i b e s

a

order of M R . to prove

word describing

lwl < ~

is

that are

idempotent

additional

of finite

is a word w 6 X* of length element

when

length-reducing

does not c o n t a i n

M R contains

systems.

it has no n o n - t r i v i a l

any n o n - t r i v i a l

s y s t e m on X, and let ~ = m a x { l / I l l presented

then this

occurs

it

problem

a non-trivial

order of M R . However,

we must

Since

R = {(/,r)}

it is d e c i d a b l e

although

char-

on X,

systems.

reduction

contains

idempotent,

non-trivial

reduction

reduction

situation

M R does not c o n t a i n

lemma p r o v i d e s Lemma

this

one-rule

complete

of f i n i t e order,

describing

in case

by

of finite

For example,

[La 74].

to the class of all m o n o i d s

complete

M R presented

trivial

such e l e m e n t s

system

to the results

potents.

R = {(/,r)}

decision

monadic

element

systems

the above

this result

or

of

has given a syntactic

reduction

reduction

n o t the m o n o i d

non-trivial

Lallement

whether

elements

or not a o n e - r u l e

for m o n o i d s

by finite

in general

non-trivial

characterization,

So let R be a finite cording

hand,

one-rule

M~ does contain

syntactic

we want to e x t e n d presented

2, it is u n d e c i d a b l e monoid contains

On the other for those

is d e c i d a b l e meets

presented

that

(i) implies

a non-trivial

then we are done.

Hence,

(ii).

element assume

So let

of finite

that

lwl > ~.

Since w d e s c r i b e s exist This

integers implies

a n o n - t r i v i a l e l e m e n t of finite o r d e r of M R, there n > 0 and k > I such that wn+k ~ w n, and w ~ e.

in p a r t i c u l a r

that n+k > 2. Since

R is complete,

the

words w n+k and w n have a common d e s c e n d a n t m o d u l o R, w h i c h means that n+k 2 w is r e d u c i b l e m o d u l o R. But }wl > ~, and so w is r e d u c i b l e modulo

251

R. Thus, w = WlW 2 = w 3 w 4 with r 6 Z* w i t h Now

Irl