Bisimple type a -semigroups-I

Semigroup Forum Voh 31 (1985)9%117 9 1985 Springer-Verlag New York Inc.

RESEARCH ARTICLE

* - BISIMPLE TYPE A m-SEMIGROUPS

- I

U. Asibong-Ibe Communicated by D. B. McAlister

i.

Introduction In

[5]

certain

semilattice tent

and

of

principally

idempotents

cancellable

inverse

to M c A l i s t e r ' s

and

(see

obtained and

their

other

Munn

A

[13])

This

inverse

semigroups.

inverse

semigroups

groups

well (see

studied The

over

the

*-bisimple

idempotent

separating

congruence

to

those

semigroups is

close

generated

interesting

subclass

structural

two

of

theorems

general.

paper

type ~* ~ ~

A

This

is

generally distinct

a

lot

the

of

and called from

with

research

direction for

of

inverse

[15],

[16],

(see the semi-

[18]),

decades.

A

m-semigroup

type

A

for

is v e r y

studies in w h i c h

is

in

[2].

form

semigroup

type

~--semigroup obtained

[8]

relationship

[13],

characterisation

of h o m o r o p h L s m

Howie

theories

[12],

type

of

essentially

structure

past

semigroups.

and

of

[ii],

inverse

conditions

Their

known [9],

an

defining

[I] , [5] , [6] , [7] , [i0]) , a l o n g already

in

analogous

internal

and

has

for

results

a

idempo-

certain

the

class

with

as

embeddings

other

[6] , u s e d

analogous

semigroups,

special

some

in

maximum

results

[16].

type

as

with

with

monoids

characterised

[12],

investigation

characterisations

projective

were

monoids,

semigroup,

Further

Preliminaries

the ~*

A

a v~ry and

useful

in

their the

m-semigroups

structu~-e = D.

of

The

study in

*-bisimple case

where

ASIBONG-IBE

Let all

x,

are

said

S be

y e S, to

if xb If

sation LEMMA

ax = ay

be

Dually, only

a semlgroup.

L*-

aR*b

S has

an

Let

the

(i)

e L* a

(ii)

ae

for

is

is

x,

y e S,

that

= by.

written

e,

for

Then

a,b

a L*b.

xa

= ya

if and

idempotent

e in

is

left

each

S such

e a~

x , y g S I, ax = ay condition

A

that

a in

a is

semigroup

caneellable

for

it

be

Right S is

in S.

ex = ey. R*.

(ii)

of

e-caneella-

called

S there

left

S can if

implies

holds

semigroup

element

idempotent

condition

e-cancellable.

notion.

cancellable

A semigroup

:

all

eharacteri-

[17] )

ar___e e q u i v a l e n t

similar

for

following

and

called

if

[14],

a ~ S satisfying

a dual

cancellable

~

the

a semigroup,

for

element

I.I

if bx

and

all

idempotent

following

B__y d u a l i t y ,

risht

if

S be

= a and

An

only

equivalent

is known (see [ 6 ] ,

Then

hility

if a n d

a, b e S such

= yb.

i.i

Lemma

Let

is

left

an

e-eaneellable. defined

is b o t h

A

similarly.

left

and

right

cancellable, Let shows

S he

that

in

a right

an

idcmpotent.

each

The

join

xl,

semigroup

contains

of

D*.

an

(see and

general

x 2 , . . . , X 2 n _ I in

each

in w h i c h

Lemma

idempotent. R*-elass each is

contains

L*-class

called

i.I Dually

and

an

[71).

In

and

an

idempotent

L*

a D * b if

semigroup.

contains

A semigroup

semi6roup,

equivalence Basically,

cancellable

L*-elass

cancellable

R*-class

abundant

a left

each

R*

equivalences L*oR*

only

S such

if t h e r e

that

on

# R*oL*

S, (see

exist

is the [6]).

elements

a L * x I R* x2,...,x2n~ I R*b.

The intersection of L* and R* is the equivalence relation H*. Thus

a H*bif

and

only

if

a L*b

and

a R*b.

Let

H*

be

H*-class in a semigroups S with e e H*, where e is an idempotent

100

an

ASIBONG-IBE

in

S.

In

[7],

it

is

shown

that

H*

is

a cancellative

monoid. Denote

by

L the

R,

respectively,

nces ~ ~

D*,

and

R

H~H =

on

~.

If

left

S.

S is

and

right

Evidently

Green's

L ~L*

a regular

semigroup,

then

The

following an

notation

element

will

a e S will

be be

used.

An

denoted

by

La and

Let potents. if

Ra

a semigroup

with

Then

S is

a type

(i)

S is

(ii)

for

In

view

condition

(i)

be

=

L*-class

of

by

(enle

For

~q

in

if

and

A

if

semigroup

a E

S, t h e n

the

remark

requirement

A

an

ea

idem-

and

only

= a(ea)*,

e

that

of t y p e

that

each

followed,

A

sem~roup

L*-class

and

idempotent.

semigroup

with

a semilattice

called

type

a type

A ~-semigroup

only

if

A ~-semigroup

if

of

E is

m _< n.

A 00-semigroup

containing

='{a

E of

E N} and

a type

idempotent L

the

S is

=

e m _> e n

ele-

semilattice

definition

S contains

Thus

E

i.i

above

a type

E.

a

e e E and

Lemma

of

S be

a*

(ae)+a.

an w - c h a i n .

L* - c l a s s

m

an .

E S[a

S,

idempotent

That

let en,

L* d e n o t e t h e n a n d R m* t h e R*-class

is

L * e n } , R* m

=

(a ~ S I a

R*e m}

so t h a t H* If H* m,n

,

cancellable

of the

R*-class

idempotents

called

every

replaced

Let

with

i

respectively.

S be

ae

and

iv=

R*a is an R * - c l a s s w i t h an e l e m e n t a + w i l l be u s e d to d e n o t e typical idempotent

ments in

each

so

L*. a a E S.

Similarly

can

and

R.

containing

and

congrue-

R ~ R*

(a I a i * en ' a R * e m } =

m,n

=

# r

evidently

H* m,n

is

101

an

L* n

R* m

H~-class

of

S.

Also,

ASIBONG-IBE

observe

that

a

= e

and

a

= e

m

Define ~a

n

mappings

the

: a+E + a ' E ,

8a

: a*E ~

a+E

by ea a =

(ea)*

f8 a =

Then

a a and

B a are

each

a s S,

(see

Let of

S be

the

there

If

LEMMA

a type

A semigroup

1.2

If

(i)

~o~

as

and

E its

semilattice

semigroup

of E,

: S + T E whose

kernel

congruence

then

interesting

an

inverse

~ on

is

S;

so

is

S/B.

connection

semigroup,

The between

see

[6],

well.

S is__ a t y p e

A semigroup

with

_a s e m i l a t t i c e

, then

a a and

8 a are

each

a e S

and

(ii)

the

mapping

homomorphism

for

[9]).

an

and

results

~

A semigroup,

shows

E o_._f i d e m p o t e n t s

isomorphisms,

Munn's

separating

of H o w i e

lemma

other

semigroup

If T E is t h e

4.9

S is

a type

for

inverse

idempotent

Lemma

following

for

an

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

maximum

(see

inverse

~])~

idempotents.

then

(af) +

mutually

onto

mutuall~

inverse

~ defined

a full

o__nn S b y

subsemi~roup

isomorphisms.

~

: a § aa

of T E

is

a

and

-I =

~

.

Using lemma,

the

the

above

proof

lemma

of which

we we

obtain omit,

the

(see

following Ill). 4@

LEMMA

1.3

t = max

If u eH~,n,V_ e H p , q ;

S be

a type

~ is

defined

a ~) b if a n d and

u v e H ~', _ n +-t , q _ p + t

(n,p).

Let relation

then

only

if

A semigroup, on a*~

and

let

a,b,c

S.

The

S by b*,

a+D

b + for

some

a*,

b*,

b+ . I)

inclusion

is

an e q u i v a l e n c e I)--~_~D*

on

relation

a type

102

A

and

semigroup

satisfies S

(see

the Ill),

a+

ASIBONG-IBE

If

S is

if

and

we

study

and

only

2.

it

by

in

an

D*

Consider contains

To

show

let

t = max

and

ts=

combine and

in w h i c h

=

let

the

M.

-n

t -p

0

endomorphism

is

an

with

an

y@

+ max(q,

(p,y,q),

r)).

The

semigroup, =

so

Let

(m,

x,

which then an

case

the

m = t

certainly

well

Reilly

and

For

by

~)

only

fact,

above

the

will of

be

~, =

if m of

r),

elements

t 3 = tl+

of

s _ r + tl )

=

(0,

in

i,

S.

(m,x,m).

M then extension

M determined

(m

x

of

M2

n)

if

Thus e is

S = N x

semigroup

x,

sake

of

clarity,

the

I03

n)

Then =

Conversely

= n and

called

0)(m,

in

x2 = x

(m,x,n)

HI x N (see

is

[.18] ).

S = N x M x N

the by

generalised 0,

and

notation

is

an idempotent.

will

Bruckbe

S = BP*(N,O).

the

S

p - q

s - r + t 2)

n-m+t)

x 2 = x.

Bruck-Reilly

extension

denoted

x,

idempotent

a~d

a subgroup

this

constructed

(m,

(m,x,m)(m,x,m

known of

an

= n,

if

H I is

In v i e w

be

=

g S;

max(q,

(r, z, s))

= (m-n+t, x 0 t - n . x @ t ' m

idempotent If

n)

n)

s)

0).

s-p+q-p+t 3)

yOt~-q-zo t2-r,

((p,. y, q)

z,

l,

~m-n+t~x@t-ny@t-p,q-p+t)(r,z,s)

(m,x,n)

(p-q+tz,

M with

(0,

outer

=

x,

which

+ t)

of

(r,

that

(m_n+ts,xOt3-n.yot3-P.zot3-q+p-r,

O)(m,

,q-p

max(q-p+max(n,p)),r),t2=

bicyclic

(m,x,n)(m,x,n)

from

S= N x M x N

identity

=

i,

set

x@ t

= (m-n+p-q+tl, (xOt-ny@t-p)@tl-q+p-tzOt'1-r'

(0,

class

H*-class

The

(m-n+t,

((m,x,n)(p,y,q))(r,z,s)

Also

D*=

a distinct

H I as of

(m,x,n),

(n,p),tz=

= (m, x, n)

follows

by

a semigroup

this,

in

is

*-bisimpie what

MONOIDS

M with

and

max(n,p-q as

REILLY

defined

H I* is

In

this

element

x,n)(p,y,q)

in

S is

A ~-semigroup

that

BRUCK

t = max(n,p)

images

then

O*~class.

r ~.

identity

operation

where

type

a monoid

the

(m,

a single

example

which

GENERALISED

with

A ~o-semigroup,

has

.-bisimple

show

those

a type if

[*(S)

or

ASIBONG-IBE

R*(S)

will

when

be used

more

than

2.1

Let

LEMMA

extension are

denote

S = BR*(M,O)

in S.

(i)

L ~ or

semigroups

of a m o n o i d

elements

and

to

one

M.

R*-equivalence

on S,

is c o n s i d e r e d . be a ~ e n e r a l i s e d

Suppose

that

Bruck-Reill 2

(m,x,n)

an d

(p,y,q)

TheB

(m,x,n)

L*(S)(p,y,q)

i_f and

only

i_f n = q

x L' (M)y. (ii)

(m,x,n)

x

PROOF.

If

(m,x,n),

(m,x,n)L'(S) of

(p,y,q)

(p,y,q), (m,x,n),

i_f and

0nly

i__f m = p

elements (n,l,n)

by d e f i n i t i o n = (p, y,

in S such

that

is a right

of L , and consequently

q)

is e q u i v a l e n t (p-q+t,

whence

y0t-q,t)

t = q and thus Similarly,

deduce

are

evidently

(p,y,q)(n,l,n) or what

(p,y,q)

R* (M)y.

and

identity

R~(S)

from

= (p,y,q), n < q.

using

(m,x,n)

t = max(n,q)

the

(q,l,q)

idempotent = (m,x,n)

(q, l, that

q~n

q) we so that

n -- q. Let

x, y g M.

ax = ay which

holds

that

is,

and

assume

then

from

that

t = max

coordinates

If

and

x

(m,a,n)(n,x,n)--(m,a,n)!n,y,n)

we let

L* (M)y.

(n,h),

gives

if

(p,b,n)(n,x,n)

if bx = by.

(h,a,k)

(xG t-n)

if

converse,

follows that (m-n+t,(xq t-n)(a

where

only

only

if and only

(m,x,n~ it

if and

if and

For the

Then

If

=

Hence

= x

(p,b,n) (n,y,n)

L* (M)y.

(m,x,n),

(p,y,q)E

(h,a,k)~ (q,b,r)

(m,x,n)

are

S, in S,

(q,b,r)

t-h),k_h+t)=(m_n+s,(xa s-n)(bU s-q ),r-q+s) s = max

t = s;

(aG t-h)

(n,q).

Comparing

thus = (xG t-n)

t = n, this gives x(aG t-h) = x ( b o t-q) since x L" y we deduce that y(a(~ t-h) = y ( b a t-q)

194

(bat-q).

the

first

ASIBONG-IBE

that

is, (y t-n)

If

(aqt-h)=

(yut-n)

t < n, then x o t - n g a g t-h = bu t

H~,

(but-q)

and hence

Thus (yg t-n ) (a(~t-h)

=

(yu t-n)

(bu t-q )

Hence,

( p - n + t , ( y q t - n ) (act t - h ) that

# - h + t ) = (p-n+s, (yu s-n ) (b~s-q) ,r-q+s)

is

(p,

y,

A similar

n)

(h,

argument

a,

k)

shows

= (p,

y,

n)

(q,

b,

r).

that

y,

n)

(h,

a, k)

=

(p, y, n) (q, h, r)

(m, x ,

n)

(h,

a,

=

(m, x ,

(m, x ,

n)

i*(S)

(p, implies

k)

n)

(q,

b,

r).

Hence

The

proof

to

LEMMA 2 . 2 (p,y,q)e

An

y,

is s i m i l a r

element

n) to that

(m,x,n)

(i). an inverse

inverse

of x i__nnM,

q -- m.

Let

(p,y,q).

of

i_nn S has

S i_~f and onl F i~f y is the

and p = n, PROOF.

(ii)

(p,

(m,x,n)e

Both

S and

(m,x,n)

suppose

and

that

(p,y,q)

(m,x,n)L(S)(p~y,q)(m,x,n)

and

are

its

inverse

regular,

is

so

(p,~q)(m,x,n)R(S)(p,y,q)

also (m,x,n)(p,y,q)L(S)(p,y,q),

(m,x,n)(p,y,q)R(S)

(m,x,n).

Consequently (m-n+t,

(xGt-n)(yut-P),

(m-n+t,

(xG t-n)

q-p+t)

L (S)

(p, y,

q)

and

where case

t = max(n,p). and t = n in the

m = q.

Thus

(p,

(m, x, n)

y,

(yat-P), By Lemma

latter, q) =

q-p+t) 2.1,

so that

(n, y, m)

(n, y, m)

(m,

105

x, n)

R(S)

(m, x, n)

t = p in the p = n. and :

former

Similarly

so (m, x, n)

ASIBONG-IBE

and (n, that

is,

y, m)

(m, xyx,

Therefore,

xyx

Con versely~ xzx that

(m,x,n)

This

completes Let

=

=

(m,x,n)

x,

and

and

yxy

the

L -class

must

proof

=

then

of the

a semilattice

= z,

other.

an a d e q u a t e

E(S)

of'

semigroup

if

of S c o n t a i n

@) be a g e n e r a l i s e d

of M d e t e r m i n e d

zxz

it f o l l o w s

lemma.

each E*-class BR*(M,

zeM,

inverse

with

S is c a l l e d

and Let

be

= (n,y,m).

y.

calculation

to the

a semigroup

Then

(n,yxy,m)

an i n v e r s e

(n,z,m)

S be

idempotents.

(n, y, m) = (n, y, m)

a straightforward

and

idempotents.

extension

n)

if x eM has

= x and by

each

(m, x, n)

unique Bruck-Reilly

by an e n d o m o r p h i s m

0 on M.

Then LEMMA

2.3

BR*(M,@)

is an a d e q u a t e

semigroup

if and o n l y

i__f M is a d e q u a t e . PROOF:

Suppose

If

x e M then

L~-related m = 0 and

to

that

S = BR*(M,@)

(0,

an i d e m p o t e n t

e L~x.

Thus

is a d e q u a t e .

x, O) g S and (m,e,m)

each

is t h e r e f o r e in S.

L~-class

By L e m m a

contains

2.1,

an idempo-

ten Dually,

each

L*-class

of

M contains

an i d e m -

pot eilt 9

(O,f,O)

Let

e,

are

idempotents

f be

(0, ef,

O) =

idempotents

in

M.

Then

(O,e,O)

and

in S, m o r e o v e r ,

(O,e,O)(O,f,O)

= (O,f,O)(O,e,O)=

(O,fe,O) hence M

e = re.

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

of

commute

showing

that

is a d e q u a t e . Conversely,

each

L'-class

idempotent (m,e,m), e,f

if M is a d e q u a t e ,

and e a c h

element. (n,f,n)

in

R*-elass

Also,

for

BR*(M,@)

then

of B R * ( M , O ) any two

106

Lemma

contain

2.1 an

idempotents

corresponding

inM,

by

to idempotents

ASIBONG-IBE

(m,e,m)(n,f,n)

= (t,eot-m.fot-n,t)

=(n,f,~(m,e,m)

since

e@ = fO = i, LEMMA

2.4:

and

BR*(M,0)

M is a type

is a

Let

monoid.

Then

us

suppose

that

S is a d e q u a t e

Thus

corresponding

for

to

(0, ea,

2.1, O) =

Similarly

a, e e M,

then

(0,

form M.

Thus,

for

if

O)

An

some

(p, x,

lemma M

(0,e,O)

A

is also

in S =

A, then

idempotent

= (t,

so that

ea = a ( e a ) *

showing

q) g B R * ( M , 0 )

that

M is t y p e A .

from L e m m a

of B R * ( M , O )

2.3

has the

e is an i d e m p o t e n t

in

then

(e0t-m)(x@t-P),

it follows

e,m)(p,x,q))*=

is a type

= (0,e,O)(O,a,0)

Hence

m s N, w h e r e

and

0)

= (O,(ea)*,O),

if M is type

t = max(re, p) ((m,

if

(O,a,O)(O,ea,O)*.

ae = (ae)+a,

(m,e,m)(p,x,q) where

i_ff and only

above

(0,a,O),

ea,O)*

is a d e q u a t e .

(m,e,m)

and by

(0,ea,O)

:

(0, a ( e a ) *

we o b t a i n

Conversely, BR*(M,G)

A monoid

S = BR*(M,

elements

(O,a,O)((O,e,O)(O,a,O))* By L e m m a

type

A monoid.

PROOF

adequate.

ef = fe.

from

q-p+t)

Lemma

2.1 that

(q-p+t,(e@t-m)(xot-P)*,q-p+t)

hence (p,x,q)((m,e,m)(p,x,q))* = (t,(xgt-p)((e@ t-m)(x0 t-p))*,q-p+t) If

t >

since

m, t h e n

eO t - m

M is t y p e

BR*(M,0) Thus

if t = m t h e n

(x0t-P))*= (x0 t-p)

argument

shows

that

= e, and

BR*(M,@)

is a t y p e

Let

a cancellative

M be

Bruck-Reilly

(xet-P)) *

((e0 t-m)

if

and u is an a r b i t r a r y

generalised

eG t - m

A, we have

((eO t-m) A similar

= 1 and

f is an

element,

idempotent then

uf =

of (uf)*u.

A semigroup. monoid;

extension

and

BR*(M,@)

of M, w h e r e

0:M+

the ,

HI,

,

and H I

is the

THEOREM

2.3

group PROOF

of M c o n t a i n i n g

S = BR*(M,@)

such that Let

H~-class

~(S)

=

u = (m,x,n),

an

is a .- b l s i m p l e

identity type

of M.

A ~0-semi-

~(S). v=(p,y,q),

107

w = (r,z,s)g

S such

ASIBONG-IBE

t hat (m,x,n) Then

(p,y,q)

(m-n+t,

=

(m,x,n)

(r,z,s).

x@t-ny@t-p,q-p+t) = (m-n+tl,x0tl-nz0tl-r,s-r+tl),

t = max ( n , p ) ,

t I = max(n,r),

m-n+tl,

that

is t I = t,

(iii)

x0t-n.y0 t-p

in w h i c h

so t h a t

= x0 t-n

(ii)

z0 t - r .

case q-p

(i) m - n + t

= s-r

=

and

If t = p t h e n

p = r,

thus x@ p - n so t h a t

y = x0P-nz,

cancellatlvity

in M f o r c e s

If t = n t h e n z@ n-r,

similarly.

x.y0 n-p

= x.z0 n-r

(p,y,q)

=

=

and

uv uf

= uw

implies Thus

= u.

that

from

let

(m,e,m),

Lemma

2.1

=

it

e@ t - n 9 z@t -r , s - r + t e,

n)

fv = fw,

(r,

z,

where

q-p+t) 1)

s).

f =

(n,e,n),

cancellable. it

can

be

shown

Since"

S = BR*(M,@)

example

if

fm =

f

f

that

S is

m n follows that To

has

of

the

S.

Let

(q-p+t,

a semilattiee fn =

e,

A

theorem fm =

of

(n,e,n)

(n,e,n)

is a t y p e

complete

idempotents

q-p+t), t=max(m,p),

q-p+t)

( p , x , q ) ((m,e ,m) ( p , x , q ) ) *

(m,e,m) S

Then

that

= (p,x,q)

(m,e,m), =

(p,x,q)gS.

(t,x0t-mxe t-p,

=

=

idempotents

for

then

(t,e,t)

= f

n

f

,

m

semigroup. let

E(S)

(m,e,m),

be

fn =

the

set

(n,e,n)e

of E(S).

if m > n f

Thus

m

f is

f

=

n

--

(m,

e, m)

if a n d

< f m

E(S)

(tl,

follows

(m,e,m)(p,x,q)

Then

y@n-P--

eot-n.y8 t-p,

argument

(m,e,m)(p,x,q)

it

hence

cancellable. Next

and

and

(t,

(n,

S is l e f t

By a s i m i l a r right

y = z.

Therefore

(n e, n)

Thus

equality

=

f

m

=

if m ~ n ;

only

f

n

f

m

which

shows

that

n

a chain

(o,

e, o)

>

(i,

e, I)

108

> (2,

e,

2)

>

. . . . . .

ASIBONG-IBE

Finally,

observing

that

H*

= {(m,x,n)IxgM}

m,n

r162

then it is evident that every pair of idempotents fm" fn in S are D* e q u i v a l e n t . elements

(m,x,n),

theorem

of the

(p,y,q)

are

Q be the

by Lemma Q =

set

where

each

the

pair

proof

of

of the

then

THE

~ G~M,

of units

in M, m, n g N}

m, n e N }

in M.

Q is a b i s i m p l e

STRUCTURE

PROPOSITION

in B R * ( M , @ ) .

3.2

G is a group

following

elements

I ( m , x , n ) Ix is a unit

[18],

3.

that

D related,

of r e g u l a r

{(m,x,n),x

From

fact

is c o m p l e t e d . Let

Then

In v i e w

3.1

(ii)

every

=

~

inverse

S be a type

conditions

D

E(S)

= E(Q).

m-semigroup.

THEOREM

Let

(i)

Also

are

A semigroup.

The

equivalent

,

nonempty

H*-class

contains

a regular

element. PROOF. that

Suppose

La** N

E*a+C~H~

Therefore If are

that

a* D* a +

so that

(ii) (ii)

elements

(i) holds.

implies

a*D

Then

a+

for ae S, we have

hence

H*a c o n t a i n s

a* D a +.

a regular

Thus

element

holds. holds,

xl,

let

a,h

e S such

x 2 , . . . , x n in S

such

that

aD*b.

There

that

a = x I L*x 2 R* x 3 L l*, . . . X n _ 1 R * x n = b. For ei, (i = l , . . . , n ) , r e g u l a r e l e m e n t s in H*x. it is well known

that

l

c l , L c 2 Rc~ L . . . O n _ I R c which

shows

that

therefore

a~b.

L EEMMA 3.2

Let

that

9' = ~.

c I D c n. Hence

(ii)

n

Since

(a,cl)eH*,

implies

S be a . - b i s i m p l e Then

P~ = t*o.R*

109

type =

(b,Cn)e

H

(i). A s e m i g r o u p such

~*ot*.

ASIBONG-IBE

.'b

PROOF A

Let

D*(S)

semigroup.

a,b c2

e

R*g~L~ a (a,b)e~*oi*.

e

by

exist # qb

where

c I,

which

Hence

S

is

proposition "

c2

such

show

a

.-bisimple

3.1 that

type

for

each

*

a* ( %

c.

eL

l~air *

Rb

~ r

that

(a,b)ei'oR ~ whenever = 0: M*oi ~ This completes t h e

i~oR ~ =

9 Let

respect S,

D(S)

Then

S there

lemma

=

S be

to

a

which

.-bisimple 0 ~=

D.

type

If

E

A

is

a

~-semigroup set

of

with

idempotents

of

then

E is

an

is

the

=

{e

e-chain

m

with

> e m-n element 9

identity The

e

notations

adopted.

Let

Assuming

Im~N}

of

u be

an

inductively u

m+ l

Hence,

for By

element

all

e

Let

if

n

< m, --

and

previous

sections

will

of

H~-class

H* o el

the

. u m Clio,m,

positive

H* 9 0,I

only

the

H*' .H * o,m o,i

proposition

in

and

element that

um.u

=

if

o

be

then

H* o,m+ I

c..

-

integers

n

3.1

then

there

a

H* 0,I

be

g

e

then

uneH * o~n

exists

such

a

a

regular

regular

element.

Then a a

e H 1., o

-x

-m m e H* . a = em m,m

~ ameH o , m , a - m e ~ , o Define

a~

=

a m .a -m = eo cH*o,o and

so that

e0 .

Define

a mapping xg

If so u o

=

a-mx I~ n that e

~: H* o,o -m n a

=

xI =

e~

H* m,n m,n following

e

N.

, m, n e N,

m,m

by

xa

a-mx2a

n then

x I e~

=

H9 then a mua -n e m,n is b o t h one-one and However,

> H

S

is

Thus,

a m -am x ~ a n a-n

eo x 2 e~

=

x 2.

H* 9 H* o,m m,n onto 9 precisely the

n,o

union

observation

:

110

a m -am x 2 a n a-n Also

H*

a

=

with

~-- H * . Therefore o,o

of above

the

H*-class

proves

the

ASIBONG-IBE

LEMMA

3.3

Let

a be

every

element

form

a -m xa n ,

ax

g

Next, let H* H* ~ o,i o,o

ax

can

Now

be

let

of

be

where

ax

=

a

regular

S admits

a

m,

n

M = H* H* o,o o,I' so

uniquely

@

a

unique ~

61, x

that

by as

defined

in

H*

Then

O~ 1

"

representation and..

and

expressed

mapping

element

s

x

M.

the

a~

E

Then

M

the

clearly

preceeding = ya,

on

in

H *o , o "

for

lemma some

yEM.

by

(xO)a.

Then

(x I x 2)0 since

a

m

a

-m

=

a = a ( x l x a) = a x l x 2=(xIO)ax2=(xIO)(x20)a e

(x I x 2 ) O Thus

@

From

xa-n n-n

a a

is

an

amx

=

n a x

=

we

O

obtain

= (xI0).(x20) endomorphism.

a m - 1 .ax xon an

=

Also = am-2(xOZ)a2=.

am-l(xO)a

a-n(xon)ana

e

S.

the

o

Let with

identity

us

..=(x@m)a m.

we also obtain

= en xa -n =a-n(anx)a n =

=

and

suppose

of that

u

=

a

-m

n

xa

n

= an(xGn),

, u

=

a

since

-Pyaq

x,yeH* 0,0

If n ~ P

then uv

If

=

a - m x a n .a -p

=

a-mx(y@

however, uv

Now Then

define

is

one

p

~(m-n+P)(x@P-n).yaq

- one

3.4

The

D~ =

(uv)~ and all

aq

=

(a-mxan)o =(uG)(va)

a-m.xa(P-n)yaq

= holds

(m,

x,

n)

from above.

Moreover

onto. the

facts

Let

S be

a

9.

The~

S

example

a - m x. (y@n-P) a q - p +n

then

=

a mapping

that

an-Pa q =

n-p)

amxana-Py

together

TH__EOREMS

<

a -m xa n-p ya q

=

=

obviously

Taking

Such

n

Ya q

below

above

.-bisimple ~

yields type

the A

following

~-semigroup

BR*(M,@).

illustrates

111

existence

of

a.-bl-

ASIBONG-IBE

simple

type

EXAMPLE. of T.

A ~-semigroup

Let

Let

where

t

both

(p, y,

=

max(n,p). product

m E n(mod

p ~ q(mod or one

(mod

2),

Thus

by the

u = (m,x,n)

2) or b o t h

case

ideal

under

m)lm

:

easily

ue = (ue)+u.

Thus

S

.- b l s l m p l e .

However

xy law

=

operation.

~q-p+t

only

if

e S

such that

x, n) =

(r,

n ~ m

z,

xz w h i c h

in T.

.

s)

yields

Hence

(n, l, n)

['(n,

(m, x, n ) R * ( m ,

we

the

(m,

we h a v e

an i d e m p o t e n t

2),

e N } if and

(m, x, n)

2),

in r e s p e c t

and m-n+t

(r,z,s)

q)

E q-p+t(mod

xyeI

(p,y,q),

(p, y,

m ~ n(mod

2) h o l d s

(n, "i, n)

q)

q - p + t)

is in S w h e t h e r

n-n+t

p ~ q(mod

cancellation

consequently

Taking

I, an

n(mod 2),m,neN},

(m - n + t, xy,

p ~q(mod

(p, y,

(n, l, n)

that

and

if m ~

(p,y,q)

1,

operation

from the

follows

xEl

in I so that

(m,x,n),

(m, x, n)

y = z

monoid

(m,x,n),

2)

{(m,

(m, i, m) ~

then

=

S is c l o s e d

=

If

2),

in w h i c h

x or y is

E(S)

and

of

2) h o l d

2).

and

q)

of m ~ n ( m o d

to w h i c h

# R.

operation

n)

The

D

set

e T if m ~ n ( m o d

to the

(m, x ,

which

a cancellative

S be the

{(m,x,n)Ix subject

T be

in

l, n).

(r,

z,

s)

Similarly,

it

i, m). e =

observe

(p, l, p) that

and

an e l e m e n t

eu = u ( e u ) * ,

and

is a t y p e

A ~-semigroup,

and

there

four D - c l a s s e s

in S~

are

is

vlz : Di, j

=

~(m, x, n)e S i m,

i, j C {0, 1}, 4.

with

so that

~

ISOMORPHISM

THE O R E M

Let

S 2 be

$I,

respect

to

and

each

n s N, m

~ i(mod 2), n~j(mod 2)},

# ~.

,-blsimple

of w h i c h

112

~ = ~,

type

A ~-semigroups

Buppose

that

ASIBONG-IBE

G:

SI +

S 2 is

following THEOREM

a mapping

4.1

Let

Sl=

type

S I ~ S 2 if

p of

them.

We

prove

the

theorem.

b_~e . - b i s l m p l e Then

between

M I onto

BR*(MI,

A

0),

and

~-semigroups

and

M 2 such

only that

if t h e r e

BR*(M2,~)

that

exist

= P~z

@p

Sz=

such

?*=

an

where

~.

isomorphism l z is t h e --l

inner

automorphism

of

M s defined

by

xX

= z x z

--

some

unit

PROOF

z in

BR*(MI,

0),

and

BR*(M2,W)

Bruck-Reilly

monoids

MI

and

monoids

M I and

then QSs

M 2.

Essentially,

loss

of

I.

Let

M 2 by

~ maps

are

the

respective

of

the

caneellative

extension

S 1 and

S z contain

the

M z respectively.

Without M I and

- -

Mz

generalised

in

for

Z

E ( S I)

generality,

and

respectively.

O:

QSI

In

SI +

denote S 2 be

the

an

isomorphism,

isomorphically

particular

identity

onto

(n,l,n)~

=

E ( S 2)

(n,l,n)

and for

n g N. Let

u =

u

:

(m,x,n)e

SI

and

uG

=(m,x,n)c~ ~ < p , y , q ) =

v.

Then +

which

~

~uo;*

implies

(p,l,p);

=

, u a

v

(n,l,n)~

that

consequently

""~u~; +

=

=

:

V

(q,l,q),

q = n and

,

and

(m,l,m)a

=

p = m.

~W

Now

H

G = H O,O

and

M2

=

H'*

.

W

so

0,0

that

Denote

the

M I ~ M 2 where isomorphism

MI=

between

H

0~0

MI

O,O

and

M 2 by By

p. definition

(O,x,O)~ As

(0,I,i)

(O,l,1)G z is

is in

(o,

=

a regular

S 2.

evidently

then

(O,xe,l)a

o).

element

Suppose a unit

xo,

in

in

S i so

a l s o is its image

that

(O,l,1)o

=

M 2.

Thus

all

for

= :

((O,xe,O) ( O , l , 1 ) ) ~ ((o,xe,o)o) ((O,l,l)~)

=

(O,xep,0)

(o,

=

(0,

i)

xepz,

113

z,1)

(O,z,1). x e MI,

Then

ASIBONG-IBE

Also

(o ,xO,l)o

= ((o, l, i) (o, x, o))~ = ((o, l, 1)o) ((O,x, o ) ~ ) = (o, z, l) (o, xp, o)

= (o, z(~p) ~, 1) Therefore,

for

all

x e Mx,

x@0z

=

z(xp)~

hence

-I

xOp where

=

1 Z

Y%z

=

Thus

we

z(xp)~

:

M2 §

z

= xO~

M 2 is t h e

z

,

automorphism

defined

by

--I

zyz

.

obtain

Op

O~XZ

=

Conversely, and m is a u n i t

in

p is

if

M 2 such

an

that

isomorphism

Qp

= P~Iz

of

then

M 1 onto

for

all

k

M2 cH

@kp = p(~X )k. Z

Let

a =

(O,z,l).

As

z is

a unit

in

M 2 then

_i

(O,z,1),

(1,z

elements

of

every

,0)

S 2.

elements

must

With of

be

mutually

respect

the

to

H*-class

so t h a t

by

Lemma

3.3

uniquely

Let

G be

~

is

from

= a-m(o,

a bijection.

isomorphism in

O~O

regular

S 2 can

P,

be

expre--i~

Evidently

aeHo,l,

a

eHl, o

element u o f S z c a n be -m( an O,xp,O) for s o m e x e M I .

u = a

a mapping

(m,x,n)o Clearly

every

as

the

H*

ssed as (O,xp, 0), for some x e M I.

expressed

inverse

S I into

xp,

We

$2,

defined

by

O)a n

show

below

that

O

is

also

a homomorphism. Let suppose

(m,x,n),

that

((m,x,n)o)

n Z

(p,y,q)

p.

((p,y,q)~)

be

elements

of

$I,

and

Then =

( 1,z

-I

,o)(o,z,1) n (1,z _I ,o) p ( o , x p , O ) ( O , z , 1 ) q. ,o)m(0,xp

_1

=

(l,z

,o)m(o,xp,O)(O,z,l)n-P(o,xp,O) (O,z,l) q .

Observing

that

(O,z,1)k(o,yp,O)

=

(O,z,1)k-l(O,z,1)(O,yp,O)

= (O,z,1)k-~(O,z(yp)

114

~, l )

ASIBONG-IBE

it

is

clear

(m,x,n)

that

other

((m,

x,

hence

group for

-1

(O,y0(~X z)k ,o1 (O,z _i , l ) k

)n-P,1)(O,z,1)q-p+n z n > p it is also evident

with =

(m,x(yon-P),

q -p

+ n)o

(l,z -I, o)m(o, xo.y9 n-p D,O) (O,z,l) q - p + n

=

((m,

(p, an

x,

y,

applied q))~

(p,

S 2 = . BR*(M2,

=

to ((m,

the x,

).

case n)~]

n <

p to

(p, y,

show

that

q)G),

isomorphism.

i is t h e

h.2

y, q)o

n)G)((p,

S 2 = BR*(M,i)

max(m, p)

,i)

,o)m(o,(xo)(~t

=

where

COROLLARY

=

-1

(l,z-1,0)m(o,xD.yon-Pp,o)(O,z,l) q-p+n

(m,x,n),

=

(O,z,l)k-1(O,(yo)~Iz,O)(O,z

z,O)(O,z-l,1)

m ( O , ( x ( y @ n - P ) ) o , O ) ( O , z , 1 ) q-p+n

,0)

y,

be

a .-bisimple

identity q)

(m, x, n) (p, y, q) t

i)

=

~ is Let

=

hand,

argument

n)

(O,z,1)k-l(O,z(yo)~

(p,y,q))O

= (1,z

A similar

=

= (t,z

((m,x,n)

that

(O,z,l)k-l(O,z(y0)~z.z -I

:

(p,y,q)~ On t h e

=

type

automorphism.

A 00-semiThat

is,

e S2 , = (m - n + t, xy, q-p + t).

Then the following holds.

S 1 = BR*(MI,0) i)

is

isomorphic

i__ffand. onl~r if 0 is an

with

inner

automorphism

of M I PROOF

If

isomorphism

80 = 01

Z

G

: S 1 = S 2 then p: M I § M 2 a n d

by T h e o r e m a unit

9

Thus -I xSp

= x o ~ z =>

x8

= Xp~zO

115

h.l

there

z g M 2 such

is an

that

ASIBONG-IBE

Let xo = x I, then x8 = Xllzp-l=(z~z-1)P-1=(zo-*)(xlp-1)(z-lp -I) --1

xe

=

(zp

--1

).x(z

--1

p

--1

)

=

z 1

x

z 1

=

xX z 1

The

converse

for inverse

follows

semigroup,

quite

naturally

(see R e i l l y

as in the

case

[18]).

REFERENCES 1.

2.

ASIBONG-IBE,

ASIBONG-IBE,

U.

U.

Structure

of Type A ~- s e m i g r o u p s

D. Phil

Thesis.

England,

1981.

The ,_ B i s i m p l e groups.-~.

3.

DOROFEVA,

M.P.

301 FELLER

E.H.

& GANTOS

Semigroup

J.B.

A

F o r u m 4(1972),

- 311. R.L.

Completely

31(1969) FOUNTAIN,

Type A e - s e m i -

To appear.

Semigroups t

5.

of York,

Hereditary and s e m i - h e r e d i t a r y Monoids

.

Univ

class

Jour.

Injective

Pacific

Jour.

Math.

359 - 366. of Right

of Math.

PP Monoids ,Quart.

Oxford

2, 28(197h)

28 - 44. 6.

FOUNTAIN,

J.B.

Adequate Proc. 22(1979),

7.

FOUNTAIN,

8.

HOWIE,

J.B.

J.M.

Semigroups ,

Edinbourgh 113

Abundant

-

Math. 125.

Semigroups

The M a x i m u m

Soc.

Preprint .

Idempotent

Semigrou~.

Congruence o n an Invers.___~e S e m i g r o u p ~ Proc.

of Edin.

2(1964), 9.

HOWIE,

J.M.

KILP,

M.

A_~n I n t r o d u c t i o n

Soc.

Press

Commutative

Monoids

116

14,

to S emigroup

Math.

7, A c a d e m i c

Principal

Soc.

71 - 79.

Theor~)Lond.

i0.

Math.

Monograph

(1976). all of whose

Ideals are P r o j e c t i v e ,

ASIBONG-IBE

Semigroup Ii.

KOCIN,

B.P.

The

Forum

Structure

Simple

McALISTER,

D.B.

Groups,

Univ.

13.

McALISTER,

D.B.

23, 7(1968)

McALISTER,

D.B.

Amer

192(1974),

227

24h.

Groups,

Semilattices

-

MUNN,

W.D.

One-to-One

Partial

Cancellative

Regular Math.

16

MUNN,

W.D.

21(1970), PASTIJN,

F.

Math.

representation

Semigroup 18

REILLY,

N.R.

Bisimple Proe. 160

of

- 251.

Glasgow

Semigroups, (Oxford)

2,

- 170.

b_yy ~ .Semigr~ Group with

Soc.

46-66.

Inverse

157

Math

231

9(1968),

Jour.

Soc.

Semigroups~

~-semigroups!

FQndamental

Inverse

Math

Translations

43(1976),

Jour.

Quart.

17.

Amer.

351 - 370.

Right

41 - 50

and Inverse

192(1974),

J. Algebra 15.

and

Trans.

Semigroups, Trans

14.

Ideal

Vestnik !

Semilattices

Semigroups

334 - 339.

of Inverse

~-Semigroups

Leningrad 12.

6(1973),

of a Semi~roup of Matrices

over

a

zero! Forum

Inverse

Glasgow

i0(1975),

238-249.

w- Semigroups,

Math.

Soe.

7(1966),

- 167.

DEPARTMEhrf OF MATHI~TICS AInU

BELLO UNI-VERSITY

ZARIA, NIGERIA. Received November 15, 1982, Received in final form July 23, 1984.

I17