inverse semigroups which are separated over a subsemigroup

TRANSACTIONSOF THE AMERICAN MATHEMATICAL SOCIETY Volume 182, August 1973

INVERSESEMIGROUPSWHICHARE SEPARATED OVER A SUBSEMIGROUP BY

D. B. McALISTER( 1) ABSTRACT.

An inverse

T is generated,

exists

x e SaPiSb

such

example,

if

cipal

and right

S.

left

In this

inverse

even

In a general semilattice

image

kernel ating

is a semilattice extensions

theoretically principle, inverse

semigroups

prinover

T which

has been the object

[13L

for example).

or 0-simple

However

one to suspect

of inverse

and a semilattice shows

These

inverse

the degree

that this

are

of much study

and 0-simple papers

attempted

semigroups

of complication

is, in general,

of groups.

semigroups

of groups.

that the maximal

a futile

ideals

fundamental

[4] and

construct

all inverse

semigroups

semigroups;

the latter,

however,

Coudron

TE

E of idempotents

idempotent

semigroups

construct

so its

separ-

has been solved,

[3] so that one could,

if one could remain

homomorphic

separating

of constructing

by inverse

of

of the semigroup

of the semilattice

The problem

by its

is a consequence

p: S —» S/p. is idempotent

of groups

by D'Alarcao

is determined

This

S is a full subsemigroup

the principal

homomorphism

of semilattices

at least,

T is separated

in some cases.

semigroup

between

The canonical

leads

which

of an inverse

For

S whose

being paid to O-bisimple

O-bisimple

the structure

of Munn [ll]

S/¡i

of isomorphisms of S.

sense,

of idempotents

a theorem

semigroups

attention

and semilattices.

cases

it is possible

for right ideals.

then

of inverse

S if

b £ S there

by a semigroup

inclusion,

the structure

a,

S.

of various

of groups

a subsemigroup

x and dually

semigroup

under

([2], [9]> [lO], [il],

in these

although

b = x~

form chains

of inverse

the structure

in terms

involved

ab~

with particular

semigroups

directly task

theory

years

to determine

ideals

subsemigroups

The structure

over recent

a~

over

by S and for each

as an inverse

we investigate

over

T is separated

semigroup,

that

T is generated

paper

separated

semigroup

as an inverse

in

all fundamental

a mystery.

Received by the editors April 10, 1972. AMS'MOS) subject classifications (1970). Primary 20M10. Key words and phrases. Inverse semigroup, shift representation, group, naturally quasisemilatticed semigroup, fundamental inverse group of (strong) quotients.

(1) This research

was supported

free inverse semisemigroup, inverse semi-

by NSF Grant GP 27917. CopyrightC 1973, AmericanMithematicij Society

85

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

D. B. McALISTER

86 In this paper, semigroups.

Suppose

semigroup

T.

generated

we shall

adopt

that

6 is a homomorphism

Then we shall

as an inverse

a more internal

say that

semigroup

is separated

concept

have been

is to investigate

over a semigroup

considered

before.

separated

over S. Clifford

Eberhart

parameter

inverse

semigroup

S of the multiplicative

Theorem semigroups

extension

semigroups

it the sets

theorem

conditions

of principal

thus an inverse latticed

an explicit

For example,

let

and

semigroup

S.

over

semigroup

S.

On

over a sub-

Thus,

construct

S is naturally

inverse

by using all inverse

to do this explicitly

quasisemilatticed

of S form semilattices

S by d then,

of S.

of all one

for all fundamental

is naturally quasisemilatticed.

T is separated

T is

reals.

We have not been able

ideals

monoid

then

T is separated

in principle,

A semigroup

of this

of T in terms

of construction

over an arbitrary

left and right

cases

the structure

of the positive

[4] one could,

semigroup

T be a bisimple

the structure

semigroup

method

over

Special

of T; if S is right reflexive

[5] have described

on S.

of an inverse

of S.

semigroup

which are separated

imposing

the structure

Any such

which are separated

D'Alarcao's without

semigroups.

3-5 gives

for some y £ Sa n Sb.

[l] has described

and Seiden

S into an inverse

S, by 6, if T is

a, b £ S,

S, in terms

and let S be the right unit subsemigroup

the other hand,

over

inverse

for some x £ aS n bS,

bd =(yÖ)_1yö

The main aim of this paper

T, which

= xfXxÖ)-1

to describing

of a semigroup

T is separated

by Sd and, for each

adiad)-lbdibd)-1 iad)-ladibd)-1

approach

under inclusion;

If S is naturally

semi-

for a, b £ S,

adiad)' lbdibd)-1 = ia Ar b)v[ia Ar b)6]-\

iaO)-ladibd)-lbd = [ia A, b)d]~Ha A, b)d, where, thus

for example,

a universal

which

a A

inverse

are separated

b in S is such that aSx n bS x = (a A semigroup

over

S.

EÍS)

An explicit

for EÍS) are given in §4 while

in the category construction

the congruences

b)S . There

of inverse and several

and ideal

is

semigroups coordinatisations

structure

form the sub-

ject matter of $5. Whenever chains

the sets

of principal

under inclusion,

group, by a homomorphic inverse

semigroup

quasisemilatticed inverse

semigroup

over S.

on S and so S can be embedded in EÍS).

(Theorem

and sufficient

The last

result

4.6) so that we can use

conditions

of a semigroup

generated,

image of S is separated

only if it can be embedded

necessary

left and right ideals

every inverse

S are

as an inverse Hence

semi-

EÍS) is the free

in an inverse

semigroup

remains

if S is naturally

EÍS)

for the embeddability

semigroups.


true

to obtain

if and

a set of

of such semigroups

in

INVERSE SEMIGROUPSSEPARATED OVER A SUBSEMIGROUP The main tools

used

in this paper

S by one-to-one

partial

Vagner-Preston

representations

tions

of cancellative The theory

consideration conditions

transformations.

c with

cipal

left and right

also

applied

It is applied semigroup

of S should

consists

of several In particular

0-simple

inverse

are traversed

the 0-simple

inverse

semigroups

semigroups

element

conditions

and sufficient

of IÍS) should

inclusion.

of inverse

be of of prin-

The theory

cone of a right

the theory gives

is

ordered

semigroups

group.

which

a method

for con-

5) ^ £). The -D-classes

in these

but noil-class

obtained

S under

are that the sets

under

examples

in which

by a semigroup

representa-

in §2.

of the positive

theory.

semigroups

both the

in §6 to give necessary

be chains

to give a characterisation

from the general

structing

generalise

when the semigroup

so that each

a, b, c £ S; the precise

ideals

of

and the regular

simplification

The final section arise

semigroups

They are described

considerable

is cancellative.

representations

These representations

of inverse

semigroups.

undergoes

on a cancellative

the form ab~

are what we term shift

87

is a subsemigróup

here are, in a sense,

so that

dual to those con-

sidered by Munn [12]. 1. Embedding it follows

inverse

a semigroup

from general

semigroup

homomorphism

semigroup.

considerations,

IÍS) and a homomorphism

erty: given any homomorphism unique

in an inverse

categorical

If S is any semigroup,

or from [8], that there

is an

77:S —>IÍS) with the following

6 of S into an inverse

semigroup

prop-

T, there is a

iff. 1ÍS) —»T such that the diagram

5.

commutes.

The semigroup

IÍS) is called

the aims of this paper

is to investigate

groups

structure

when the ideal

when the sets

of principal

It follows and IÍS)

easily

relationships

between

IÍS)

factored one-to-one.

any homomorphism

We shall

assume

Let S = S

and sufficient

be a semigroup.

conditions

properties;

Hence,

semi-

in particular,

/(S1)

in studying

throughout

the

assume

that

this paper.

semigroup

can be uniquely

semigroup

if and only if r¡ is

proof of Schein's

for embedding

Then a nonempty

One of

under inclusion.

loss of generality,

the latter

in an inverse

We can use this to give a short

on S.

of S , S° and IÍS) that

isomorphic.

of S into an inverse

through r/, S can be. embedded

gives necessary semigroups.

special

properties

ate naturally

semigroup

of IÍS) and some related

of S form chains

S and IÍS) we may, without

S has a zero and an identity.

Because

of S has certain

left and right ideals

from the functorial

and IÍS ) and

the free inverse

the structure

subset


theorem

semigroups

[l6] which in inverse

H of S is strong

it

88

D. B. McALISTER

ax, bx, ay e H together strong

subsets

Let

imply by £ H. Clearly,

subset

for example,

H/x

transformations

WH is clearly

of S = S

(KH)

of

and define

if and only if H'x = H'y

= [u e S: x u e H\.

on S [2, § IO.2] and can be used

partial

the intersection

is strong.

H / D be a strong

x =y where,

if nonvoid,

to construct

in the following

an J\„-class

Then

3\H is a right congruence

a representation

way [2, §11.4].

on

of S by one-to-one

Set W^ = [x e S: H 'x = □ }.

of S, and let XH be the set of J\„-classes

different

from WH. For each a e S, define xp Then the mapping transformations verse

that,

for each

p : a —>p

of X„;

semigroup Recall

= xa

thus

HjCA

x e !X„ such that ~xâ e Ji^.

is a representation p

of S by one-to-one

is a homomorphism

partial

of S into the symmetric

in-

on Jl„.

if T is an inverse

semigroup,

the natural

partial

order on T is

defined by x < y if and only if x = ey for some

Lemma

semigroup which

1.1.

d be a homomorphism

T and let a e S.

contains Proof.

also,

Let

Then

e = e

e T [2, §7.l].

of a semigroup

S = S

into an inverse

K - [x e S: ad < xd\ is a strong

subset

of S

a.

Suppose

bx, by. ex e K. Then ad