Semigroup Forum Vol. 26 (1983) 349-373 9 1983 Springer-Verlag New York Inc.

RESEARCH ARTICLE

THE ORDERABILITY

OF R E G U L A R

T6ru Dedicated

60th

Communicated

A semigroup a linear The

order

purpose

ability

for

in T h e o r e m s Let

V(x)

the

D(x)

V(T)

ordered

is to g i v e

if

S

The main

admits

semigroup.

a condition

regular

LEMMA

i.i.

semigroup.

of

S.

inverses

= Uxe T V(x).

First

let

to be o r d e r a b l e

semigroups.

be a r e g u l a r

dered

group,

R. R e i l l y

it a l i n e a r l y

idempotents

the D - c l a s s i.

by N.

this paper

set of all

we w r i t e

birthday

is s a i d

regular

G. L. G l u s k i n

of o r d e r -

result

is g i v e n

A and B.

S

set of all

S

making

of

Sait6

to P r o f e s s o r

on his

SEMIGROUPS

of

S

we g i v e

For of For

x

in

by

For

the

of

by

T ~ S,

we d e n o t e

the e l e m e n t

lemmas

E

we d e n o t e

S.

x e S,

containing elementary

We d e n o t e x e S,

by

x.

linearly

or-

semigroups.

Let x,

S

be a l i n e a r l y

z e S

and

let

x'

ordered

regular

e V(x).

semi-

Then

(x'zx) 2 = x,z2x. PROOF.

We h a v e

First

suppose

either zxx'

zxx'

~ xx'z.

(x'zx) 2 = X ' ( Z X X ' ) Z X =

and

so

x'z(zxx')x

(x'zx) 2 = x ' z 2 x .

we o b t a i n

the

same

~ xx'z

or

xx'z

~ zxx'

Then

~ X'(XX'Z)ZX

= x'z2x

~ x'z(xx'z)x

=

In the c a s e

relation

(x'zx)

where

in a s i m i l a r

way.

2

xx'z

s zxx'

SAITO

LEMMA put

1.2. P =

Also, and

p E P\E < pp'

PROOF.

ordered

{x e S; x ~ x2}.

if p'

In a l i n e a r l y

Let

and

Then

p'

regular V(P)

e V(p),

=

semigroup

S, w__ee

{x E S; x 2 ~ x}.

then

p'

< p'p

< p

< p. p ~ P

and

q E V(p).

We h a v e

either

q ~ p

or p ~ q. If q ~ p, t h e n by [4; L e m m a 6], we h a v e 2 q ~ q. Next suppose p ~ q. T h e n a g a i n by [4; L e m m a 6], 2 we h a v e p ~ p and so p e E. But by [4; C o r o l l a r y of Lemma

I],

we h a v e

E is a s u b s e m i g r o u p q ~ E.

Conversely we can

Thus

let

prove

q2

p'

E V(p).

and

result 2 = p < p

p'

< pp'

< p

Now

and

V(P)

let

P ~ p2.

c V(P).

Then

the

pp'p

~ q

that

{x E S; x 2 ~ x}

of S and

we o b t a i n

by

p c V(q).

Since

a b o v e , we h a v e 2 p' < p' = p'pp'

and

and

we g i v e

the

< p'p

we h a v e and

[2; L e m m a 2

p'

p'

1.3],

similarly

p c P\E

proved

and

Then

q c V(p),

Finally suppose 2 p < p and by

we h a v e

[2; L e m m a

c {x c S; x 2 ~ x}.

< p'.

1.3]

Thus

so

< p.

former

h a l f of our

fundamental

theo-

rem. THEOREM S

A.

If a r e g u l a r

satisfies

i.

S

is o r t h o d o x ,

is a s u b s e m i @ r o u p 2.

if

x

3.

every

semigroup

S

is o r d e r a b l e ,

then

the c o n d i t i o n s : i.e.

the

is a p e r i o d i c g-class

set

E

element

of

o_ff i d e m p o t e n t s

o_ff S

o_ff S;

of

S

is e i t h e r

S,

then

x

i-unipotent

2

= x

3

;

o/_r R - u n i -

potent; 4.

there

P u V(P) and

P n V(P)

x'

c V(x), and 2 that x s P\E

such = ex' 5.

exists a subsemi@roup = S,

and

the

making

x2e

= E,

if

P

x E S,

and

of

S

x'Px ~ P

xe,

x'

such

for

e V(x)

ex

e E

then

E

admits

that

every

x c S

and e e E 2 ex = ex

= xe = x'e;

idempotent

semigroup

it a l i n e a r l y

ordered

idempotent

a linear

order

semigroup

satis-

fying a.

if

then b.

x e S, x'ex

if

x'

~ x'fx

x e P\E,

c V(x), in x'

e,

f E E

and

e ~ f

in

E,

E; e V(x),

e e E

350

and

exe

E E,

then

SAITO

ex'e

~ exx'e

Remark by

i.

Let

[2; L e m m a

x'ex, 2.

x'fx

Let

ex'e

1.3],

x'e

x' V(E)

c

PROOF.

Suppose

group 9

Then

tion

by

i,

P =

{x 9 S;

Lemma

i],

have

V(P)

P

n V(P)

Then = x

we

and, we

x

since

P

Hence

we

exx'e,

f 9 E.

ex'xe

of Lemma

we

have

and

~ x

x'

x' s e

and

< x'e.

~ x.

x'e

Hence

that

e

or

e

c p.

[4; L e m m a

by Lemma

~ x.

By

way

x '2

x '2

(x2x'2x)x

which

~ e(ex')

we

have

we

= x 2.

obtain

we

Hence the

= ex'

ex'

(ex)x

2],

have

~ ex'x = ex'x

ex 2 = ex

~ ex = ex

other and

= ex'

and

= ex'

Hence hand,

so

ex In

x

Then

so

< e. since

so and

x'x

9 E

(x2x') (x'x)

so

x3 =

and

so

x

since

= ex'x

the

in a s i m i l a r

351

and

follows

suppose

and

x2 = x3 = x4

the

it

< x.

x2x'2x

assumption.

On

and

9 V(ex),

a contradiction 9

First

(x2x'2x) 2 = x2x'2x

contradicts

ex

ex 2 =

=

we

contra-

< x'e

Similarly is

of

x'e

i],

1.2,

~ x'e

and

=

[4; L e m m a

x3x'2x

Suppose 2 x 4 Q

S

Hence

< x'.

p 9 P.

~ x'p2x ~

Hence

< e. which

and

x'px

by

we

and

Then

9 E

9 E. xx'

[4;

1.2,

9 E.

Then

ex

< xx',

< e

of

~ x'x

x '2 9 V ( x 2 ) ,

= S

x'Px

xe

x 9 P\E.

and

assume

x

so

by

By

9 V(x)

i.i,

ex,

2 and

By Lemma

x'

Condi-

put

and

=

so

semi-

have

x 2 ~ x}.

= P u Q

and e,

We

S.

by Lemma

9 P

so

3.

x 9 S,

and

we

Condition

of

P u V(P) Let

regular

i],

{x 9 S;

a subsemigroup

and

have

by

and

e E 3 c E.

9 P

so

Then

Then

By way of contradiction, we assume ex' 3 3x, x s e, we have x ~ ex' < x = x x ' x x 2 x , < x ,x. Now we have x 2x, = x ( x x ' )

Hence

E.

e V(fx)

ordered

Q =

x'px

x '2 4 E

e ~ x

9 E.

and

~ p2

is

~ xx'

from

exe

Condition

9 V(x)

Q

= x'

2],

Hence

x 2 4 E,

have

and

a subsemigroup

Hence

x'xx'

x'f

have

n Q = E.

x'

and

Corollary

we

= Q.

we

in

e,

and

a linearly

Theorem

x 4 Q

diction,

we

[4;

is

have

x'

is

~ x 2}

P

since

have

by

= P

have

and

9 V(ex)

c E

S

3.1],

9 P\E,

s exe

_

[4;

(x'px) 2. 2

9 V(x)

9 V(x)

m

Theorem

x'

~ ex'xe

E.

9

V(exe)

[5;

ex'e

x 9 S,

x 9 S,

9

and

~ exe

case way.

9 E.

(x3x'2x)x x 2 9 E,

~ ex' x'

and

s x'x

= ex'. where

so ~ x,

Hence e

< x'

Similarly

SAIYO 2 x e = xe = x'e.

we c a n p r o v e The

idempotent

semigroup linearly dition

< x

and

so

~ exe.

2.

latter

The

B.

of

Let

ex'e

half

1-5

semigroup

PROOF.

Let

S

1-5.

steps.

For dual

DEFINITION x, y e S,

a Lemma

i.

(x'y)

2.

(x'y) 2 e E,

for

some

e P\E

x'

3.

x ' y , y'x

in

E

for

4.

x'y

~ y'y

5.

(yx') 2 e E, some

x'

6.

yx',

xy'

in

E yx'

xx'

~ yy'

define

for

in

S

5 can

B by d i v i d i n g

be

a linearly in

satisfying

by L e m m a

More

a linearly

in C o n d i t i o n makin@

S

the

relation

into a*

S}.

Condi-

several

the m u l t i p l i -

if and o n l y

~ on

S

by:

if a n y o n e of

some

x'

E V(x);

D ( ( x ' y ) 2)

for

is

L-unipotent

D(x'y) x'

E

e V(x)

for

and

for

the

x ' y e P\E

is L - u n i p o t e n t and

y'

and

y'x

is L - u n i p o t e n t ,

y ' x = x'y and

y'

is R - u n i p o t e n t

and

x'

< x'y

e V(y);

some

D ( ( y x ' ) 2)

r V(x)

and

c V(y); yx'

c P\E

e V(x); c E,

some

e E,

E

is s a t i s f i e d :

D(x'y)

in

for

satisfies

a.

in

e E,

some

e E,

which

E V(x);

x'x

7.

We

theorem which

is o r d e r a b l e .

semigroup

a, we d e n o t e

of L e m m a 2.1.

S

and

5b.

P = {x e S; x ~ x 2

Theorem

conditions 2

that

S

< x

as f o l l o w s .

makin@

@iven

on

be a r e g u l a r

x ~ y

following

Then E

Con-

and

and

semigroup

on

order

such

< x'x

fundamental

A.

by the

e e E

~ exe

A is s t a t e d

semigroup

We prove

cative

~ ex'xe

x'

4.

ordered E

we h a v e c l e a r l y

we h a v e

of our

order

to a l i n e a r

ordered

Then

on

we h a v e C o n d i t i o n

in T h e o r e m

idempotent

induced

x' e V ( x ) ,

be a r e g u l a r

a linear

tions

1.2,

Thus

S

precisely,

extended

S.

Theorem

Conditions

ordered

is a l i n e a r l y

x s P\E,

by L e m m a

is the c o n v e r s e

E order

semigroup

~ exx'e

THEOREM

linear

Suppose Then

< xx'

ex'e

the

ordered

5a.

exe E E. x'

subsemigroup

under

T h u s we h a v e C o n d i t i o n

D(yx') x'

e V(x)

D(yx') E

for

is R - u n i p o t e n t and

y'

xy'

some

and

352

xy'

< yx'

E V(y);

is R - u n i p o t e n t , x'

and

e V(x)

= yx' y'

and

E V(y).

SAITO

LEMMA

2.2.

PROOF. c E

The

Let

and,

x r S.

since

By C o n d i t i o n

PROOF. some by

Q

Let

a,

of

Then

S.

for

some so

a e Q,

S,

xx',

x'x

= D(x'x)

L-unipotent

or

by d e f i n i t i o n .

Hence

c V(x'yx) Let

and by

S,

i,

S

[2; L e m m a

T h e n the f o l l o w i n g 2 9 E; 2 (ii) (y'x) 9 E; 2 (iii) (x"y) E E; 2 (iv) (y"x) e E; 2 (V) (yx') e E; 2 (vi) (xy') e E; 2 (vii) (yx") e E; 2 (viii) (xy") c E.

and

so

Q

is

Conversely

P = V(Q). Then

for

Hence

a = axa = a x b x a

let

and

so

c EPE

Finally

a c V(y)

for

1.3],

c V(P)

x',

and

b c V(y)

x E Q = V(P)

x' e V(x).

x, y e S,

= Q

p c V(Q). some

Hence

c V(x'Px)

and

is o r t h o d o x .

c V(P)

Hence

c p.

P = V(Q)

x' e V(x).

a e V(x)

for

c V(y). (i)

in

of

Clearly

b e P. V(Q)

x e S

y e P.

Then

a e V(x)

and

2.4.

is e i t h e r

and

ab e V(yx)

c p3 c p

LEMMA

Then D(xx')

x < x

By C o n d i t i o n

x c V(b)

x'ax

e V(x). we h a v e

= D(x'x)

x e S

b e Q.

1.3],

a subsemigroup

some

x'

we have

every

x, y e P.

[2; L e m m a

let

take

is r e f l e x i v e .

is a s u b s e m i g r o u p

for

a e V(Q).

S

Q = V(P).

2.3.

x ' Q x _c Q

-< on

R x L x'x ,

D(xx')

Hence

We put

We

xx'

3,

R-unipotent.

LEMMA

relation

= Q.

x" 9 V(x)

conditions

are

and

y',

y"

equivalent:

(x'y)

Also

under

these

conditions,

we have

D((x'y) 2) = D((y'x) 2) = O((x"y) 2) = O((y"x) 2) = O((yx') 2) = O((xy') 2) = O((yx") 2) = O ( ( x y " ) 2 ) . PROOF. and Also

We h a v e

(y'x) 2 ~ V ( ( x ' y ) 2 ) ,

(y"x) 2 9 V ( ( x " y ) 2)

and

so

(x"y) 2 ~ V((y'x) 2)

(i)<=>(ii)<=>(iii)<=>(iv).

D((x'y) 2) = D((N'X) 2) = D((x"y) 2) = D ( ( y " x ) 2 ) . 2 (v)<=>(vi)<=>(vii)<=>(viii) and D((yx') )

Similarly

= D((xy') 2) = D((yx") 2) = D ( ( x y " ) 2 ) . Then

(yx') 3 = y(x'y) 2x ' = y ( x ' y ) 4 x ' =

353

Suppose (yx') 5.

(i) holds. Hence

SAI~O by C o n d i t i o n 2 (yx') 9 E. if

(i) and

i (yx') 2 LEMMA and

Thus

2.5.

2 =

(yx')

(i)=>(v)

(v) hold, and

let

(yx')

2,

so

3 =

and

then

(yx')

4

similarly

(x'y) 2 R

and

so

(v)=>(i).

Also

(x'y) 2x ' = x'(yx') 2

D((x'y) 2) = D ( ( y x ' ) 2 ) .

Let

e 9 E

e' 9 V(e).

such

Then

that

D(e)

e'e = e,

i__ss i - u n i p o t e n t ,

ee'

= e'

and

e R e'. PROOF. and

Since

e'ee'

since

set

have

Here

we

for

E

in the

S.

E.

in

E

E* = E/D E and

the p a r t i a l is d e n o t e d

only

if

only

if

e L f of

e R f E.

e R f

For

for

e 9 E,

is

i-unipotent DE(e )

LEM_~

2.6.

and

9 V(z) z'x'z

PROOF.

We

if

e,

f 9 E,

and

e L f

in

S

hold

E,

Suppose and

z'xz

~ z'x'xz have

zz'

in

x 9 PiE, 9 E. s z'xz E E

5b, we h a v e

Then in and

D(e)

of

semilattice

X ~

Y

if and

in

S

if and

either

if

one

D E ( e ) = DE(f ) .

but

the c o n v e r s e

It is c l e a r in

E

S,

that,

then

is R - u n i p o t e n t

DE(e ) in

S,

E. x'

9 V(x), z'x'z

z 9 S,

~ z'xx'z

~ z'xz

E. zz'xzz'

zz'x'zz'

354

the D E-

if and o n l y

that

is i - u n i p o t e n t if

S

n E

in the

semilattice

f 9 E,

implies ~ D(e)

in g e n e r a l .

and

in the

in the

e R f

e,

the

The

is a s e m i -

The

in

for

are

by D E in

semilattice

X, Y 9 E*,

DE(e)

D(e) in

e.

ee'

i,

S.

DE(e )

defined

is R - u n i p o t e n t

by C o n d i t i o n

element

by

naturally

e [ f

not

E

Thus

E,

we h a v e

does

then

in

of

are D E - C l a s s e s

associated

For

e'e,

By C o n d i t i o n

semigroup

for

But,

[4] w h i c h

is d e n o t e d

which

In p a r t i c u l a r ,

and

e 9 E,

inclusion

z'

order by {.

E

= e

e R e'

[3] and

we d e n o t e

the

the

X ^ Y = X.

in

bands

containing

in

e, e', also

the D - e q u i v a l e n c e

idempotent

is c a l l e d

and and

ee'e

i e'.

a subsemigroup

semigroup

e 9 E,

= D(e'),

ee'

discussions.

it f r o m

For

E*

= e'

forms

of r e c t a n g u l a r

semigroup class

The

and

notions

following

idempotents

semigroup

ee'

some

to d i s t i n g u i s h

lattice

i e

D(e)

is i - u n i p o t e n t

remind

D-equivalence order

we h a v e e'e

e = e'e,

the

of

Hence

= D(e')

we

needed

9 V(e),

= e'.

D(e)

9 E,

e'

9 zEz'

~ zz'xx'zz'

c E.

Hence

~ zz'xzz'

SAI~O

in

E and so by Condition 5a, z'x'z = z' (zz'x'zz')z z'xx'z = z'(zz'xx'zz')z < z'xz = z'(zz'xzz')z in E.

Similarly we have

z'x'z < z'x'xz ~ z'xz

in

E.

LEMMA 2.7. Suppose x, y e S, x' e V(x), y' e V(y), (x'y) 2 E E, x'y ~ P\E and D((x'y) 2) i__{si-unipotent. Then (i)

(x'y) 2, (y'x)2(x'y) , (x'y)2(y'x),

(y'x) 2 c E;

(i=~i) (y'x)2 R (x'y)2(y'x) R (y'x)2(x'y) R (x'y) 2 and, particular, (y'x) 2, (x'y)2(y'x), (y'x)2(x'y), (x'y) 2

in

DE((x'y)2) ; (iii) (x'y) (y'x)2(x'y) = (y'x)2(x'y) and (y'x) (x'y)2(y'x) = (x'y) 2(y'x) ; 2 2 2 (i=~v) (y'x) < (x'y)(y'x) < (x'y) (y'x) < (y'x) (x'y) < (y'x) (x'y) < (x'y) 2 in E; (v)

(y'x) 2 < x'x < (x'y) 2(y'x) -< (y'x)2(x'y) < y'y < (x'y) 2 i__nn E; (v~_) y'yx'x = (x'y)2(y'x) and x'xy'y = (y'x)2(x'y); (vii)

yx', xy'

E,

yx' = yx'yy'xx'

xy' = xy'xx'yy'; (viii) xy' < xx' < yx'

and

and

xy' < yy' < yx'

i__nn E;

(ix) xy'xx' = xy' and yx'yy' = yx'. 2 2 PROOF. (i) By assumption (x'y) E E and (y'x) V((x'y) 2) c V(E) c E. By Condition 2, (x'y) 2 = (x'y) 3 and so ((y'x)2(x'y)) 2 = (y'x) 2(x'y) (y'x)2(x'y) = (y'x)3(x'y)

= (y'x)2(x'y). (x'y) 2 (y'x) c E.

Hence

(y'x)2(x'y)

e E

and

similarly (iii) By Lemma 2.5, (x'y) 2 R (y'x) 2 Also we have (x'y) 2 = (x'y)2(y'x) (x'y) R (x'y)2(y'x) and similarly (y'x) 2 R (y'x)2(x'y). (iii) By Condition I, (x'y) (y'x)2(x'y) 2 = ((x'y) (y'x)) ((y'x) (x'y)) c E and by (i) , (y'x) (x'y) c E. Further (y'x)2(x'y) = (y'x) ((x'y) (y'x)2(x'y)) i (x'y)(y'x)2(x'y). But by (ii) and by assumption, D((y'x)2(x'y)) = D((x'y) 2) is i-unipotent. Hence (x'y) (y'x)2(x'y) = (y'x)2(x'y). Similarly we have (y'x) (x'y)2(y'x) = (x'y)2(y'x). (iv)

By (i) we have

(x'y) (x'y)(y'x)

355

= (x'y)2(y'x)

e E.

SAI~O Hence

=

by

(iii) and L e m m a 2.6, (y'x) 2 = (y'x)2(x'y)(y'x) 2 (x'y) (y'x) (x'y) (y'x) = (x'y) (y'x) (y'x)

< (x'y) (y'x) (x'y) (y'x) 2 = (x'y) (y'x) in E. imply which

=

(x'y) (y'x) < (x'y) (x'y) (y'x) 2 But (y'x) = (x'y)(y'x) would

x'y = (x'y) (y'x) (x'y) is a c o n t r a d i c t i o n .

= (y'x)2(x'y)

Also

c E

(x'y) (y'x)

=

by

(i),

(x'y)2(y'x)

would

imply x'y = (x'y) (y'x) (x'y) = (x'y)2(y'x) (x'y) 2 = (x'y) 9 E, w h i c h is a c o n t r a d i c t i o n . Hence we have 2 2 (y'x) < (x'y) (y'x) < (x'y) (y'x) in E. Similarly 2 2 (y'x) (x'y) < (y'x) (x'y) < (x'y) in E. F u r t h e r by (ii) (y'x) (x'y) (y'x) (x'y) = (y'x) (x'y) < (x'y) 2 = (y'x)2(x'y) 2 =

(y'x) (y'x) (x'y) (x'y)

in

E

and so by C o n d i t i o n

have (x'y) (y'x) < (y'x) (x'y) in E. (x'y)2(y'x) = (x'y) 2(y'x) (x'y) (y'x) < (x'y)2(y'x) (y'x) (x'y)

5a, we

Hence

( x , y ) 2 ( y , x ) 2 ( x , y ) = (y'x)2(x'y) 2 2 in E. Thus we o b t a i n (y'x) < (x'y) (y'x) < (x'y) (y'x) 2 2 < (y'x) (x'y) < (y'x) (x'y) < (x'y) in E. 2 2 (v) By (iv) we have (y'x) (x'yy'x) = (y'x) < x'yy'x =

(x'x) (x'yy'x)

=

(x'yy'x)(x'y)

< (x'y)2(y'x) < y'y

< (x'y)

(vi)

By

and 2

(x'yy'x) (x'x)

(y'x)

in 2

=

in

E.

in

E

and

Similarly

so

- - x ' y y ' x < (x'y)2(y'x) 2 (y'x) < x'x

we have

(y'x)2(x'y)

E.

(iii) and (v), we have y'yx'x 2 2 -< (y'y) (x'y) (y'x) = (y'y) (y'x) (x'y) (y'x) 2 2 2 = (y'x) (x'y) (y'x) = (x'y) (y'x) = (x'y) (y'x) (x'x) 2 < y'yx'x in E and so y ' y x ' x = (x'y) (y'x). Similarly we have (vii)

x'xy'y By L e m m a

tion i,

xy' = xx'xy'yy'

9 E.

Also

have

yx'

(viii)

= (y'x)2(x'y). 2 2.4, (xy') e E

xy' 9 E

By

and by

= x(y'x)

2

(x'y)y'

(vi) and C o n d i 2 = (xy') (xx'yy')

= (xy')2xx'yy ' = xy'xx'yy'. and

yx'

Similarly

= yx'yy'xx'.

(v) and (ii), x ' ( x y ' ) x = x'(xy') 2 2 2 2 = x'x(y'x) <- ((x'y) (y'x)) (y'x) = (y'x) < x'x = x'(xx')x E.

Hence

Similarly (ix) On

By

(vii),

< (x'y) 2(y'x)

= x'(yx'yy'xx')x

by C o n d i t i o n

5a,

we have

< yy'

(vii)

the other

and

hand

xy'

(viii), by

we

(vii)

xy'

< xx'

< yx' xy'

and 356

=

in

(viii),

x

= x'(yx')x

< yx'

(xy')

2

in

in

E.

E. 2

< xy'xx' xy'xx'yy'

in = xy'

E.

SAI~O

< yy'

= yy'yy'

Hence

xy'xx'

so

xy'

LEMMA x'y

e E,

(i)

x'y

(iv) (a) y'y

if

x'x

yy'

e E

~ yx'

yy'

in

by C o n d i t i o n = y'x

=

(y'x)

and

= xy'

in

E

and

= yx'yy'.

e V(x),

y'

x'y

c V(y),

= y'x.

= yy'xx'

then

~ yy'

in

E,

Then

~ yx'

First

~ xx'

y'y

in

= y'x

~ y'x

= x'y

E.

either xx'

= x' ( x x ' ) x

s x'y

i_~n E;

suppose

= y'(xx')y

e E;

x'x

then

5, w e h a v e

x'x

E.

= D(xy');

E,

~ xy'

~ xy'

in

yx'

and

yx'

in

~ xx'

E. 5a 2

x'

= D(yx')

S yy'

By Condition

~ xx'

have

< yy'

= y'yx'x;

= D(y'x)

yy'

and

PROOF.

we

is L - u n i p o t e n t

xx'

if

xy'xx'

y e S,

= x'xy'y

xx'

and

(b)

x,

= xx'yy'

D(x'y)

so

Similarly

D(x'y) = y'x

and

(xy'xx') 2 ~ xy'xx'yy'

Suppose

xy'

(iii)

=

E

= xy'xx'.

2.8.

(ii)

in

xx' ~ yy'

~ x' ( y y ' ) x

~ y'(yy')y

~ yy'

or

in

E. Then 2 (x'y) = x'y

=

= y'y

in

E.

Also

x'xy'y

~

(x'y) (y'y)

= x'y

=

(x'x) (x'y)

~ x'xy'y

and

y'yx'x

~

(y'y) (y'x)

= y'x

=

(y'x) (x'x)

~ y'yx'x

in

and

so

x'y

= y'x

= x(x'xy'y)y' E.

Further

= D(xy')

is

yy'xx'

LEMMA

by L e m m a

2.4,

i-unipotent

yy'

2.9.

e E

and

only

i_ff x ' y

and,

First

since

x'y

= x'x

x'x

= y'y.

D(yx')

D(x'x)

LEMMA S

PROOF.

Then

9

Then Since

x'x

=

r

yx'

x ~ y

(xy')y

we

x,

Then

x'x

E.

The

y'

way.

e V(y),

x = y

if

L x = y i y'y

suppose

we x'y

have

xy',

Hence

by L e m m a

(yx'xy')y

y e S,

s P in

=

in

L-unipotent,

Conversely 2.8,

= xx'xx'

in a s i m i l a r

c V(x),

Then is

xx'

= xx'yy'xx'

= y'y.

L-unipotent.

Suppose x'y

x'

x = y.

by Lemma

= x(y'y)

2.10.

and

= D(x'y)

is

= yy'

treated

= yy'xx'

= D(yx')

2.5,

i__{s L - u n i p o t e n t.

= y'x.

= D(x'y)

be

xy'

yx'

= D(y'x)

by L e m m a

y c S,

= y'x

suppose

= y'y

x = x(x'x)

in

D(x'y)

D(x'y)

~ yy'yy'

can x,

Moreover

similarly

( x x ' y y ' ) (yy'xx')

= xy'

Suppose

x'y

PI~0OF.

=

~ xx'

and

and

and

= xy'yx'

= xx'yy'

where

= y'yx'x.

e E

= yx'

xx'yy'yy' case

= x'xy'y

= xx'yy'

E

x'

yx'

have = y'x

and

e E

and

2.5,

= yy'yy'y

e V(x)

and

= y.

x ~ y

"

S,

357

any

one

of C o n d i t i o n s

1-7

SAITO

in D e f i n i t i o n 2.1 is s a t i s f i e d . 2 C a s e : (x"y) ~ PiE for some x" Take and

y'

c V(y).

2.3,

(x'y) 2 e PiE.

we have

diction,

we a s s u m e

(x'y)2x'x (x'y)2x'x (x'y) 3 =

e PE

c P

for

Take

y'

y'x

by L e m m a

= D((x"y) 2) yx'

by L e m m a

c x'Qx

c Q.

(x'y)2x'x

(x'y) 2 =

2.4,

by L e m m a

By w a y of c o n t r a 2.3, But

e P n Q = E.

((x'y)2x'x)2x'y

=

Hence

(x'y) 5.

(x'y) 3 =

Hence

(x'y) 4

Hence

is s

Then

c V(Q)

y'x

= P.

x'y

we h a v e is

Then

and

= Q

and

yx'

so

e P.

and

x" 9 V(x).

and

2.4,

x ' y E P.

D((x"y) 2)

some

4 E

so by L e m m a

is a c o n t r a d i c t i o n .

e V(y).

x'y 9 V(y'x)

have

so

2, we h a v e which

so

e Q.

=

and

(x'y) 2 4 Q

c x'Q2x

and

(x"y) 2 9 E,

x " y 9 PiE

so

yx'

(x'y) 2 x ' x x ' y

by C o n d i t i o n 2 (x'y) 9 E, Case:

and

= x'(yx')2x

= p

Hence

x'y 4 Q

~ V(x).

(y'x) 2 e V ( ( x " y ) 2) c V(P)

(x'y) 2 e V ( ( y ' x ) 2) c V(Q)

we h a v e

c V(x"y)

Also, and

4 E (x'y)

2

i-unipotent.

since

so E

c V(P)

= Q

x " y 4 E,

x ' y e PiE. and

Hence

and we

Further

D((x'y) 2)

by L e m m a

2.7,

we h a v e

c E c p.

Case:

x"y,

y'x

< x"y

for

some We

y'x

9 E,

in

E

have

9 V(yx') and

y'x = x " y and

y'

x ' y E V(y'x) yx'

c V(QiE)

4 P.

x 9 V(x")

and

x"y

in

c PiE.

have

yx' The

LEMMA

remaining

yx',

y' y'x,

9 p n Q = E. = D(y'x)

The

Suppose

9 V(x),

x'y,

and

either

< y'y

in

E

e V(y).

But

By w a y of c o n t r a d i c -

yx'

r QiE

by L e m m a

and

2.4,

is [ - u n i p o t e n t . and

so

xy'

(xy') 2 9 E

Further

so by L e m m a

2.7,

which

is a c o n t r a d i c t i o n .

cases

can

we have

Hence

we

9 P.

2.11.

PROOF.

x"x

c E c p.

y 9 V(y') E,

and

Then

D((xy') 2) = D(x"y)

< y'x

is i - u n i p o t e n t

D(x"y)

or

x" E V(x)

t i o n we a s s u m e

x'

Then

relation

< o__nn S

x, y s S,

x < y

c V(y).

Then

xy'

and

Also

= D(xy').

9 P

by L e m m a First

be t r e a t e d

i_ss a n t i s ~ m m e t r i c . and

by L e m m a so

x'y,

2.4,

suppose

358

similarly.

y -< x 2.10,

yx',

D(x'y) D(x'y)

in

S

and

we h a v e

y'x,

xy'

= D(yx') is not

R-uni-

SAITO potent.

Then

Definition y'y

and

Lemma

2.9,

not

2.1,

~ x'x

we h a v e

L-unipotent,

Lemma

2.5,

x ~ y

in

E.

have in

E.

y'y

~ x'x

if

in

LEMMA

2.12.

either

and

E.

Hence

and

also

x'x = y ' y

In the c a s e w h e r e x = y

in a s i m i l a r

way.

x'y

x ' y = y'x.

R y'x

and

so

x'x

~ y'y

E,

then

~ yy' E.

in Hence

since

y ~ x x'x

in

= y'y

x, y c S yx'

and

~ P\E.

have

we h a v e by L e m m a

~ V(x)

2.9,

in

we

that S.

First suppose x ' y ~ P\E. T h e n by C o n d i t i o n 4, we 2 2 2 (x'y) c P. If e i t h e r (x'y) ~ P\E or (x'y) ~ E t h e n we h a v e

by d e f i n i t i o n .

N o w we c o n s i d e r

the c a s e w h e r e

and

is R - u n i p o t e n t .

D ( ( x ' y ) 2)

since

x ~ V(x') that

xy',

Lemma

2.4,

x ~ y

in

implies

S

2.13.

y ~ x

in

in

For

take

4, we h a v e

x, y ~ S,

x'

have

x ~ y

then

y'x

in

Similarly

E.

yx'

S ~ E

Then

from Lemma

is R - u n i p o t e n t

w__ee hav___~ee i t h e r

yx'

~ V(x)

and

~ P\E

or

or

yx'

y'

Also and

by

so

c P\E

x ~ y

or yx'

~ V(y).

o__rr

~ Q\E

~ P\E,

or

then

S.

If e i t h e r

x'y ~ Q \ E

or

xy'

and

suppose by L e m m a 3,

Suppose

x'y,

yx'

2.4, D(x'y) D(x'y)

~ E.

D(x'y)

so

Then

x'y,

or

x ' y ~ E. ~ E.

If

2.12,

we

yx'

c Q\E,

in

S.

y'x,

yx',

= D(y'x)

L-unipotent

is L - u n i p o t e n t .

359

yx'

y ~ x

= D(yx')

is e i t h e r

or

by L e m m a

in

~ P\E

By C o n d i t i o n

x'y ~ Q\E

c P\E

By C o n d i t i o n potent.

< yx'

(x'y)

E V(y).

it f o l l o w s

xy'

x ' y E P\E

x ' y c P\E

and

and

y'

in 2

S.

either

either

Finally

c E

= D ( ( x ' y ) 2)

take

x ~ y

S.

We

either

yx'

We

y ~ V(y'),

by d e f i n i t i o n .

x ~ y

PROOF.

and

D(yx')

LEMMA

E

we

~ y'y

such

x ~ y

is L - u n i p o t e n t ,

Also

2.8,

and

x'

~ yy'

x'x

S,

Then

By Since

xx'

by L e m m a

we a l w a y s

Hence

or

or

is

Final-

R-unipotent.

either

so by

D(x'y)

D ( ( x ' y ) 2)

2.7*

x'x and

and

Suppose

x'y E P\E

y'x = x'y

and by

is L - u n i p o t e n t

in

Similarly,

x = y.

have

xx'

is L - u n i p o t e n t

E.

have

we h a v e

~ y'y

have

PROOF.

we

we h a v e

But x'x

have in

x = y.

D(x'y)

S,

= D(y'x)

we m u s t

y'y

ly s u p p o s e

in

D(x'y)

xy'

= D(xy').

or R - u n i -

By C o n d i t i o n

5,

SAI~O

we

have

either

y'x

< x'y

we h a v e

or

in

E.

Also

If

y'x

< x'y

in

E,

x'x

then

If

x'y

< y'x

in

E,

then

x'y

~ y'y

or

y ~ x

in

S

x'y = y'x

and

~ y'y

in

E,

~ x'x

in

if

x'y = y'x

The case treated

y'y

D(x'y)

in a s i m i l a r

LEMMA

2.14.

Then

fe = ge

PROOF.

Let

f

and

^ DE(g)

f

or

then E,

f, g E E

g

in

E.

~ DE(e)

and

Hence

in

that

f ~ ge

and

or By

in

s g

such

and

so

fe = ge

in

ef = eg.

2.15.

PROOF.

so

by

either

y ~ x

and S.

be

DE(fg).

E.

But

E

and

= DE(f)

which

in

so by

lies

^ DE(g)

contradicts

E.

or

in

would

[3; L e m m a whence

suppose

e ~ ge

E

the

e < g

First

2], we h a v e

f s ge

e 4],

4]

s g imply

DE(fg)

DE(fg )

In the r e m a i n i n g

e,

f 9 E,

suppose

e ~ ef

x 9 S

e ~ f

~ f

in

E

~ x'fx

in

E.

= x'e(fxx')x

= x'(xx'e)2fx

= x'exx'efx

x'exx'fx

cases

we o b t a i n

similarly. and

x'

9 V(x).

Then

= x'efx.

~ x'efx

= x'efx.

the 2.16.

same Let

in and

E.

Then

x,

Hence = x'efx

x'exx'fx

in a s i m i l a r and

5a,

~ x'efxx'fx

= x'(xx'e)fx

~ x'exx'fx

z 9 S

[3; L e m m a

so by C o n d i t i o n

In the c a s e w h e r e

equality

by

in

E

f ~ e

and

so

in

E, we

way.

z' 9 V ( z ) .

Then

(z'xz) 2 = z,x2z. PROOF.

S in

can

that

[3; L e m m a

e < f ~ g

g ~ f < e

= x'e(fxx')2x

LEMMA

in

that DE(efg ) ~

DE(fg)

[3; L e m m a

ef = eg Let

First

we h a v e

obtain

i-

w h i c h is a c o n t r a d i c t i o n . Hence ge < f ~ g 2 ~ fe ~ ge in E. Hence fe = ge and

and

x'exx'fx

x'ex

then

is If

ge = ge

similarly

LEMMA

x ~ y

we a s s u m e

= DE(f ) A DE(g ) ~ DE(ge ) ~ DE(e), = DE(efg)

= D(x'y)

is R - u n i p o t e n t

Then

we h a v e

E.

e ~ eg ~ g

E.

ef = eg.

f ~ g < e

e < f S g

in

by d e f i n i t i o n .

D(y'x)

= DE(e ) A DE(f ) ^ DE(g ) = DE(efg), assumption.

~ x'x S

way.

e,

and

y'x = x'y

by d e f i n i t i o n .

= D(yx')

By w a y of c o n t r a d i c t i o n

between DE(f)

and

where

in

since

we h a v e

or

y'y

x s y

unipotent,

x'x

< y'x

We

take

x'

9 V(x).

First 360

suppose

x'zz'x

2]

SAI~O

< x'xzz'x'x <

E.

(x'x)zz'(x'x)

we

have

ma

3],

and =

in

and

x'zz'x we

so

case

z,x2z

=

x'zz'x

where

x'xzz'xx'

=

=

=

D(x'xzz'x'x) z,x2z

=

LEMMA

2.17.

Then

ex

2 that 2

= ex

=

x

4,

we

2.18.

Then

xz

PROOF.

in

and

obtain

suppose Since

we

since

have

In

zz'xx' = xzz'x'

and

so

the

case

Hence

z'x2z

R-unipotent,

x'

proof

S,

any

take and

xz

y'

so

Also

since if

y,

e V(x)

one

such in

only

where

we

obtain

and

e,

xe,

ex

c E.

some

r V(y)

and

r P.

x' z' If

in

S

by Lemma

we

have

zz'x'y

c PiE, e PiE

we

that

and

that

= x'xe.

x ~ y

i__nn S.

xz

1-7

~ yz.

Since

in D e f i n i t i o n

E V(x). e V(z). z'x'yz

2.12.

We

have

e PiE, If

zz'x'y,

D(z'x'yz)

361

Hence

= ex 2 = ex'x

= xx'e

( z z ' x ' y ) 2, have

= x'2e.

= ex'

Conditions

for

x2e

S.

show

2.4,

x'y

ex

= x'e

z c S

of

and

so

xe

s zy

we

z'x'yz

~ yz

by Lemma

and

have

zx

the

then

Hence

x,

satisfied. 2 (x'y) c PiE

We

have

can

In

way.

e x 2 = e x '2

we

Let

~ yz In

~ y

PiE

we

= xzz'x'.

zz'xz

is

r PiE,

have

Similarly

LEMMA

Case:

so

D(x'xzz'x'x)

(z'xz) 2.

= e x '2 = e x 2 = e x ' x

= exx'

is

and

[3; L e m -

(z'xz) 2.

E,

= xx'zz'xx'

a similar 2

=

Finally

then

=

= D(xx'zz'xx') in

in

If

zz'xx'

z'xzz'xz

z'xzz'xz

= xx'zz'xx',

= zz'xx'xz

Let

=

i-unipotent,

have

(z'xz) 2

Condition

2.1

E by

= ex' = e x x ' = e x ' x and xe = x'e = xx'e = x'xe. 2 Since x e PiE, it f o l l o w s f r o m C o n d i t i o n s 4 and 4 x c PiE. Also by Condition 4, w e h a v e e x = ex' 2 2 2 and x e = x ' e = x e. Hence ex , x e e E and by

PROOF.

x

in

Hence

(x'zz'x)zz'(xzz'x')

xx'zz'xx'

R xzz'x'

is we

(z'x) (xz)

=

way.

= D(xx'zz'xx').

( x z z ' x ' ) (xz)

exx'

Then

L xzz'x'x

L xx'zz'xx',

E.

< x'zz'x

in a s i m i l a r

= D(xx'zz'xx')

xz

( x z z ' x ' ) (xz)

= x'xzz'x'x.

D(x'xzz'x'x)

in

(z'x) ( x ' z z ' x ) ( x z z ' x ' ) (xz)

(z'xz) 2

x'xzz'x'x

= x'zz'x

< x'(xzz'x')x

< xzz'x'

(x'zz'x) (xzz'x') =

(z'x) ( x ' z z ' x ) z z '

the

(x'x)x'zz'x(x'x)

x'zz'x

< zz'

have

z'x2z

Then

x'y

then

z'x'yz

we

e E,

( x ' y z z ' ) 2 e E. x'yzz' is

e P.

L-unipotent,

SAI~O

then

by L e m m a

2.7,

z'y'xz

< z'x'yz

tion.

If

then

E

in

we a s s u m e

that

x'yy'x

=

e E,

(x'yy'x) (zz'x'y)

= D E ( ( x ' y y ' x ) (x'x)). x'xzz'

= x'yy'xzz'

Hence

x'xzz'

by L e m m a Hence

r E

Hence

x'y =

Also

we h a v e

and

(xzz'y')yx'

yx'(xzz'y')

=

2.14,

D(zz'x'y)

in

S.

by L e m m a 2.9*,

suppose

2.8*,

we h a v e z'x'

E E.

so

xzz'x'

= y(x'xzz')y'

= y(y'yzz')y'

= yzz'y'

x'y

e P\E We

for

take

and

so

(yx') 2 ~ P\E. c E

by C o n d i t i o n = xzz'y' = yx'(xzz'y')

= yzz'x'.

Hence

D((x'y) 2) x'

e V(y)

is

i-unipotent

and

E V(x). and

z' E V(z).

By L e m m a

2.4,

9

we have

(y'x)"

we

S.

some y'

xzz'y', = z'y'.

= y(x'xzz')y' Hence

= xy'(xzz'y')

in

D(zz'x'y)

element

and

yx'(xzz'y')

so

Then

((x,y)2x,x)2x'y.

= x(y'xzz')y'

s yz

= y'xx'yzz'

Next

we h a v e

and

we h a v e = y'xx'yzz'.

is i - u n i p o t e n t .

(x'y) 5 =

E E,

Hence

is i - u n i p o t e n t .

is not a p e r i o d i c

= xy'(xzz'y')

(x'y) 2 e E,

would

= z'(x'yzz')z

= x(x'xzz')y'

xz

by

= z'(y'yzz')z

= D(zz'x'y)

= x(zz'y'y)x'

have

Then

DE(x'yy'x)

= x'yzz'

(x'y) 2x'x c P\E

xzz'y'

Case:

e E.

y'yzz'

= z'(y'xzz')z

(x,y) 3 #

x'(yx')2x

by L e m m a

= y'xzz'

by L e m m a

2,

we

in w h a t

= x'yy'xzz'

= z'(x'xzz')z

s yz Then

((x,y) 2 x , x ) x , y

in

e P\E,

is a c o n t r a d i c t i o n .

similarly

suppose

xz

and

By C o n d i t i o n

Hence

z'y'xz

is R - u n i p o t e n t . yzz'x'

which

D(z'x'yz)

we have

~ yz

Thus

x'yzz'

= y'xzz'

and

xz

x'yzz' way.

= DE(ZZ'x'yy'x ) ~

and

z'x'xz

First

e E

D(yzz'x')

we have

where

by d e f i n i -

x'y = x ' y y ' x x ' y

= x'yy'xzz'

2.4,

S

is R - u n i p o t e n t ,

2.4,

Hence

zz'x'y,

so

D E ( ( Z Z ' ) (x'yy'x) (x'x))

Also

in

and

(x'yy'x) (zz'x'yy'x) (x'yy'x)

=

and

E E

DE(ZZ'x'yy'x ) = DE(x'yy'x )

and

= z'y'yz.

by L e m m a

x'yzz'

= x'yy'xzz'x'yy'x

= z'x'yz

xz ~ yz D(z'x'yz)

in a s i m i l a r

z'x'yz,

But

z'x'yz

xzz'y' , yzz'x'

and

S

we h a v e

= y'xx'yzz'.

= y'yzz'.

so

and

In the c a s e

~ yz

2.17,

imply

and

is R - u n i p o t e n t .

xz

follows Lemma

z'y'xz,

we h a v e

in

by d e f i n i t i o n .

obtain

E

e P\E

2.7*,

< yzz'x'

= D(z'x'yz) S

in

zz'x'y

by L e m m a

xzz'y'

we have

c E.

Also

we h a v e 362

z'x'yz

e P

and

by

4,

SAIqD Lemma

2.16,

V(E) have

(z'x'yz) 2 = z ' ( x ' y ) 2 z

~ E.

If

xz ~ yz

r E.

Then

(y'x)

2

z'x'yz

e P\E,

S.

In what

in

z'y'xz

< x'x < y'y

z'y'xz

e V(E)

either

=

= z'x'xz

E.

< z'x'yz

= z'y'yz.

then we have

in

Hence

xz ~ yz

E

~ z'x'xz

if

in

or

by L e m m a

2.16,

~ z'y'yz

E.

Hence we have

z'y'xz

D(z'x'yz)

S

we

z'x'yz

2.7, we have

Hence

in

2.12,

we assume

By L e m m a

(z'x'yz) 2 = z'x'yz

z'y'xz

(z'y'xz) 2

then by L e m m a

in

= (z'y'xz) 2 = z'(y.'x)2z

z'(x'y)2z

and

follows

c E.

< (x'y)

r E

= z'x'yz

is [-unipotent,

by d e f i n i t i o n .

Next suppose

D(z'x'yz) is not i - u n i p o t e n t . By L e m m a 2.7, we have (y'x)2 R (x'y) 2 and so (y'x)2(x'y) 2 = (x'y) 2 and 2 2 2 (x'y) (y'x) = (y'x) Hence by L e m m a s 2.16 and 2.15, z'x'yz = (z'x'yz) 2 = z ' ( x ' y ) 2 z = z ' ( y ' x ) 2 ( x ' y ) 2 z = z,(y,x)2zz,(x,y)2z

=

(z,x,yz)2(z,x,yz) 2

= (z'y'xz) (z'x'yz)

and s i m i l a r l y

z'y'xz

= (z'x'yz) (z'y'xz)

and so

R z'x'yz.

D(z'x'yz)

is R - u n i p o t e n t ,

so

= z'x'yz

zlylxz

2.9*,

we have

z'x'

z'y'xz we have

z'y'xz

= z'x'xz

= z'y'yz.

= z'y'.

Also

and also

y'xzz'

2.16,

r E.

Hence

by L e m m a

= z'x'yz

Hence

by L e m m a

x'yzz' , yzz'x' , zz 'x'y e E

But since by L e m m a

2.8*,

zz'y'x,

x'yzz'

and

xzz'y',

= x'yzz'x'yzz'

= x,yzz,(zz,x,yzz,) 2 = x,yzz,zz,(x,y)2zz , = (x'yzz') (x'y)2zz '

Hence

DE(X'yzz')

^ DE((x'y) 2) ^ DE(ZZ' ) ~ D E ( ( x ' y ) 2 ) . is i - u n i p o t e n t potent,

and

we have

D(x'yzz')

= DE(X'yzz') But since

= D(z'x'yz)

DE(X'yzz' ) # DE((x'y) 2)

DE((X'X) (x'y)2(x'yzz'))

D((x'y) 2)

is not

i-uni-

and so

= DE((x'y) 2(x'yzz'))

= DE((x'y) 2)

^ DE(X'yzz' ) = DE(X'yzz' ) < DE((x'y) 2) = DE((X'X) (x'y)2). Hence

by L e m m a

2.14,

= (x'y)2(x'yzz

')

= (y'x)2zz '.

Also

DE(x'yzz' ) ~ Lemma

2.14,

= y'xzz'. have

= x'yzz' = y'yzz'.

=

(x'x) (x'yzz')

and s i m i l a r l y

y'xzz'

DE((x'y) 2) = D E ( ( Y ' X ) 2 ( x ' y ) 2) x'yzz'

But since =

x'yzz'

D E ( ( N ' X ) 2 ( x ' y ) 2 z z ') = D E ( ( x ' y ) 2 z z ')

we have

y'xzz'

we have

= (x'y)2zz '

=

(y'x) 2 < x'x < y'y < (x'y) 2,

(y'x) 2zz ' ~ x'xzz'

= y'xzz' Further

and so by

(x'y)2zz ' = (y'x)2zz '

and so by L e m m a

x'yzz'

s (x'y)2zz '

= y'xzz'

= x'xzz'

2.7, we have

363

we

~ y'yzz'

xy',

yx'

c E

SAIYO

and

xy'

< yx'

= xy'xzz'y' E

and

xzz'y'

yx'

take

we have < x'y

unipotent and

z'x'xz by L e m m a

~ z'x'yz,

D(z'x'yz)

D(z'x'yz)

(xzz'y') 2 e E But

z'x'yz

< z'y'xz e Q

and

2.12,

suppose

have

y'x

xzz'y'

z'y'xz

Next

z'x' y'x

< y'y

in in

in

S

Then

= z'x'yz

by L e m m a

~ yz

in

S.

in

=

E.

t h e n we have

in w h a t

fol-

yzz'x'

e E.

By L e m m a

2.15,

S

we

have

z'x'yz

5a, we have = D(z'x'yz)

xzz'y'

xx'

E.

Also

= xzz'y'

z'x'xz 2.9*,

< yy'

= xx'xzz'y'

in

is

by d e f i n i t i o n .

by L e m m a

we h a v e

364

we

< z'x'yz

T h e n we h a v e

Hence

xzz'x'

2.7*,

z'y'xz

= z'y'xz

D(yzz'x') in

= yzz'x'

then

(z'x'yz) 2 R z ' x ' y z z ' x ' x z

= z'x'yz.

Hence

Hence

e P\E,

and

by C o n d i t i o n

s yz

2.7*,

imply

Thus

is R - u n i p o t e n t

= z'x'yz.

2.4,

is R - u n i p o -

would

yzz'x'

Since

xz

t z'x'yz.

is a c o n t r a d i c t i o n .

z'x'xzz'y'xz

E.

= yzz'y'

c P\E

and and

by d e f i n i t i o n .

If

Hence

E

z'y'xz

z'y'xz in

is R - u n i p o t e n t ,

By L e m m a E.

z'y'xz

~ z'x'yz

so by L e m m a

E.

z'y'xz

in

= y'x

e P.

z'x'yz

Hence

= z'y'xz

yy'yzz'y' = yzz'x',

and

we h a v e

= z'y'.

xz

Also

< yzz'x'

suppose

which

is R-

= z'y'xzz'x'yz

then

< z'x'yz

D(z'x'yz)

R-unipotent,

= z'y'yz

E,

yzz'x'

< y'y

s E,

Hence

xzz'y'

D(z'x'yz)

= z'x'yzz'x'xz

y'x

2.7*,

y'xx'y

z'x'yz

xz ~ yz

2.7*,

= x'xy'x

since

~ z'y'yz

in

= z'x'yzz'x'xz.

we have

= z'y'xx'yz

so

and

= D((yx') 2)

is R - u n i p o t e n t .

so

= z'x'xzz'y'xz. and

2.5",

By L e m m a

< x'y

z'y'xz

we h a v e

lows we a s s u m e

is R-

and

D ( ( x z z ' y ' ) 2) = D ( z ' x ' y z )

by L e m m a

First

< x'x

= D(y'x)

z'y'xz,

and

and

xzz'y'

by L e m m a

z' c V ( z ) .

is t - u n i p o t e n t ,

Suppose

in

by d e f i n i t i o n .

= z'x'yzz'y'xz.

= z'y'yz

= D(z'x'yz)

S

Hence

z'y'xz

= z'x'xz

tent.

y'x

so by L e m m a

z'x'yz

= yzz'x'

= xzz'y'

is R - u n i p o t e n t

and

D(x'y)

= x'y.

similarly If

Also

2.15,

xzz'x'

x' e V(x).

x'y E E,

and

x'yy'x

D((yx') 2)

some

E.

in

= xx'xzz'y'

= yzz'y'

D(yzz'x')

xz ~ yz

y' e V(y)

y'x, in

= yy'yzz'y'

Since

(yx') 2 e E, for

xzz'y'

then

we h a v e

e P\E We

Hence

= yzz'x',

= yzz'y'.

unipotent, Case:

E.

~ yx'xzz'y'

if

= yzz'x'

in

we h a v e and

~ yy'xzz'y' if

xzz'y'

= yzz'x'

SAI~O

= yzz'y'.

Since

we h a v e Case:

D(yzz'x')

xz ~ yz

in

S

x'y , y 'x 9 E,

< x'y

in We

by L e m m a Hence

E

for

take 2.5,

some

2.15,

z'y'xz

= z'x'yzz'y'xz

y'xx'x

= y'x

y'xx'y

and

E.

by C o n d i t i o n

Hence

E,

we have

~ z'x'yz

D(z'x'yz)

Thus

is not

L-unipotent.

since

z'x'yz

= z'y'xz

R z'y'xz,

~ z'y'yz

in

= D(z'x'yz)

is not

L-unipotent,

we h a v e

have

x'xzz'

= x'yzz'

since

we have xzz'y' =

and

= D E ( ( y ' y ) (y'x)) = y'yzz'

Hence

(xy') 2 = yy'xy'

E.

Also

Case:

yx',

< yx'

in We

= xx'xzz'x'

= xy'

Also

9 E,

D(yx')

in

some

S

yx'xy'

= yx'.

E

so

E.

Since

yx'

D((x'y) 2) = D(yx')

z'x'

we have

= D(x'y)

and

= D(x'y)

is

2.14,

we

x'xzz'

= z'x'yz = z'y'

Since

(xy') 2 i yy'xy' is i - u n i p o t e n t ,

= yy'

in

E.

Hence

= x(y'x)2zz'x ' = yzz'x'

= yzz'y'

is R - u n i p o t e n t

and

by d e f i n i t i o n . is R - u n i p o t e n t

x' c V(x)

z' 9 V(z).

and

= z'y'yz

= D(z'x'yz)

xy'

in

S

= D(z'y'xz)

by L e m m a

= xy'xzz'x'

in

for

in

D(z'x'yz)

~ z'x'yz

similarly

= yy'yzz'x'

~ yz

and

= yx'xy' < yy'

D(yzz'x')

E

~ yz

if

DE(Y'X)

we have

~ y(y'y)y'

xz

take

xz

Hence

we a s s u m e

and

z'x'xz

2.9*,

(xy')2xzz'x ' ~ yy'xzz'x'

in

~ z'x'yz, E.

D(y'x)

and

D((xy') 2) = D(y'x)

so we h a v e

in

is R - u n i p o t e n t

= DE(Y'XZZ') X

by L e m m a

= xzz'x'

< y'y

D(y'xzz')

yy'(xy') 2 = y(y'x)2y ' = yy'xy', and

< x'y

y'x

in

z'x'xz

Since

Since

D E ( ( Y ' X ) (x'x) (zz'))

= y'xzz'

and

we h a v e

and

= y'x

and z'y'xz

follows

i-unipotent

= y'yzz'.

= z'y'xz

x'yy'x

< x'y

Also

= y'x.

R z'x'yz.

D(z'x'yz)

E.

= D E ( ( y ' y ) (y'x) (zz')) = DE((Y'X)(x'x))

z'y'xz

x'yy'x

then we have

Then

9 E.

= z'y'xzz'x'yz

~ z'y'yz

in w h a t

9 V(y).

z'y'xz

and

and

z'y'xz

y'

y'x

z'x'yz

5a, we have

and

and

= x'y

x'x

is i - u n i p o t e n t ,

by d e f i n i t i o n .

and

z'x'yz,

so

< x'y = y ' x x ' y

in

L-unipotent

Then

we h a v e

= x'yy'y

z'x'xz

is

x' 9 V(x)

we have

is R - u n i p o t e n t ,

by d e f i n i t i o n .

D(x'y)

z' 9 V(z).

by L e m m a

= D(z'x'yz)

By L e m m a Hence xx'

2.5*, xx'xy'

< yx'

9 E,

and

and

we have

is R - u n i p o t e n t 365

and

xy'

y' 9 V(y). we h a v e = xy'

xy'yx'

< yx'

similarly xy' 2 (x'y) 9 E. and

since

SAI~O

(x'y)

2

R

(x'y)x'x,

= x'(yx')2x y'y

=

in

PiE,

E.

2.7*,

tradiction. z'x'yz S.

Hence

x'y

then

Suppose that

would

< xy'

e P

and

e E

imply

(x'y)2(y'x) 2 =

z'x'yz

and

similarly

t z'y'xz. z'x'yz

and

E E

D(z'x'yz)

so

Hence

= z'y'xz

=

xz ~ yz

and

in

true,

then

from Lemma < z'x'xz

that

a contradiction. P\E,

then

yzz'x'

we have

x'yzz',

zz'x'y

< yzz'x'

in what Then

E

E.

xzz'y',

and

D(y'xzz') have

xzz'y'

since xz

we a s s u m e

in

and

c E

= D(z'y'xz)

y'xzz'

S

= y'xzz'y'x

which

e P.

If

~ yz

in

Since

S. x'y

x'yzz'

we have

=

Now e P, e PiE

xzz'y' is R-

zz'x'y

also

is

yzz'x'

= D(z'x'yz)

and

Thus E E.

y'xzz'y'x

(y'xzz') (zz'x'y)

R y'xzz'y'x

R y'xzz'x'y.

Now we

and

Since

is R - u n i p o t e n t ,

= y'xzz'x'y. 366

E,

x'yzz',

= y'xzz'y'xzz'

= D(z'x'yz)

e QiE

it f o l l o w s

by d e f i n i t i o n .

y'xzz'x'y

(y'xzz') (zz'x'y) (y'xzz')

e E,

D ( ( x z z ' y ' ) 2)

if either 2.7*,

yzz'x',

zz'y'x

e E

y'xzz'

But

xz e E.

D(yzz'x')

~ yz

z'x'yz

yzz'x'

in

yzz'x'

we have

c P.

E

(z'x'yz) 2 = z'x'yz

< x'x

2.12,

by L e m m a

in

z'x'yz

if

since

Then

then

y'xzz',

Moreover =

we have

zz'x'y

(y'xzz') (zz'y'x)

y'xzz'

Hence

we have

follows

= 2

(x'y)

(z'x'yz) 2

Suppose

But

and

=

we have

~ z'y'yz

is R - u n i p o t e n t ,

z'(x'y)2z

e E.

c P\E, in

unipotent,

~ PiE

Hence

by L e m m a

suppose

z'x'yz

Since

(xzz'y') 2 E E.

xzz'y'

E.

so

~ z'(x'y)2z

by d e f i n i t i o n .

= D(z'x'yz)

2.7* in

and

is R - u n i p o t e n t .

(yzz'x') 2,

= D((yzz'x') 2)

or

S

z'x'yz

2 = z'x'yzz'y'xz

is i - u n i p o t e n t ,

z'x'xz

in

from L e m m a

we have

(z'y'xz) 2 = z ' ( y ' x ) 2 z

D(z'x'yz)

we have were

= z'y'xzz'x'yz

If

~ yz

(y'x)2(x'y) 2

(z'x'yz)2(z'y'xz)

z'y'xz

so

is i - u n i p o t e n t .

it f o l l o w s

since

= z'y'xz.

= z'x'yz and

But

xz

and

is a c o n -

e P.

we have

D(z'x'yz)

(x'y) 2

y'x

which

(y'x) 2. H e n c e by L e m m a s 2.16 and 2.15, (z,x,yz) 2 = z , ( x , y ) 2 z = z,(x,y) 2 ( y , x ) 2 z =

that

E,

so

and

(y'x) 2

is R - u n i p o t e n t in

2.12,

and

~ x'yx'x

similarly

= D(yx')

(x'y) 2 E E

= z'(x'y)2zz'(y'x)2z

=

and

yx'

by L e m m a

z'x'yz

(y'x) 2,

x'x = x ' x x ' x

(x'y) 2

x'y e QiE

we have

e P\E,

have

2.5* = =

Now

=

D((y'x) 2) = D(xy')

by L e m m a

We

we have

(x'y) 2x'x

show

we that

SAITO

xzz'y'

~ yy'xzz'y'

in

E.

y'x 4 E.

We have

^ DE(XX')

= DE(yy'xx').

would

imply

Hence

have

y'x

~ x'y

= y'x

and

Hence

y'x = y'y

so

= yx'xy' = yy'

= yx'

~ xx'

tion.

Hence

=

E.

in

E.

=

=

in

(y'x) (zz') (x'y)

in

E,

whence

in

E.

Moreover

yy'xzz'x'

unipotent, = yzz'x'yy' xzz'y'

xzz'y'

and

we s h o w

E

and y'xzz'

xzz'y'

we have and

yy'xzz'x'

y'xzz'

= yzz'x',

=

E, =

E.

then

(y'x)2zz ' < y'xzz' 5, y'xzz'

(y'y) (y'xzz')

and

in

(y'x)2zz ' ~ y'yzz'

by C o n d i t i o n

yzz'x'

since

< zz'

~ y'yzz' R yzz'x'yy'

D(yy'xzz'x')

= D(z'x'yz)

= xzz'y'xx',

is R-

yzz'x'

= yy'xzz'x'yy'.

Hence

= y(y'xzz'y'x)x'

= y ( y ' x z z ' ) (zz'x'y)y'

= yzz'x'yy' then

~ y'yzz' (y'x)2zz '

= y'xzz'

y'xzz'

=

= xx'xzz'y'

we h a v e

in

yx'

the a s s u m p -

that E,

s yy'xzz'y'xx'

= yy'xzz'x'yy'

Hence

xzz'y'

= D(yzz'x')

xzz'y'

E.

y'y

so

R xzz'y'xx',

R yy'xzz'x'yy'

= D(xzz'y')

in

Hence so

= xy'.

and

in

y'xx'y

~ y'y

contradicts

(y'x) (y'xzz')

y ( y ' y z z ' ) (zz'x'y)y' if

E

then

we have

= xzz'y'xx'

= yy'xzz'x'

xx'

(y'x)2zz ' < y'xzz'

(y'x) (y'xzz') (x'y)

and we

= y(y'yx'xy'y)y'

(y'x)2y'xzz ' ~ y'yy'xzz'

in

and

= y'x

= DE(y'y)

in

we have

x'x = x'y.

which

Next

2.5*,

~ y'xx'y

= yy'yy'

< yy'

(y'x)2zz ' = y ' x z z ' ,

= D(y'xzz')

By L e m m a

(y'x) 2 ~ y'y

(y'x) 3zz ' =

If

= y'y

E,

If

We

~ y'yy'y

similarly

E.

Since

e E.

By w a y of c o n t r a d i c t i o n

similarly

in

xx'

yy'xzz'y' in

and

= xy'

y'x

y'x = y ' x y ' y E.

yy'

xzz'y'

= D(yx')

= DE(Y'X)

Hence

we h a v e

= D(xy')

E.

and

= DE(X'y)

is a c o n t r a d i c -

D(y'x)

in

in

2.14,

since

y'y = y ' y y ' y

= y'yx'xy'y.

which

N e x t we s u p p o s e

and

~ xx'

c E,

DE(XZZ'y'yy'xx' ) = DE(XZZ'y' )

so by L e m m a

we have

x'x

yy'

DE(X'X)

y'x = y ( y y ' x x ' ) x

= yy'xzz'y'.

is R - u n i p o t e n t , similarly assume

so

R y'xy'y

DE(yy' )

DE(XZZ'y' ) = DE(yy'xx' )

= y'xzz'y'x

and

= xx'xzz'y'

we s u p p o s e

(yy'xx') (xzz'y') (yy'xx')

we h a v e

DE(yy'xx')

But

=

and

= y(yy'xzz'y'xx')x tion.

first

DE(XZZ'y' ) = DE(XX'XZZ'y'yy' ) ~

yy'xx'

= yy'xzz'y'xx'

In fact,

xzz'x'

= yzz'x'

in

E.

= xzz'x'xzz'x'

xzz'(x'y)2zz'x ' = x(zz'x'yzz')2x ' = xzz'x'yzz'x' 367

Also

SAITO

= xzz'x'xzz'y'

= xzz'y I = yzz'x'

= yzz'y'xzz'y'

= y(zz'y'xzz')2y

yzz'yy'zz'y'

= yzz'y'

= yzz'y'yzz'x'

' = yzz'(y'x)2zz'y

in

E.

Hence

'

xz ~ yz

in

S

by d e f i n i t i o n . Case: x'x

x ' y 9 E,

~ y'y

D(x'y)

in

E

We t a k e = z'x'yz

i-unipotent,

some

z' 9 V ( z ) .

and

D(z'x'yz)

for

is

z'x'xz

x' 9 V(x) We have

~ z'y'yz

is i - u n i p o t e n t ,

tent.

2.8,

By Lemma

xy'yx' Lemma

we h a v e

= yx'

2.8*,

D(xzz'y') have

and

we h a v e

E.

Hence = xzz'y'

yy'yzz'x' xzz'y'

and

= yzz'x'

in

by

= xx'xx'

in

E.

By

since we

= yzz'x'xzz'y'

similarly

yzz'x'

~ yzz'y'

= xzz'y'yzz'x'

E.

Thus

~ yzz'y'

Also

xx'

and

yzz'x'

S

= xzz'x'xzz'x'

~ xy'yzz'y'yzz'x'

~ yzz'x'

9 E. so

9 E

in

is R - u n i p o -

is R - u n i p o t e n t ,

and

xzz'y'

z'y'xz if

xz ~ yz

~ yy'

yzz'x'

xzz'x'

A l s o we h a v e

= xx'xzz'y'yzz'x'

xy'

and

9 V(y).

9 E,

Hence

yx' and

= D(z'x'yz)

= xzz'y'yzz'x'

2.5*.

xzz'y'yzz'x' in

xy',

xzz'y',

y'

D(z'x'yz)

= xy'yx'

similarly

= D(z'y'xz)

xzz'y'

by L e m m a

we h a v e yx'

E.

then we have

N e x t we s u p p o s e

2.5,

and

z'x'yz

in

by d e f i n i t i o n .

Lemma

y'x = x'y

in

E

= xy'yzz'x' we h a v e and

so

xzz'x' xz

~ yz

in

S

by d e f i n i t i o n . Case: xx'

yx'

9 E,

~ yy' We

x ' y 9 E, = xx'yy'

D(yx')

in

E

take

z'

Lemma

2.15,

we h a v e

x'x = x'xx'x E.

= z'y'xz

and

so

so

2.15, yzz'x'

and

z'y'xz

in

Next

and

E.

2.8*,

xy' 9 E.

Hence

we h a v e

D(z'x'yz)

368

Hence

and Also y'x

by

z'x'yz we h a v e ~ y'y Then

~ z'x'yz xz

~ yz

in

S

is R - u n i p o t e n t .

= z'y'yx'xz

= yzz'y'yzz'x'xzz'x

By L e m m a

is i - u n i p o t e n t . z'x'xz

z'y'xz

y'x, = yx'

= x'y.

similarly

and

9 V(y).

we h a v e

z'x'yz x'yy'x

= yx' y'

and

so

suppose

we h a v e

and

i z'x'yz.

and

D(z'x'yz)

= z'x'yz

xy'

= z'y'xzz'x'yz

= x'y

suppose

~ z'y'yz

z'y'xz,

z'ytxz

and

z'y'xz

by d e f i n i t i o n . By L e m m a

By L e m m a

x'y = x'xy'y

= y'x

~ x'yy'x

First

x' 9 V(x)

Hence

y'xx'y

= z'x'yzz'y'xz

have

e V(z).

= yy'xx'.

we h a v e

in

some

y'x = y'yx'x,

2.5*,

we

for

is R - u n i p o t e n t ,

= z'y'yzz'x'xz

I = yzz'y'xzz'x'

9 E

SAITO

and

similarly

= xx'xzz'x' larly

xzz'y' ~ xx'yzz'x'

xzz'y'

ment

in

the

is

R-unipotent

9 V(x)

and

Hence

have

= D(z'x'yz)

is

LEMMA

Let

only

i_ff

PROOF.

e

efe

in

fe E.

f 9 E,

we

D(ef)

have

= efe Hence

is =

fe. in

we

efe

have

e

suppose then

by

the

and

so

by Lemma

in

S

e

is a c o n t r a d i c t i o n . D(ef)

is

LEMMA

2.20.

For

and

x 9 Q

i__nn S

9 E,

some

x'

in

xzz'x'

D(yzz'x')

xz

s yz

~ f

in

Then

e

have

e e V(e)

~ f

in E

S. if

in

S.

and

only

~ f =

E, f2

definition. in we

E

have

have

e =

f,

E.

The

case

only

x2 ~ x

f which

in a s i m i l a r

if a n d if

in

e2 = e

f < e

in

treated

f

since

e s f

above,

we

s f

x e P

if a n d

If

2.11,

be

x c S,

by

proved

e

can

if

S

and

Then

Also

in

result

Hence

R-unipotent

Hence E.

~ f

were e

xy'

for

or

Since

have

argu-

~ yzz'x'

i-unipotent.

~ ef 2 = ef

Conversely true,

E

simi-

same

yx',

in

xzz'y'

E.

we

f 9 E.

Case:

< yzz'x'

in

and the

i__nn S. e,

we

of

in

< yx'

xzz'y'

xzz'x'

= yzz'x'

obtain

~ yzz'y'

Suppose

[ fe,

then

have

Further

part

we

either

e,

we

xy'

R-unipotent,

~ f

Since

9 V(f).

E.

and

9 V(y),

= xzz'y'

2.19.

in

corresponding

y'

we

yzz'x'

Also

~ yy'yzz'x'

~ yzz'y'

D(yx')

E.

as

9 E.

where

way.

if

x

~ x

2

i__nn S.

PROOF.

We take x' 9 V ( x ) . First suppose x 9 P. If 2 x 9 E, then we have clearly x ~ x in S. We have x 2 x , = x(xx') 9 P and x' x 2 = ( x ' x ) x 9 P. If e i t h e r 2 , x 2 x x 9 PiE or x' 9 PiE, t h e n by L e m m a 2 . 1 2 , w e h a v e 2 x ~ x in S. Now suppose x 9 P\E and x 2 x ', x , x 2 9 E. First x'xx

suppose

x'(xx')x then

we

Hence

Similarly Now

x

D ( x 2 x ') and we

is

is

can

conversely

from

in

= x'2x ~ x2

we

i-unipotent.

follows

s x'x 2

x'x

have

= D ( x 2 x ')

it

= x'x

have

we

where

S.

D ( x 2 x ')

= x ' x 2 9 e,

E.

= x'x 2

in

S

by

R-unipotent, obtain

prove

that

suppose

if x

369

Also and

2.6 if

so

since

that

x'2x x'x

have in

x 9 Q, 2 ~ x in

S

x'2x

= x ' x 2,

= x ' 2 x 2.

definition. we

x s x2

Then

Lemma

In

the

D ( x ' x 2)

case

= D ( x 2)

in

a similar 2 then x ~ x S.

If

x

< x

way. in 2

SAI~/O

in x

c

can

S,

then

by

P.

Also

if

prove

LEMMA S.

that

2.21.

Then

and

ex

exe

Since 2 ~ e =

in

S.

On

S.

xe

the

E P

Hence

then

~ x,

e P

s e

e

in

in

other and

by

by

in

by S.

have

we

Lemma

2.20,

Hence

we

and

have =

we

in 2

x

have xe

~

(xe) 2

~ e

have

e

xe

= x

e x 2

in

= x

~ ex 2 ~ e = e

xe

E P

xe

have

so we

s e

xe

x

and

xe

and

x

E,

Hence

x

have

that

in

2.18.

we

we

such

have

2.20,

~ e,

hand

we

4 Q

Similarly

c Q.

s xe

Lemma

x

c P.

s e

since

xe

since

ex

hand,

we

Also

x

Lemma

2.11,

in

have

E E

e E E

S,

S

S,

2.19.

other 2 ~ x e

x

we x

then

and

~ e

xe

the

x2 x

result,

x 2,

by Lemma

Since

xe

if

Let

so

have

above

x =

x e , e x c E, x e s e, 2 ex . Also we have x

=

PROOF.

the

2

in

S.

and

xexe

~ xe

e E, 2

we

(xe) and

in

xe

c E.

E

by

Lemma

~ xe 2 ~ x

in

S.

e.

By

e

and

so

way

of

On

con-

tradiction we assume xe < x in S. Then we have xe 2 2 x e ~ x in S. On the other hand since x ~ e in S, 2 2 we have x ~ xe in S. Hence we have x = xe < x in =

S,

which

x

~ xe

a

similar

contradicts in

S.

The

S.

2.22. Then

and

ex

LEMMA

so

Lemma in

We

By

Lemma

2.20, and e

in

and

Case:

S

we so

P\E, so

we

assume 2 (x'y) e

By

by

Lemma

for

ex

can

2.20.

Hence

be

proved

that

e

in

x

prove

c Q

and

way

of

relation

take

x'yy'z,

S

can

x, y ,

S.

zy'yx'

lows

The

Suppose in

E V(z). and

we Let

2.23.

z

e P

e E

E

such

x e , e x c E, e ~ xe, e ~ ex in 2 ex . Also we have xe s x and

=

PROOF.

x

way.

Similarly LEMMA

that assertion

2.10,

have by

is S

we

such

have If

we

have

x

< z

in

zy'yx'

x

~ V(y)

x'y,

y'z,

=

=

x

=

x

in

and

and yx',

e S.

Thus

y

z' zy'

c P

then

by

< x'yy'zx'yy'z

< z

zy'yx'

S.

s y

e PiE,

x'yy'z

have

zy'yx'xy'yx'

x'yy'z,

that y'

x'yy'z

x'yy'xx'yy'z 2.18,

xe ~ x

in 2

transitive.

e V(x),

e P.

Lemma

then we

z e x'

zy'yx'

~

E, ex

~ x

in

S.

If

< zy'yx'zy'yx' in

what

fol-

~ E.

P\E. contradiction

we

370

assume

y'z

~ x'y

in

S.

SAITO

Since

y ~ z

S

by L e m m a

=

and

in

hand

(x'y)2y'z

~

=

(x'y)

4 E.

= x'yy'y

x'y = x'yy'y

we h a v e

since

y'z in

(x'y) 4

( x ' y ) 2 = ( x ' y ) 3, 2

have

(x'y) 3 ~

(x'y) (x'yy'z)

(x'y) 3

(x'y) 3y'z =

we

2.21,

(x'y)2(x'yy'z)

the o t h e r

=

S,

=

~ x'y

S. and

in

Hence

S,

we h a v e

we

in

S,

On

we h a v e (x'y)2y'z

2, we h a v e

the

< y'z

S.

assumption in

S.

Since

x'y

f r o m L e m m a 2.21 2 = x'yy'z(x'y) 2 But in S and s i n c e (x'y) 4 E 2 2 and x ' y y ' z x ' y 9 E, we h a v e x'yy'x(x'y) = (x'y) 2 < x'yy'zx'y = x'yy'z(x'y) in S. H e n c e by L e m m a 2.18, that

s x'yy'z

x'y

in

(x'y) 3 =

by C o n d i t i o n

in

(x'y) 3y'z

(x'y)2y'z

which contradicts

Hence

~ x'yy'z

it f o l l o w s

x'yy'zx'y 9 E and 2 (x'y) ~ x'yy'zx'y

have

x < z

in

x'yy'zx'y

S.

We c a n o b t a i n x < z in S 2 2 (y'z) 9 PiE, C a s e : (yx') 9 PiE

similarly

in Case: 2 Case: (zy')

and

9 PiE. Case:

(x'y) We

x'yy'x

and x ' y e P\E. 2 3 x ' y y ' x x ' y = x'y < (x'y) = (x'y) and so 2 (x'y) in S. A l s o we h a v e x'y = x ' y y ' y 2 in S. Since x ' y < (x'y) in S, we h a v e 2 (x'y) y'z in S. O n the o t h e r h a n d we h a v e

<

~

(x'y)2y'z in =

S.

=

and Case:

x'yy'z

~

=

~

(x'yy'z) (x'yy'z)

(x'y)2y'z

< x'yy'z

in

S.

Hence

x'y,

y'z,

by L e m m a

yx',

2.10,

9 Q.

Since

Hence

by L e m m a

x'z other

so

we h a v e

= x'yy'z (x'y) 2

Hence in

by L e m m a

E

x < z

and in

so S.

We can o b t a i n x < z in S s i m i l a r l y in Case: 2 2 9 E and y'z c PiE, C a s e : (yx') e E and yx' 9 PiE 2 Case: (zy') 9 E and zy' e PiE. zy'

9 E.

By w a y of c o n t r a d i c t i o n Then

and

(x'y) 2y'z = x ' y y ' z in S. 2 x ' y y ' x < (x'y) ~ x'yy'z

we h a v e

x'yy'x

(y'z)

(x'y) (x'yy'z)

Hence

(x'y)2y'y

2.19,

e E

have

x'yy'z x'yy'z

2

in hand

y ~ z

S.

2.22,

since

we h a v e in

Hence

we a s s u m e xz',

S,

we

x ~ y

have

in

z'x c P

we h a v e x'yx'z

x'yx'zz'x S,

371

z < x and

x'y

~ x'z

e E

and

~ x'zz'x we h a v e

in

S.

zx', in

x'z S.

x'yx'z

in

S.

On

x'x

~ x'y

the and

SAITO

so

x'zz'x

have =

=

(x'x) (x'zz'x)

x'zz'x

= x'yx'zz'x

(x'yx'zz'x)(x'z)

e E.

First

x ~ y, x'y

~ x'z,

in

S

and

and y'x

= x'z

and

xz'

in

y'x

we h a v e

2.5,

and

and

= zz'

so and

so

Then

D(yy')

x ~ z

in

original

= D(y)

~ x'z

~ y'x ~ x'x

= D(y'y)

is

~ zz'

~ xz'

= xz'z

= D(y'y)

in

Hence and

zz'

E.

Hence

= zz'z

= z,

D(y'y)

is R-

is R - u n i p o t e n t

in a s i m i l a r

way.

Hence

S.

2.2,

2.11,

2.13,

2.18,

2.19,

2.20 a n d

2.23

B.

Acknowledgement. to the

yz'

Next s u p p o s e

a contradiction

y'z

x'x = x'y

y ' y = z'z.

~ zz'

x = xx'x

~ x'y,

y'y = y'x = x'yy'x

s yz',

~ yz'

~ x'y

zx'

Since

s y'z,

D(y'x)

similarly xz'

xz'

we h a v e

x'x

Hence

Hence

Theorem

thanks

x'x

E.

by L e m m a

Now L e m m a s prove

y'y

= y ' y = y'z.

and we c a n o b t a i n we h a v e

in

H e n c e we

(x'zz'x) (x'z)

we h a v e

we h a v e

is a c o n t r a d i c t i o n .

unipotent.

S,

s y'y,

~ y'x

we h a v e

S.

Similarly

and

S

= yz'

which

x'z =

y'x

Also

in

is L - u n i p o t e n t . in

2.19,

~ y'z

= x'y = x'x

x ' x = z'z. xz'

~ x'x,

so

e E.

D(y'y) z < x

by L e m m a

s y'y

L-unipotent = x'yy'y

and

x'z

and

= x'yx'z

suppose

y s z

~ x'yx'zz'x

I would

referee

version

of

for

his

like

to e x p r e s s

suggestions

this p a p e r .

372

given

my h e a r t y for

the

SAITO

REFERENCES i. Clifford, A. H. and G. B. Preston, The algebraic theory of semigroups I, Amer. Math. Soc. 1961. 2. Reilly, N. R. and H. S. Scheiblich, Congruences o nn regular semigroups, Pacific J. Math.

23(1967),

349-360.

3. SaitS, T. Ordered idempotent semigroups, J. Math. Soc. Japan 14(1962),

150-169.

4. SaitS, T. Regular elements in an ordered semigroup, Pacific J. Math. 13(1963),

263-295.

5. Sait6, T. Ordered regular proper semigroups, J. Algebra 8(1968), 450-477. 6. Sait6, T. The orderability of idempotent semigroups, Semigroup Forum 7(1974),

264-285.

Department of Mathematics Nippon Institute of Technology Miyashiro,

Saitama

Japan

Received 20, November 1981 and, in final form, October 26, 1982.

373