SemigroupForum, Vol. 7 (1974), 200-263,
LEFT COSET EXTENSIONS Pierre Antoine Grillet
Dedicated to Professor A. H. Clifford for his 65th birthday
i° Let
A
be a semigroup,
functor on the set assigns to each with
~ ¢ aA 1
A
and
G
be a group-valued
preordered by left divisibility; G
a~A
a group
a homomorphlsm
G ~
and to each pair
:G
~
G~ • The main con-
cept in this paper is that of an extension of is a semigroup
S
together with a congruence
S/C --- A , and, for each tion of
Ga
ecA,
holds in
S
G C
by
A :that
such that
a simply transitive group ac-
on the corresponding C-class
q0aa~g.ab = (g.a)b
(g,~)
whenever
C,
such that
gcG~,
a~C
, bgC~o
The main results are as follows. Given any semlgroup S, certain congruences on gruences,
S,
which we call left coset con-
give rise to a description of
S
as an extension
of a suitable functor by the quotient semigroup. All congruences
contained in ~
converse holds general.
if
S
are left coset congruences;
is regular,
or commutative,
explicitly construct by
A;
but not in
We call any extension which arises from a left co-
set congruence a left coset extension. G
the
Then we show how to
(up to equivalence)
all extensions of
this is completed by necessary and sufficient
conditions for equivalence,
and by a characterization of
left coset extensions among all extenslons~
Finally,
these
results are applied to the problem of explicitly constructing semlgroups.
Call a semlgroup reduced (left reduced)
© 1974 by Springer-VerlagNew York Inc.
200
in
GRILLET case it has no congruence contained in ~ congruence)
other than the equality.
tions that reduced semlgroups
(no left coset
There are some indica-
can be constructed in terms
of partially ordered sets alone. Any semlgroup can be explicitly
(and practicably)
constructed in terms of groups
and a reduced semigroup; any finite semigroup can be (explicitly, at least in theory,
but not yet practicably)
con-
structed in terms of groups and a semigroup which is both left reduced and right reduced. There are a number of theorems
which explicitly con-
struct all semigroups of a certain kind in terms of groups and/or sets,
semilattlces. A great many of these theorems
depend on congruences contained in
~ , perhaps
~
itself,
and can be recognized as constructing particular cases of left coset extensions. This
is particularly obvious
the Rees-Suschkewitsch theorem; verse unions of groups groups
[18];
Clifford's theorem on in-
[3]; Leech's
theorem on bands of
Reilly's theorem on bisimple ~-semigroups
and its subsequent generalizations al.); see also, neral,
for:
for (0-)blslmple
[Z5]
by Warne ([29],[30] et
inverse semigroups
the theorems of Reilly-Clifford
([~6],
in ge-
where the
extension is done at the RP-system level) and McAllster [20], and for 0-slmple
inverse semlgroups Munn ([Z3],Mac-
lean's results). This is an arbitrary selection of significant results;
the reader will find a bibliography
and topological examples the general
in [16]
in [2]° In each of these examples
theory might not provide the best proof,
but
it explains why a construction is possible, and gives a way to find such a construction as well as the general form of the multiplication. gun to yield new results;
The general theory has also be-
in particular,
all finite commutative semlgroups on
a construction of
[13] (partly by induction
S/~ ) which also applies to the finitely generated com-
mutative semigroups
considered in [ii]
(into which any fi-
nitely generated commutative semlgroup has an explicit embedding);
a construction of arbitrary
inverse semlgroups
[IZ] (which extends [~0]) and will be extended to regular semlgroups
in forthcoming papers. It should also provide 201
GRILLET valuable insights on finite semigroups
in general.
Z. There are also examples of left coset extensions general that are already known, sions
in
beginning with group exten-
(with which the general extensions have much in com-
mon). A construction of semigroups
from semigroup-valued
functors on a semigrou~ (but without factor sets) was also considered by Kolibiarova [14]o The first general example of left coset extensions
is the construction by Anderson
and Hunter of split extensions arising from is a congruence
~,
in case
[i]. There are also specific constructions
of inverse semigroups from a semilattice of groups and the corresponding reduced semigroup which are covered by the general theory ~irjaev
(Coudron [5], D'Alarcao [6], Preston [Z4],
[27], and,
in the 0-bisimple case, Munn [ ~ ] ) ;
so
is Leech's generalization of group extensions to monoids
[17]. In its present form, the theory was developed independently, and
in somewhat different directions,
(for congruences
contained in
by Leech [18]
~ ) and later by the author
(as announced in [9], much as in what follows).
It is felt
that the two approaches are sufficiently different to justify independant publication. self-dual ctors on
First the approach in [18] is
from the start, using A
(essentially)
instead of just one.
In addition,
two dual funa number of
results will be found in [18] and not here: a more thorough conceptual
investigation in terms of certain very worth-
while categories; a construction of all congruences ned in
~
contai-
in terms of Sch~tzenberger groups; a treatment of
cohomology in full generality.
Conversely,
some results
will be found below which are not in [18]: the characterization of left coset extensions
(including the case
C ~ M ),
and the appropriate modifications and additions to the cohomology in the commutative case. The main difference, ever,
how-
is that we allow for congruences not contained in
~.
Of course all the important applications of the theory so far concern regular or commutative
semigroups,
this case is vacuouso Future applications, 202
for which
however,
may not
GRILLET be so restricted; congruences ness
is clear,
a left
coset
does):
this
ger group
it may
group
get group
need not
first
We call
basic
morphisms
idea
and
these:
Cm m D
for
9g. cm =
(g.c)m
whenever
basic
definitions
a congruence valently, The left
C
such
functor ger)
on
Various
C,
elementary
them in s e c t i o n
when
~p : G(C)
C, D
I,
homo-
are
left
is a canonical
m G(D)
such that
g e G(C) , c e C, Cm ~ D • of e x t e n s i o n
C ~ £
classes
and
xa = a C b
cosets;
implies
of the C-classes
then
give
us a group
(= g e n e r a l i z e d
on left
of
S
coset
coset
con-
congruence
are left
and a d e s c r i p t i o n results
and left
2. A left coset
groups
homomorphisms
of
in Z-clas-
and the natural
m e S I, there
all of whose that
study
groups
A = S / C , the left GS
functor
a different
out with subsets
instance,
in section
Sch~tzenberger
associativity
and
for some
homomorphism
are g i v e n
in the S c h ~ t z e n b e r -
nor even be contained
"associativity"
The
which
the usual S c h ~ t z e n b e r -
may be c a r r i e d
Sch~tzenberger
between
from
sharp-
(much as
and so gives
is that
them left cosets
left
on the group
to a S-class
be c o n t a i n e d
not be Z-classes
with their
to build
The added
insight.
construction
which n e e d
gruence
level,
attaches
(it may even be greater)
3. Our
cosets
as possible.
at a more basic
congruence
kind of s t r u c t u r a l
ses.
then be advantageous
that are as large
is
equixb = bo
and
their
valued Schutzenber-
as an extension.
congruences
eompl~
te the section. The When
S
select
next
three
is an extension p~ ¢ C a
then be w r i t t e n know all
= ~[,~hopap~ o ,~
(gope)(hep~)
of
d e a l with G
by
for each
a¢ A
uniquely
as
a s,~ e G ~
commutation
~,
sections
maps
=
and every
element
defined
,~h , the m u l t i p l i c a t i o n
203
of
S
a~ A, g¢ G
all products;
(q0:~g a ,~ X [ , ~ h ) . p ~
theory.
S/C = A , we can
p~p~ : a , ~. p ~
) G ~
, we can describe ca, ~
with
g.p~ , with
such that
w~,~ : G~
A,
the general
by:
can
• If we
a n d all pe(h.p~)
=
putting takes
the
• In the group
form
exten-
GRI LLET sion case,
one would use
w
rather than
every example we mentioned before,
X ; otherwise,
in
the multiplication has
essentially that form. The structure theorem for extensions then consists
of associativity
conditions for
o
and
X o
The basic algebraic fact here is that the "right hand side" maps
X
can be recovered from purely "left hand side" con-
cepts, which yields a sort of a posteriori duality. In section 3 we also point out that the not be homomorphisms.
X
maps need
This phenomenon (bad behavior)
can
happen in left coset extensions and thus cannot be ignored. In section 4 we show that in a left coset extension bad behavior does not happen: nite combinatorial; C ~ M • In fact, functor on
A
GS functor of
inside £-classes;
when
A
is fi-
when the extension splits; and when
in the case
C ~ ~,
which is naturally
the
X
maps yield a
isomorphic to the right
C , showing that in this case the theory is
actually self-dual.
Section 5 completes the general
with a study of equivalence,
normalization,
theory
split exten-
sions, and a characterization of left coset e~tensionso This last yields a characterization of left GS functors which (unavoidably)
is not very nice, except e.g. for fi-
nite commutative semigroups
in case
C = ~ •
Section 6 deals with the greatest congruence contained in
~
and the greatest left coset congruence,
as reduced and left reduced semigroups,
as well
and establishes
the few easy facts mentioned in paragraph i0 The last section deals exclusively with the commutative case, especially with the cohomology
theory for that
case. The equivalence classes of extensions of
G
by
A
can then be naturally organized into an abelian group Ext(A,G) ; this in turn can be described as a second cohomology group of coefficients
A
with coefficients in
G.
Because the
lie in an abelian category of possibly high
homological dimension,
there is no universal coefficients
theorem that satisfied the author,
in the general case.
This can be remedied in case only one group is necessary
204
GRILLET in ths construction; rectly
irreducible
this happens
commutative
homology has a good d e s c r i p t i o n groups G
of
commutative
theory
if only the groups
[18]. A l t h o u g h
for these extensions
as the E i l e n b e r g - M a c L a n e
cohomology,
more relevant,
because
precisely
[8]; then the co-
in terms of the h o m o l o g y
A o All this will go through
are a s s u m e d
homology
e.g. with finite subdi-
semlgroups
the general
co-
does not work as well it should be rather
of its ties with semi-
group construction° The general can be used
in building
of semigroup excepting
theory).
semigroups
(a classical o b j e c t i v e
The reader should bear in mind that,
c e r t a i n cases such as inverse semigroups,
is very likely cess;
theory shows to just what extent groups
to be the easier part of the building pro-
the d e s c r i p t i o n
significantly
of reduced
semigroups
seems to be
more difficult.
4. The m a i n results have been announced ver, when c o m p a r e d left cosets,
this
and
in [9]. Howe-
to [9], we have put more emphasis
(because of [18])
adding the last part of section ~,
on the cases
on
C ~ ~,
the third part of sec-
tion 4 and most of section 6 • The main result consists Theorems
of
2.3 , 3.3 , 4ol-4.5-4.6 , 5.1 and 5.6 ; at first
reading one m i g h t skip parts
Z , 5 , 6 , 7 of section l, part 5
of section Z, part 4 of section 3, parts 1,3,5 of s e c t i o n 4 and parts
2,3,5 of section 5.
Dots will be used extensively plicatlon
to denote group multi-
as well as group actions;
in no case will this
create ambiguity.
i. Cosets and S c h u t z e n b e r g e r
i. A left coset of a semlgroup set
C
of
(CI) If xc = yc
S
with the following
x,y E S i f o r all
and
205
is a non-empty
sub-
properties:
xc = yc ~ C
cE C ;
S
groups
for some
c ~ C , then
GRILLET
(C2)
C
(C3)
If
If
is c o n t a i n e d x~ S I
C
and
in an g - c l a s s ; xC A C
is a left
coset
~ ~ , then
of
S,
xC ~ C °
the
submonoid
= Ix E S I ; xC ~ C ] = {x E S I ; xC N C ~ ~ } cally
on
induces
C, on
by
left m u l t i p l i c a t i o n .
T(C)
the
xc : yc
for all
quotient
monoid
congruence
cc C G(C)
nonical
action
on
x~T(C)
, c e C , where
PROPOSITION group,
and
acts
Proof. I = ~i) and
and
more
implies which
l.l.
When and
and
g.c = h . c c E C,
b = xa
that
some
such
h e G(C)
c e C.
and
hg = I • Thus
left
inverse,
and
this
left
with
case,
and
2.
is c l e a r
It
cosets
groups
(i.e.
still
with
its
group
by t h a t
the a c t i o n
from
T(C)
induces
that
groups
acting
there
hg.c
let is
= l.c
G(C)
,
has its
canonical
a
ac-
of
name on
C.
action
on
More
generally,
any g r o u p
isomorphic
C
induced
by
transitive
the
group
is a g a i n d e n o t e d
by
isoac-
~
in
~.
on the p r o o f
in o r d e r
Finally,
In p a r t i c u l a r ,
t h e n be a s i m p l y
the p r o j e c t i o n
(C3)
(by p e r m u t a t i o n s ) . "
Schutzenberger
must
then
If further-
showed,
of the m o n o i d
left
(which
c e C,
C.
= c ; then
to call
is
g = ~x ~ G(C),
on
just
is a g r o u p °
(g.h).c
for some
element
action
=
(C2);
h.(g.c)
G(C)
identity
x e S I, by
that
G(C), t o g e t h e r
G(C), t o g e t h e r
morphism
we
is a
action
every
is a g r o u p
it is c o n v e n i e n t
some
what
group
C , is the
(with
g = h.
transitively
By
G(C)
C.
for some
i.e.
b = g.a
G(C) acts
any
tion);
for
a ca-
for a l l
(i.e.g.(h.c)
for all
iff
T(C)
coset, on
g , h ~ G(C) , c ~ C)o The
x ¢ T(C) , so that
The
C
in turn
c ~ C ). T h e
from
a monold
= h.c
C, t h e n
C
is a left
on
canoni-
is the p r o j e c t i o n .
transitively
as a m o n o i d
g c G(C) ; p i c k
to
C
:
x -= y
~x.c = xc
G(C)
g, h e G(C)
shows
t i o n on
~
by:
for some
by:
is i n d e e d
f o r all
g.c
a,be
~ : T(C)
acts
action
inherits
well-deflned
simply
acts
if
(CI)
C,
This
SI
defined
xc : yc
= T(C) /-z
First, G(C)
l.c = c
simple: by
(iff
=
of
T(C)
they
simply 206
of
have
I.I that
we d e f i n e d
left S c h u t z e n b e r g e r
and
transitively
on
C
GRILLET by r e s t r i c t i o n s
of
conditions
and
fect.
(CI)
Condition
(in fact,
inner
left
(C2)
(C3)
for all
clearly
is a u t o m a t i c
it c h a r a c t e r i z e s
particular
translations).
are
left
precisely
to
for c o n g r u e n c e
classes
cosets
More
necessary
that
ef-
classes
of left c o n g r u e n c e s ) ,
considered
in the
in
following
sectl ons • Any M-class any
subset
quence usual
of G r e e n ' s left
cosets
as
that
below);
we
groups The
properly shall
are
Thus,
left
the
corresponding
group,
if
contained in the
of a left
are
not
within
coset
it s a t i s f i e s
but left
(see 1.8
(example
covered M
I)
by one ~-
in order
to e~-
and
can be s i m p l i f i e d
subset
(CZ),(C3)
xc = c
for
C
thus:
is a left
coset
and
some
c ~ C, then
zc = c
cg C. First
CI)
implies
If c o n v e r s e l y
(CI'), (C2), (C3)
be such
xa = ya = b • By
that
x b g C o By
(C2), a = ub
: ya = ~
; hence
uza = a , w h e n c e
(CI')
(let
hold,
let
y = 1
y u z b = xb , by
and hence
ug S 1 • Then (C±').
uzb = b ; t h e r e f o r e
(CI)).
x,y~ S I , a,b¢
(C3), x e T(C)
for some
in
yuxa
C za,
= yub =
O n the o t h e r
hand,
xb = yuxb = yb • Thus,
holds .i We c o n c l u d e
to left
cosets
these m i s c e l l a n e o u s
a well-known
PROPOSITION group
is the
action
to c o n s t r u c t
section
for
conse-
into a n M - c l a s s
nezt
that stay
holds
group
its usual
is easy
1.2. A n o n - e m p t y
zg S I
Proof.
(CI)
with
It
(CI)
is a c l a s s i c a l
from a s e m l g r o u p .
if and only
for all
(condition
(C3)
cosets
definition
If
see
we need not
PROPOSITION
(CI')
coset and
a left action.
are
and
that t h e r e -class.
Lemma);
Sch~tzenberger
now w r i t t e n
tract
is a left
of an g-class,
1.3. A left
is n e c e s s a r i l y Proof.
When
C
C ~ T(C) • F u r t h e r m o r e
a group
remarks
property coset
C
(and
then
is a s u b s e m l g r o u p C
is left 207
by e z t e n d i n g
of M - c l a s s e s : which G(C) of
is a s u b s e m i -= C ).
S, we h a v e
cancellative:
if
ab = ac
O RI LLET holds
in
then
e =
C , then
cancellative,
Define then
e(ye)
C
implies
f : T(C) = ye
= f(y)
fore,
tains uses
assumes an
xe = y e )
abe C
must
be
is e n o u g h
3. C e r t a i n
Cm ~ D
all
Proof. and
(by
that C
canonical
1.4.
First,
C, D
xC ~ C
ce there
exists
T(C)
for
C
one
if one con-
essentially
show
that
C
case,
instance,
some
it (by
congruence°] between
a homomorphlsm
cosets
exists
of
left
S
that
for
Cm ~ D .
xCm ~ C m , xD N D
for all dE D
~
x,y c T(C)
c ~ C, xcm = ycm
and
~ : G(C)
such
a homomorphism
~ T(D) • If now
some
WDX
= wDy
~ G(D),
o Hen-
unique
diagram --~
G(C) commutes. (Clearly CE C, i t
that
~0g.cm = (g.c)m
such
xc = yc
T(C)
if
work
exist
be left
implies
xD ~ D ; thus
c E C, xd = yd
, g
(CI),
in an ~ - c l a s s
modulo
that
m¢S I
for all
gcG(C)
(xy)e , i.e. By
x m y ; there-
For
is c o n t a i n e d
such
, c E C, and
the
=
in the g e n e r a l
means.
me S I • There
WC x = W c y , t h e n
that
case,
all
T(C),
not
to first
homomorphisms
Let
satisfy
such
group
C~T(C));
C
x,yE
to
will
a, b E C (e.g-
is a class
f o r some
(C3))
proof
by o t h e r
~ G(D) , unique
gEG(C)
is e q u i v a l e n t
for
groups.
PROPOSITION that
(xe)(ye)
a ¢ C,
is left
e •
is surjectlve.
[This
(so that
or that
Sch~tzenberger
q0 : G(C)
and
In the ~ - c l a s s
established
to a s s u m e
1.8 below),
= xe • If
SchStzenberger
is a s u b s e m l g r o u p this
f(x)
C)
C
eb = b
by:
for some
idempotent).
the right
and
element
(C2), f
m G(C) oI
b = c • If
identity
yeE
By
yields
ae = a ; since
a e b = ab
~ C
(since
(i.e.
C ~ T(C)/m
merely
that
has a left
is a h o m o m o r p h l s m .
f(x)
(ha) -I
is such
this
b E C , so t h a t
f
applying
(~a)-l.a
and
9 ~g
follows
does are that
not
T(D)
> G(D) depend
induced ~g.cm
208
on
by the = x(cm)
m ). same
For
each
xET(C)
~T(D);
= (xc)m = (g.c)m
•
G RI LLET Since
G(D)
acts
simply
quely determines We tall morphlsms
have
the above some
the following
C,D,E
re,n,
then
> G(E)
(note that
this property
in turn uni-
homomorphism.
composition
Qp is the
are left cosets
then the
G(D)
D,
the assoclatlvlty
C = D,
thermore ~:
~
on
qo.i
~0 maps This
if in
G(C) ; if fur-
Cm ~ D, Dn ~ E
~0~ : G(C)
compose to yield
Cmn ~ E ) .
property:
identity on with
These homo-
~ G(D) q0C : G(C)
is immediate
for
and ~
G(E)
on the commutative
square above. 4. Right
cosets
are defined
groups.
T'(C)
; Cx ~ C}, the right Schutzenberger
= {x¢Sl and
G'(C)
act on
the projection C
C,D
unique
These
associatlvity
zenberger
of
group
two groups
group by
~' ; we have
of
S
and
a homomorphism mc.@g'
= m(c.g')
m e S--'i--such that
homomorphlsms
mC ~ D oi
also enjoy
the com-
S
is a subset
of
S
A coset
which is both a C has a left Sch~t-
and a right Sch~tzenberger
have
commuting
actions
group.
(g.(c.g')
These
= (g.c).g'
c ~ C, g ¢ G(C) , g' e G'(C) ) and are isomorphic
[4], lemma
pc C • Define
Proof.
1.6. Let
Cp - G(C)
o Then
both groups
(of.
2°23). More precisely:
PROPOSITION ~
P
First
C
be a [two-sided]
~ O'(C)
coset
__bY: p.epg = g.p
and for all
is an isomorphism. Cp
is well-deflned
and bijeetlve,
act simply and transitively
G(C) . Since poep(gh)
exists
such that
left coset and a right coset.
geG(C)
by
property.
5. A coset
for all
G'(C)
are right cosets
c ~ C , g' ¢ G'(C), and
position
>
The dual of 1.4 is:
m e SI~, there
~ G'(D)
for all
T'(C)
1.5. If
for some
:G'(C)
is a right coset we denote
on the right.
PROPOSITION mC ~ D
C
and so are right
Sch;Jtzenberger G'(C)
If
dually,
the two groups
on
C.
Let
since g,h
have commuting actions,
= g.h.p = g.p. eph = p . ¢ p g . ~ p h , 209
so
ep(gh} = epg. eph.
GRILLET This groups° G(C)
a third
the
differ
P ry).
by an
position
it is a f f e c t e d
after.
for all
while
property,
together.
This
In f e w
~-classes,
kind
of map
1.7.
First
C,D
C
as
com-
q0,@,e t a k e n there-
extent
(for
in the
special
follows.
such
other
hand,
m
and D.
it f o l l o w s
that
cosets
of
. There
that
S
such
exists
~g.mc
an
= m(g.c)
jections implies
such
and
for all
D.(In
Then
that
inverse nmc¢
WC u =
C,
and
D
de D.
Thus
particular,
the
is a left
mutually
so
(Wc(nm))-l;
unD ~ uC ~ C o On
left m u l t i p l i c a t i o n ) C
so that
mutually
c~ C.
race D
m
coseb
and
un
inverse
bi-
the h y p o t h e s i s
mC = D .) now assume
for
all
that
mC = D , nD = C , nmc = c
cE C, de D.
m x n D = mxC = mC = D , so that x , y ~ T(C)
induce
Pick
= mc ; since
between
We m a y mnd = d
n
c a n be chosen
ae C , and
mund = d
(through
n
u c T(C)
for all
munmc
that
and
n m ~ T(C) ; c h o o s e unma = a
=
the
felt
to some
m, n e ~ l
~ G(D)
by
between
then
and
n d ~ C ),
for all
can be a r b l t r a -
true of
be left
for some
we show
left m u l t i p l i c a t i o n
mapping
maps
c ~ C , g ~ G(C) •
bijections
(as
p : shows
two
q0 h a v e
c a n be p r o d u c e d
cosets,
Let
~ : G(C)
Proof.
induce
any
be a m p l y
it can be r e p a i r e d
mC ~ D , nD ~ C
that
not
of
[i0]).
PROPOSITION
for all
(which
will
when
choice
g, k c G(C) ; thus
is u s u a l l y
of £ - e ~ u i v a l e n t
isomorphism
Schutzenberger
= p.¢pg.epk
the h o m o m o r p h i s m s
this
cases
6. A f o u r t h situation
by the
= g.k.p
lack of c o h e r e n c e
see
between
is not c a n o n i c a l :
inner a u t o m o r p h l s m
Furthermore,
that
of m a p
two, ~p
p.¢pk. C k . p g = k o p . ~ k . p g = ~p( k-i g k ) ,
Ck.pg
kind
other
is n o t abelian,
indeed
¢
yields
Unlike
~C x = ~ c y , i.e.
• : G(C)
~D(mxn) ~
G(D)
x e T(C) • W h e n
mxnmyn ; however,
when
Then
take
mxnE
then
x E T(C) o N o t e
= ~D(myn)
: mynd o Hence
such
that
for all there
210
C
W~cX
and hence
de D
exists
a
= ~D(mx~
x , y ~ T(C), we n e e d not h a v e de D, ynd¢
th~b
T(D) • If f u r t h e r m o r e
mxnd
unique
and
mxyn =
nmynd
=
GRILLET = ynd , mxnmynd lows
that
w
= mxynd
perty
in the
wg.mc
= mxnmc
statement:
well-defined = nmxc
larly,
W~WDZ
= xc
c , w
nD ~ C , mutation
; it f o l -
has
the
pro-
•
• If
c,xc~
~ : G(D)
x ~ T(C) , c c C , t h e n
C), i.e.
z ~ T(D)
~ G(C),
~WWcX
= WC x
, so t h a t
w
; simi-
and
C
are
isomorphisms.ll
think
depends
that
w
commutation
isomorphisms
to
is n o t
choice
show
mapping
formula
the
similar
o n the
4 will
the
w
g = W C x ~ G(C), c ~ C , t h e n
for all
properties
3 in s e c t i o n
= wD(mxyn) that
is a h o m o m o r p h i s m
(as
= WD z
would
have
if
Note
CWDZ = ~C(nzm)
inverse
One
as
there
by:
nmxnmc
would
wD(mxnmyn)
= mxc = m(g.c)
Similarly,
mutually
and
is a h o m o m o r p h i s m .
wgomc
of
that,
which
q0. T h i s
m ; furthermore,
without
is
so:
still
= m(g.c)
w just
example
the h y p o t h e s i s defined
need
by
that
the
com-
n o t be a h o m o m o r -
phism. 7. A refore, show
left
coset
a coset
that
show how
must
a left
necessarily
left
PROPOSITION Y-class i)
H.
C
~i) Then
has
K
K =
Proof. take
x~ S I
x ¢ T(H) gu.h = :
C
If
; let v.h
and
implies Now
xc i)
in an Y - c l a s s . contained
(and hence,
Let
C
a non-empty
h~ H
and
Sch(Jtzenberger
that
g.C
First,
we
retrieved:
subset
of an
subgrou p group
i.e.
that
properties that
C
gu ii)
•
C ; furthermore,
C = Koh = [k.h ~ k~ K], • Then also
xH fqH ~ ~ , i.e. exist
u,v ~ K
= v , g~ K, g.C : gK.h holds.
(CI),(C~) is a l e f t 211
of
K < G(H)
~ C ] •
xC N C ~ ~
implies
T ( C ) , so
assume
c a n be
is
equivalent:
g : ~x ~ G(H) • T h e r e
as
now
(C3); some
]]i) h o l d s ,
; this
We
the-
in a n Y - c l a s s
a coset).
group
be
are
[ ge G(H);
such
is
in an g - c l a s s ;
coset;
for
is a l e f t
necessarily
contained
following
property
C = K.h
contained
which
coset
1.8.
The
be
Schutzenberger
is a l e f t
ii) C
be
coset
a right
its
must
It
are coset.
with
= Koh =
is c l e a r
that
i0
inherited
from
H.
Note
that
CI ~ H
G RI LLET (where into
1 E SI); by 1.4 there
G(H)
unique
e ~ C • If
tion-preserving
C
of
onto
K
C o Since
g ¢ G(C),
g = i,
i.e. ~
C = K.c
and so K
for any
G(C)
for all
g.c = l . c , hence
isomorphism
of
~
K
is an ac-
is a left
then acts ee C.
is
transitive-
Thus
our three
are equivalent.
Since
G(H)
such that = L.h
~ g . c = g.c
~
~(G(C)) = K < G(H) , then
group
we have
properties
acts simply
C = L.h
implies
gg K
then
If we let
Sch~tzenberger ly on
such that
~ g = i,
inJective.
is a h o m o m o r p h i s m
clearly
must
g.C ~ C , then as
H , any subgroup
be the a b o v e
h = l.h g C implies
on
K
and hence
(note that
C = K.h ).
goC ~ C ; if c o n v e r s e l y
in the first
part
L < G(H) C =
Also
g ¢ G(H)
of the proof
and
we see t h ~
g E K .i We can now prove: PROPOSITION
1.9. A left
is also a r i g h t
coset.
Proof°
C,H,K
some
cg C.
Let Then
Proposition
K' = ecK < G'(H)
Left
i. A left congruence
following
PROPOSITION congruence either
coset
every
class
of
for
.R
G(C) (in case an a p p e n d i x
o n a semigroup coset.
has property
C
on
if it is c o n t a i n e d properties:
212
S
is a
Since (C3),
we nothe
1.2:
2.1. A c o n g r u e n c e
following
and extensions
is a left
class
is clear from
if a_~ud only
of the
C = coK'
of [18];
congruences
congruence
result
and
C = Koc
used h e r e as a n alternate.
coset c o n g r u e n c e
whose
ted that every
version
in an M-class
so that
1.8 is the d e f i n i t i o n
the d e f i n i t i o n
2.
contained
be as above,
C g H ) in the p r e l i m i n a r y gives
coset
S in
is a left £
cos~
a n d has
GRILLET a) if
x , y ~ s l , a , b c S, xa = ya C a C b
b) if
x ¢ S l , a , b e S,
With ences
a), this
in [9]
ences).
left
coset
~
A congruence
from
in
coset
the definition
must
are not
contained
in
left
coset
has classes
~
is a
eoset
conversely,
it
with both
~.
are
left
coset congruences
For example,
a
a a o c c b b c c c c c c o c
left zero
number
in Tamura's
{e,d},[a,b},{c};
{a,b]
let
S
be the
on
a) (or
aa = a
set congruence let
.Neither
example
will
are
two n o n - t r i -
£ ; the
other, C,
is c o n t a i n e d which
is not
in
semlgroup,
left
even
be any left
coset
In what follows,
though
coset
we put 213
~.
contai-
be used a g a i n ° ) | on the other
C
is trivial
a = ab = b ) ; thus,
is trivial, C
coset
[28]).
b)) is not a mild condition.
every
implies
are
S : one is
of a left zero
that
catalogue
the B-classes
~ = ~ . There
is a left
This
semlgroup,
2. N o w
)08
[e},{d},[a,b},{c]
example
shows
S.
in
edabc debac
, so that
ned in an ~-class.
group
in
is then both a left
e d
congruences
(In p a r t i c u l a r ,
a,b¢ C,
contained
edabc
are
{e,d],[a],{b],[c}
The
~
~.
1
is s e m l g r o u p
The £-classes
hand,
congru-
and hence
that a congruence
there
b c
vlal
congru-
semigroup: EXAMPLE
(this
properties,
congruence;
be contained
O n the other hand,
following
of left coset
congruence
contained
and a right
properties
which
these
;
xb = b .l
of left S c h u t z e n b e r g e r
has
~.2. Every
xb = yb
congruence.,
congruence follows
the name
that
PROPOSITION
implies
is the d e f i n i t i o n
(under
We know
xa = a C b
implies
£
every
(when left
has but one
congruence S/C = A
In a
co-
clas~
on a semi-
a n d denote
the
GRILLET elements and
of
A
by
prevents
with
f-la
ambiguity
is a left
coset
Ca
with
When
unique
for
only
on that
~ e aA l
~
(with
~
A.
C
of
(the
S
I)
each
corresponding
on t h e s e
A
facts,
and
G
GS
then :a
Ga
the
acts
the
A
regard ~ ~
this
relative
than on
214
C
to
how
Two
geneof
S
~,
S
c a n be re-
relations
a left
GS f u n c t o r
in ca-
in [i])°
facts
process
being
left
left
transitively
reconstruction
corresponding
~ _~ ~ ), and
functor,
considered
and
rela-
Schutzenberger
first
simply
is
as a cate-
the
functor
G.
when-
is a g r o u p - v a -
functor
sections
the
A
when
or left GS
functor
~oya
is a p r e o r d e r
(a _~ ~)
in the n e x t and
On
generalized]
it was
rather
the
is all
i implies
~0y~0~ a =
~ _~ a )
C a ; 2) the a s s o c i a t i v i t y
• In fact,
for
~, a o
Y e aA: ). This
language.
functor, [not
that
implies
write
(a,~)
left
see
from
by
G a~
~
= (g.a)b
1.4
, which
ge G a , aeC
in s e c t i o n
and
we can
is its left
We shall
constructed
ruence
Oa
We call
is a c o n g r u e n c e ;
(g.a)b
seen
(which
we also
on
and
~ G~
~a~ : Ga a
~g.ab
on
one m o r p h i s m
to
possible:
that
x~ C 8
be d e n o t e d
for all
a ~ ~ G a , (a,~) J ~ ~0~ G
Ga
will
a,~ e A ,
in c a t e g o r i c a l
Schutzenberger
3.
if
is a c o n g r u e n c e ,
and h e n c e
property
preorder);
functor
functor
we also
•
(which
(the usual
relative
only
Ca
~ = ~Y , we have
C
= (g.a)m
Y ¢ ~A 1
expressed
ralized
f ,
= Ga
~ S I ; note
homomorphism
identity
and
~ ¢ aA I
see that
by
itself,but
convenience
C I = [i]
since
~
~ ga . a m
composition
~0aa is the
lued
by
and s a t i s f i e s
that
gory
For
for a r b i t r a r y
ever
tion
a
CaC ~ ~ Ca~ ] • E a c h
~ e aA I , say
and
, a ¢ C a , b ~ C~
easily
~ A
T(C a) = T a , G(C a)
xC a g C a ) if and
m e C¥,
a
defined,
The
=
and
any
In p a r t i c u l a r ,
g~G
S
is really
8a = a •
such
always
put
~ G
l aeA l
us an a s s o c i a t i v i t y
depends
[this
in s t a t i n g
(i°e.
a,~ e A
Cam ~ C~
projection
Ca
:T
a a=
x¢ T
8 e AI
gives
e.g°
~
when
in g e n e r a l
by
and we also
projection
define
se
a,~,..°, the
the C - c l a s s
make
thi~
on the ~0a~g.aab
depends coset of
only cong-
S ;
GRILLET hence
we shall
only
i),2)
group
and
study
G
more precisely,
transitive
) A of
(goa)b
G
was to o b t a i n
2.3.
S,
e.g.
an e x t e n s i o n relative
If
that
extensions
we o b t a i n tensions
8. S
is closed under
by
We conclude
and
f :S ~
~,
on a se-
then
Sch~tzenberger
S
is
functor
The
which
arises
manner. A
(with projections
that to
in case
f'o 8 = f C
(which
g~ G
is
and,
sends
for Ca
is a c t i o n - p r e s e r v , a ~ C )~ We call
8
of e q u i v a l e n c e s
of
class
composition
there
and
inverses;
on the class
thus
of all ex-
this
C
all
e~tensions
of
G
by
A
and
it
to do so up to equivalence.
coset
in p a r t i c u l a r
in
A o
suffice
: S/C,
congruence
f'o 8 = f)
relation
to construct
of left
8
for all
of eztensionso
G
by
such of
since
= g. Sa
clearly
perties
G
~-S'
We want
4o
coset
contained
in this
of
a n equivalence of
1,2
fact:
~- A ) are equivalent
C' = f,-l([~]),
extensions
o
the main goal of sections
is an e x t e n s i o n
a E A , the restriction
an e q u i v a l e n c e
re,
S,S'
~ A , f' : S'
8(g.a)
such that
, b~C~
S/C .g
congruence
iug (i.eo
, a¢C
is a left
extension
will
Ca = f-l({~]), gcG
coset
upon
S vor,
a ~ A, a simply
coset
each
is a semigroup
with a s u r j e c t i v e
A left
an isomorphism
by
together
from a left
f :S
(preordered
, (a,~) I ~ qo~
for each
any congruence
C , by
A
where
be a semi-
S
on
C
A
of its left g e n e r a l i z e d
to
Two
by
the now obvious
THEOREM migroup
on
A
and,
whenever
It can be detected
functor
G
situation
let
G :~ i > G
a semigroup f :S
action
=
of
general
precisely,
as above),
• A n extension
homomorphism
~goab
More
be a g r o u p - v a l u e d
left d i v i s i b i l i t y (~ ~ aA I)
it in the more
are known°
section
with some elementary
congruences. denotes A
a left
The
notation
coset
is the projection.
215
pro-
is as befo~
congruence,
A =
GRILLET PROPOSITION a subgroup ea = a
~
=
of
whenever
~
so
For every
idempotent
identity
elemenb
a e C a , ca = a
¢
e
of
; be = b
of
A,
C¢
C¢
is
satisfies:
whenever
b e C~ ,
•
Proof. ae C
2.4.
S ; the
That
, ca = a ~ae = i
Ce
is a s u b g r o u p
; then and
follows
ee Ta i n e
ea = a .
On the
~¢ = g.b ~ ° T h e n b e e C~, so that Now = be = (be)e = (g.b)e
from
lo3.
Assume
must
be an
idempotent,
other
hand,
take
b e C~ ,
be =B g.b for some = q0h¢gobe = g°(g°b)
g e~.bo_ =
F
it f o l l o w s
that
g = g~ , g = I
It f o l l o w s set of left
from
idempotents
coset
congruence
ment
S
upon
~o5.
always
This
follows
[15].i
(The
conclusion
5° T h e
Proof. let
inverse
By
x¢ of
a
in
e,¢
z
g ~ 1
2.7. ~ of
P r o o f . (A g r o u p Let
x¢ T
a
hence:
every
left cos~
result
of L a l l e se-
in
of
be
when
that
the Ce
is
is a s u b g r o u p i~ trivial.l
the c o n d i t i o n : A
is finite), element
lies
( g = ~ x). Since
216
~
the m o n o i d
inclusion
by a g r o u p
is one g
f - l ( H c) °
xyz = e , with
f-l(H¢)
satisfies (e.g.
element induce
element
of S°
= He .
x y e C¢ ; since
converse
is i n d u c e d
of
meH c • Let
C c • Hence
hence
A
f-l(H¢)
element
Then
in
inverse;
When
2.4,
with
every
periodic G
also involve s u b g r o u p s
o Thus
f-l(H e) g H e • The
implies
In p a r t i c u l a r ,
for c o m m u t a t i v e
identity
y e C~.
a right
6a = ~
holds
as
and
has
element
of the
idempotents~
semigroup,
also
inverse
He an
PROPOSITION
A.
a well-known
x¢ Ca
f-l(H¢)
group.)
from
say
= C~ g f-l(H¢)
and
of
separates
is an
yz e C~¢
S
that
a bijection
~ •
With
~.4,e
has
induces
properties
2.6.
f-l(H¢),
a group, xy
be = b . |
£ = M .)
remaining
PROPOSITION
of
in
Proof.
as then
f
On a r e g u l a r
is c o n t a i n e d
migroups,
Now
that
of
congruence
COROLLARY
2.4
and
in some
every of
S.
sub-
g ~ i, x ~ I
GRI~ET i.e. x ¢ S , and theiis, is in of
x ¢ C 5 with
6n = ¢ H e • Note
Co,
then
2.8°
implies
5
and
and 2o4,
When
5a = a.
for some
ea = a = 6 e m .
wee = 1 , by
COROLLARY 6e = e
that
y = x e ¢ C6¢
Furthermore
finite
5¢ A
is idempotent
y
n;
If
e
6 n+l = 6e
is the
idempotent
is a group
and
By the hypo-
then
element
A
satisfies
the condition:
periodic
of p e r i o d
i (e.go
combinatorial),
every
by 2°6o
g = wax = waxe = ~ y o|
non-trivial
when
element
A
of
is G
is e
induced
by a n element
Proof. has p e r i o d
Same
of a s u b g r o u p
as for ~.7,
I) and so
yg C
save
Cc
o__f S (with
that n o w
5c = ¢
ce = ~)o (as
5
.| C
COROLLARY every
left Proof.
each
Ge
~o9.
eoset Since
Then
Our
last
congruence
only of
result
yields,
Proof°
of
idempotents,
Card C a = Card
PROPOSITION
combinatorial
shows
G
= i
that,
If
It suffices
S
to prove
In case with
2)
property
we can
then G
C
~, a left
coset
per $ - c l a s ~
in case: i) a g ~ ; ~0~, ~
inverse
is a congruence)
mC e ~ C ~ , nC~ g C a , and
is
G a z G~
~ G~
are m u t u a l l y
(since
and hence e .m
one group
2) ~ £ ~ . In case I) we h a v e homomorphisms the c o m p o s i t i o n
S,
idempotent,
by 2.7,
much like
e ~ ~,
are
for all
up to isomorphism,
2o10.
semigroup
is trivial.
all group elements
consists
trivial°
On a finite
congruence
then the result
which by
isomorphismso find
m,n¢ S 1
follows
from
1.7.S Note class
of
2.4 shows
that A.
be a s u b g r o u p
in the case
obtained
ourselves
of the S c h ~ t z e n b e r g e r
I (continued).
yields
a group
one group
group.
S-classes,
Either
per S-
to a ~-class
of a regular S-class
true for irregular
EXAMPLE congruence trivial.|
in fact
If we restrict
that
need not be
we have
of
this
However, as s h o w n
non-trivial
S, will
this by:
left
of order 2 for D a , whereas
coset H
is a
217
GRILLET
]. Extensions:
In this group
A
section, G
(preordered
tension of
G
io The ted from gruence each
by
A o The n o t a t i o n
and
A
Since
G
every e l e m e n t of some unique jection
f,
and
S
is an ex-
is as in paragraph S
of the con-
i.e. select one element
can be written
in the form
; in other words,
of any disjoint
union
t] G
Zo3.
can be r e c o n s t r u c -
acts simply and transitively S
~ c A, g ~ G
6
functor on a semi-
is to choose a cross-sectlon
induced by
C.
is a g r o u p - v a l u e d
by left divisibility)
first step in showing how
G C
structure
pe
in
on
Ca,
g.p~
for
we obtain a bito
S o Note that
EA
makes a commutative jection
[J G ~A ~ and transitively g.(h.pe)
G
for all
It remains form
g.p~
ments
p~
for each
of
of
g¢ G
each
G
acts
8
, we conclude
to find a suitable S
description
when its elements
multiply.
simply since
that
the res-
of the mul-
are written under
the
need to know h o w the ele-
We know that
there
pro-
is action-preserving.
• For this we first
~,~
f and the obvious
on itself by left multiplication;
~ C
tiplication
with
> A • In addition,
= gh.p~
triction
triangle
is a unique
p~p~ ¢ C C~ ~ C ~ ; hence ~ ,~ ~ G
~
such that
• Next, for each g ~ G B we k n o w that PaP~ = ~ , ~ p m ~ p~(g.p~) g C ~ ; hence there is a uniquen b e G A~ such that p (g.p~) = h.p~p~ be a c o m m u t a t i o n
; we denote
p~(g.p~) ~ G ~
•
The double
on
a
appearance
aria
~
.]
When
G
and
w ~ , ~ g , showing
with
w
= w [ , ~ g , p~p~
[We could also denote
the group e x t e n s i o n case, w ~ (by a n a l o g y
by
w
to
operator:
(3.1) w~,~ : G~
h
of
save that ~
in
218
h
by
g~
,
it also depends
as in on
w~,~. allows to write
q0a~ ) in case
are known,
,
it should depend
we can calculate:
~ • it
only
GRILLET
(3.Z)
( g . p ~ ) ( h • p13) =
~
o
= ~g
o
=
,~h o
~o¢~go
•
,~ o p ~
So far we have followed the same procedure group extensions, under a similar
and of course
form. To put
a trivial m o d i f i c a t i o n define
X~,~ : G~
(3.3)
a~cl ~
]. T h e n
(3o4)
this
obtain the m u l t i p l i c a t i o n For each
X~
~,~
•
in case
it depends
= (~[g.
~
T = __~VA(G × [ a } ) . multiplication
on
on
T
(3.4)
o
union of the
T
O a ::
says that we can define
a
by :
= (~g.
% , ~ • ×~,~h , ~
) •
~ S , (g,~) ! ~ g.p~ , is an isomorphism.
particular and
Formula
(g,~l(h,~) 6 :T
only
~ . X~,[h)op¢~
We now select the following disjoint
Then
a,~ ~ A ,
(3.2) becomes:
(g.p)(h.p~)
(3.5)
only
by:
X ~ , ~ g = - ~,~ i " ~,~g
[again we write
as for
it under sandwich form,
is necessary.
~ G ~
o p~p~
is a semigroup.
Furthermore,
for all
In g,h¢ G
k ¢ G~ •
• ~[3g .
.
. ~o [~g . qo ~h
.
~c~,~
= (go(h,~))(k,~) since
~
G
A o It is clear that
by
sions,
is a homomorphismo 0
Thus
X~,[3k , ~ ) =
, T
is an extension
is an equivalence
of
of exten-
and we have proved:
THEOREM
3.i
Every e x t e n s i o n T = ~A(G
of
G
by
A
theorem for extensions).
is equivalent
x [~}) = [G,A;o,X] , where
ous projection, = (gh, a)
(Weak structure
~ ,~ ~ G(L~
Two remarks
to an e x t e n s i o n ~ A
is the obvi-
the group actions are given by: g.(h,a)
and the m u l t i p l i c a t i o n
for suitable
T
and
on__ T
X~,~ " G~
are in order before
219
is given by
=
(3.5)
~ G ~ om
we find what "sulta-
GRILLET ble" means. cation of only
A
First, S
3.1 completely
(up to isomorphism)
and a bunch of groups.
determines
The second remark is a com-
parison with the two-sided approach approach
is self-dual and hence
in [18]o The two-sided
intrinsically
whereas we shall have to recover self-duality C ~ ~ ) in a rather devious
the multipli-
in terms of data Involving
more elegant; (in case
way. On the other hand,
ve shows that the needed maps
G~
of purely left-sided concepts;
that is one of the central
facts of this paper, with the case
and what makes
arise in terms
it possible to deal
C ~
2. We now complete
theorem
3.1 by giving necessary and
sufficient
conditions
that the multiplication
sociative;
with these conditions,
struct all extensions, The basic
> G ~
the abo-
(3°5) be as-
we can theoretically
cor~
in view of 3oi.
condition
for associativlty
is obtained
by
calculating :
rf - .X ~,~y ~Y (~o~yh. oiB ,y'X~,y Y k) ,=l~Y) (g,~)((h,~) (k,y)) : (~0~yg'O~,Dy ((g,m)(h,~))(k,¥)
~ ~ = (~m~y(qo~g.o
,~ .X~,~h).o ~,y.X~,Y yk,~y);
since the ~'s are homomorphisms perty,
we obtain the necessary
(3.6)
for all
~,~V. ~,~v
with the composition and sufficient
pro-
condition:
08,
~,~,y ~ A, he G~ , k~ Gy •
This condition will break down further in case the X'S are also homomorphlsms One might expect
with the composition property°
(3.6) to imply this;
that it does not is
shown by the following EXAMPLE 2. Let
A
be trivial and the functor
values the symmetric group identity
thereon.
Let
S
S3
(table below left)
G
have
and the
be the semigroup having the same
220
GRILLET elements
as
G
G
eabpqr
S!eabpqr
e a
eabpqr aberpq beaqrp pqreab qrpbea rpqabe
eieababe a abebea b beaeab p pqrqrp q qrprpq r rpqpqr
b P q r
and the m u l t i p l i c a t i o n
associativity S
is checked
is g e n e r a t e d
gives that
the a c t i o n
Finally,
of
notation)
right
the same or
on
is it the map,
e
e.g.,
(wp~
particular not have tension).
for all
in
trans-
is indeed an extension;
g ¢ G, x , y ¢ S .
y
insures
it is simply
action
by
G
of
g
Observe
has,
(in
S )
by
has, in
G
g
in
formally,
Pa
i.e.
• Thus on
w
also
I,
and c a l c u l a t e the
e(g.e)
= wg.ee
= wg.e •
w(p 2) = we =
is not a homomorphism;
G o This
G,
by e i t h e r
as single
element
that
formally,
must commute.
(3.1),
that
also
the two operations
an extension
applies
well-behaved
for some
more
choice
extensively
we shall show that
that property For
nor
to the single
3.2. G o o d
of the elements
Proof.
Assume
its
left coset
X
maps
p~ • This
section;
in
e x t e n s i o n s do
2 is not a left
coset
ex-
we just note: is not a f f e c t e d
by the
pm •
that each
yields
in the next
behavior
are homomorphisms; • This
when all
of the elements
some
(example
the time being,
PROPOSITION
P¢'¢Ca
this
e.g.
left
in v i e w of (3o3)oi
will be s t u d i e d
X~
S ; associativity and clearly
right:
noting above
we = e , wa = up = a ; then
identity
We call
all
below
as left m u l t i p l i c a t i o n
w , using
are h o m o m o r p h i s m s
choice
table
as right m u l t i p l i c a t i o n
We now p i c k
= e ~ b = a~ =
test,
• The
the left
x
b ; therefore
One finds,
on
multiplication
effect
only m a p p i n g
×
G
action,
p
we show that
the same e f f e c t while
by Light's
and
it, g.xy = (g.x)y
(due to our
a
a
it is a group
itive. that
by
table
pC
has been
choose
new elements 221
chosen
another
so that
cross-section
G'e,~ and maps
¢'~a,~ '
GRILLET X '~,~ • For each ~¢ A there is some p~' = u .p~ ; h e n c e , by (3.4), ' ' = (u . p ) ( u ~ . p [ ) p~p~
using ( 3 . ~ )
= ~ ~a ~ u
o
~,~
again and the hypothesis
m'~,~g • p~p~'' = p~'(g.p~) =
~ u
=
= k. X[,~g" k -I homomorphlsm.
•
.o ,~
on
o
such that
k~ ~ ,~ u ~
• pa~
;
X~,~ :
= (u . p ~ ) ( g u ~ . p ~ )
=
-x~,~gox~,
k o X~ , ~g"
k : ~U
where
o
ua~ O
k-i
and
for all
" P ~'P ~ ' ,
g¢ G a . This implies
ge G~
By (3.]), X ~
; therefore
~,~ ~,~ g =
m~9~_, is a
is also a homomorphism.|
3. We now resume our investigation of associativityo First we consider the general case. A useful eonsequence of (J.l) for all
is that
always, hence
m,~ , by (3.3)° We use this to analyze
If first we let
(3.7)
~[,~i = I
o~ , ~ Y
If we only let
•
h = i, k : i
×13Y
o
~,I~Y ~ , Y
=
~13
X[,~I=I
(3.6).
in (3.6), we obtain:
~p~yO
,~ o o ~,y
k = i, we obtain:
o ,~y o Xa,~yGP~y
,
~DY a,~
since, by (3°7), ~a,~Y = q0a~yOa,~ • Oc~,yO (x~Y ~,~yO~,y) -i the last formula is equivalent
(3°8)
(×13Y
o
to:
y) -i • ×~Y ~,~y(~yh
o O~,y) =
Finally, from (J.6) and its particular case above, we obtain:
,
=
222
k -- 1
a, l~Y(m~vh'c~, y) °X~13'
GRILLET which,
t o g e t h e r with
the maps
X
it is clear
that (3°7),(3.8),(3.9)
A pair O
O ~
~
X[,~I = 1 , shows that,
(o,X)
and
of families
(X~,[)¢,~e A
is a factor cosystem [We keep factor
Conversely,
together imply
(o ,~) , ~ A
o f mappings
(3.6).
of elements
Oa~
X~,[ : O[
(3.7),(3.8),(3.~.
in case it satisfies
system for the corresponding pair
If now we are in a w e l l - b e h a v e d
(3.8')
collectively,
are not too far from homomorphlsms.
extension,
(m,o).]
(3.8)
reads
X~Y a -1 ~Y ~ h • X~Y ¢¢,13Y 13,Y ' Xa,l~Yr'Pl3Y c~,13Y°13,Y :
-~ ,y " wad3yZ,~. ~ ~t3 ,~ ' ~h • °a~,y Oc~
= In view of
(3.8")
(3.7),
this is easily seen to be e q u i v a l e n t
~ h o o -1
13Y
oa , ~ Y
to
= ~o~..o -~ "~ ~ h . ~ " ~~Y o o.,t3 -1 ~!a¥ ct,!3 ° w~13YXo., Either formula
shows that
automorphismso
[As we shall see
fact that they
commute only up to inner automorphisms
connected e
Still
(3°9') [There
and
X
commute
i , at least
Xt3Y
c~,t3Y
XY
Y
t3,Y = X ~ , y
is no evidence
that
~ = i ~ A = Al
to define
X~,~_
X~,~.
on Leech's
section.]
of the c o r r e s p o n d i n g
C ~ ~ .] (3°9) simplifies
to:
is the identity on
When it holds,
[18]). S i m i l a r situations in the next
9,@,
•
as the identity,
ved£ in case all its
is
G~
though we shall see this holds
left coset extension. yields a functor
case,
the
between the maps
in the case
in the well-behaved
up to inner
in the next section,
with the lack of coherence
in section
when
~
or when
(3.9')
category
A ~ A I allows
shows that
]L(A) (defined
X in
are commented on more thoroughly
We call a factor X
in a
maps
oosystem w e l l - b e h a -
(or, equivalently,
factor system)
223
all
w
are homomorphisms.
maps
GRILLET Formul~ rephrased
(3.7)
to (3.9) and their variants
in terms of w
rather
than
reader to state the correspondiug fication and
occurs;
w
(not merely
but then the partial for
w
composition
T = [G,A;o,X] (3.5)). then
T
T G
property
A.
so that
(l,~)(l,~)
is valid
given
(~,X)
an ex-
is well-behaved,
To see this,
cosystem
by
as we saw,
(P~)~eA' : namely,
(o,X) • Indeed,
' ' = P~P~
in fact,
extension.
cross-section
rise to a factor
show is just
(3-9')
cosystem and construct
If furthermore
is a well-behaved
simplinow shows
only.
be a factor
is a semigroup;
by
an appropriate This gives
(3.8")
as in 3.1 (with multiplication
Then
tension of
(o,X)
is that
up to inner automorphisms);
up to inner automorphisms
We now let
it to the
f o r m u l ~ . Little
the most striking
commute
can also be
X o We leave
we s ~ e c t
Pa' = (l,a)o
(c',X') 9 which we
by (3°5):
(o ,~ , ~ )
=
= ~ ~,~ ' P ~
,
o'~,~ = o ,~ ; furthermore,
p~(h.p~)
: (l,=)(h,~)
= (a=,~ o
= °a,~°X ~,~h " P ~ for all
h e G~ , so that
implies
X~
X~,~
=
"
X~,~h,
mS) =
~ ^h.a-l^ ' ' ' = °a,~ oX a,p ~,p " P~P~
w ~,~ '~ = m~ It follows
,~ ; finally ' (3.3) that
T
is a well-behaved
extension.
Let
We have
proved:
THEOREM
3.3
G
(Full structure
be a ~roup-valued
functor
tiered by left divisibility) (resp. a well-behaved G • Let
T = [G,A;o,X]
with m u l t i p l i c a t i o n (3.5)
Then
T, to~ether
group actions well-behaved
(a,X)
cosystem)
(preor-
be a factor
cosy~em
on
A
union
X~,~h,
with the p r o j e c t i o n
g.(h,a)
A
with values in U
x ~})
~).
(g,m) i ~ m
= (gh,m) , is an extension of
(G
by:
= (~g.%,~.
extension)
for extensions).
on a semigroup
be the disjoint
defined
(g,a)(h,~)
and
factor
theorem
G
by
224
A.
C0nverselF,
and (resp. ever F
a
GRILLET extension
(resp.
equivalent
well-behaved
to an extension
4. This
extension)
[G,A;~,X]
of
G
by
A
is
built as above.i
theorem must be completed by a c r i t e r i o n
two extensions
[G,A;o,X]
giving sufficiently wise we have
be equivalent,
many cases
to build in terms
momorphisms).
Such results
4 , respectively,
that
and by results
of good behavior of mappings
(as other-
rather than ho-
will be given in sections
5 and
and together with 2.3 and 3.3 constitute
our main theorem° We conclude 3.3. First,
this section with some remarks
as a structure
theorem,
ted than most of the structure introduction.
This
simple
semigroups,
constructed
whereas general
m,~ ), in which case
extensions
tion there often are s i m p l i f i c a t i o n s ~,X,~
(e.g.
structure
be a c c o u n t e d
for by a general
that such descriptions
(i.eo and
many
ca,~ = 1 (3.8')
for
becositua-
in the d e s c r i p t i o n theorem,
w-semigroups) theory;
cannot be
in any specific
in the R e e s - S u s c h k e w i t s c h
t h e o r e m for bisimple
First,
of (0-)bi-
In addition,
(3.7) disappears
Last but not least,
in the
of reasons°
are of split extensions
mes very simple.
mentioned
are constructions
in terms of only one group.
of the examples all
theorems
is due to a variety
most of the existing examples
concerning
it looks more complica-
of
or in the
which
cannot
indeed it can be said
are now the main point of these the-
orems o The m u l t i p l i c a t i o n group e x t e n s i o n clearly
(3.5)
form (using
the best
could have been g i v e n w
rather than
form for group extensions,
X )' This
in the is
for then one
can arrange we give
that w~ depends only on ~ . But the form ~,~ is b e t t e r for semigroups. This is shown by non-
-split examples
such as completely
bands of groups; cance of the
X
no s u c h result
but the strongest maps in case
for
w
unless
simple semigroups
C ~ ~ w = X).
225
and
reason is the signifi( lemma 4.~ ; there
is
GRILLET
4. B e h a v i o r
i. We b e g i n can h a p p e n a left
by an example
in left coset
coset
extension)o
this can be partly of this
section;
example
may
extensions
explained
by
(3.9)
from
in example 2. This
complicated;
and the other
results
to find a s i m p l e r
behavior
3. A g a i n we start
and its table
is rather
failure
that good
that bad b e h a v i o r
(as example 2 was not
The example
the author's
indicate
EXAMPLE
showing
is more
common.
the symmetric
time we let
S
group
have
S3
ele-
,
!
!
merits en,an,bn, en, a'n,b'n (n _> 0) and Pn, qn, rn, Pn, qn, r n (n ~_ i)o The m u l t i p l i c a t i o n is given by the table next page, which
is written
m _~ 2 ; and each closest
with
blank
the
following
in the table
to it on the same llneo
conventions:
stands
E.go
n _~ i ;
for the entry
elb n
is blank,
so
elb n = ela n = el+ n • Associativity S
is g e n e r a t e d
midable and
as
tered
by, say,
it seems.
is f o r m e d
way that
only
while
The
table
C
within
Light's
table
of each block
on
S
shows
that
C
A = S/C
(n _>
table
i) ; we see on the
element,
while
Ta = Ce U C 6
readily with G
G
= Ta
seen Ta
, with
multiplication
6~ = ¢, 6~ = ~ .
that
(under
is indeed w
the action in
Next
for all o t h e r C
need
a
be en-
with classes
is indeed a c o n g r u e n -
C m £ • The s e m i g r o u p
,
in such
, C n 6 = [e'n,a'n,b'n,~n, ~' qn" r~]
6 , ~n
entity
3 × 3 blocks
and all
ce and that Bn 6
is not as for-
into
test.
, C 6 = [e' , a' ,b']
n > I o The
task
noting that
each block
be the congruence
C~n = [e n,an,bn,pn,qn,rn]
test,
decomposes
the top left entry
performing
= [eo,ao,bo]
with
by Light's
ao,al, e'o • The
by permutations
We n o w let
C¢
is verified
)
ae A,
for all of
G
S
226
on
that
we find
a left
consists ¢
is an
¢ , id-
T¢ = T 6 = C¢,
and then
coset
of
it is
congruence,
m o We may as well take Ca
then given
by left
--...a
+
4D
+
+
+
÷
4D
+
~
÷
~J
q.-
"~
+
~
+
+
~
+
'~
-t-
CY
.+
-t-
CY
q-
C~
~
~
4-
+
+
q-
~
+
@
q.-
-t-
eD
q.-
CD
+
+
+
÷
+ 5
,.O
+
÷
"S
Jr
q-
+
q..~
~
4-
q-
+
+ D
0"
+
+
+
+ D
~
+
q.-
+
+
®
,÷
:4~
"D
÷
~r~ q'-
÷
÷
+
~
4-
Jr
43
÷
+
÷
CO
q-
+
-'k
CY
q--
+
+
+
~ 4:~ CO CY I.-~ ~ ~ I-.- I~ 4-t-t+ +
÷
+
+
~3~
I,-~ +
t-~
~
+
+ ~
qS
4:3
+
q~
q5
CO
÷
q'D
q..5
CY
+
-tD
q5
+
'~ 5
÷ D
+
I'~
®
cT
I.-
4D 'D cY ~ i-~ tt--, q+ Jr 4-
'D
I-~
Cb
~
~
4D
~
S
0
0
0
O~
,",s
®
@
CY
5
0"
o"
el:,
CD
~
0
0
0
~
0-0-0-0
0-0
0
0 - 0 - 0 - 0
0
0
Os
c£
o"'
4D
S
o
0
0
¢'
0
cT
5
~ ®
~
5
5 cY
@
@ cT o - o - 0 -
o - o - 0 -
c:T ~ cb o - 0 - 0 -
0
@ 0 " el) 0 " ~ -0 -0 0 0
0
0 - 0 o 0 - 0
D o"
cD"
o" o-
o-
(I) o-
0
0
0
~3
Q SS
GRILLET In each
Ca, let
p~ = e I , p ~ instance,
p~
be the "e" element of
e~ • It is easy
to calculate all maps
e°
(eo)z
and
upon
e'o,ao,'b'o
e o , whereas
a homomorphlsm cidentally, At
sends
eo,ao,b °
ao • We see that
(w~,~e~) 2 = bo • Thus
and the e x t e n s i o n
it sends ~,~
is not
is not well-behaved.| [Inof t h e o r e m 3-3]
this point we might as well make sure that some are well-behaved.
EXAMPLE
1 (continued).
In example
congruences, groups,
w~,~
upon
this example was built by means
extensions
ways.
~ • For
ele I = ela I : elb I = e£ ~ eoe 2 , elP 1 : elq I =
-- elr I = aZ = aoe 2 , and hence upon
C a ; e.g.,
A n easy example We know that
i, relative
all classes,
is:
X~,~I_ : 1
al-
to any of the three left
coset
and hence all S c h G t z e n b e r g e r
have at most two elements;
thus no
X
can fall to
examples.
The e a s i e s t
be a h o m o m o r p h i s m . 2.
We now turn to more general
and most
important
is g i v e n by:
T H E O R E M 4ol. All extensions contained
in
Proof.
M
a r i s i n g from c o n g r u e n c e s
are well-behaved.
Let
~
the c o r r e s p o n d i n g
be any c o n g r u e n c e left GS functor,
tation be as before. C ~ ~ , we have
Pick any
k.p~ = p~y
in
~, G
be
and the rest of the no-
~,~ ¢ A
for some
w~ (hk).p~p~ : pa(hkop~) ~,~ =
contained and
h , k ¢ G~ • Since
y ~ S i • Hence
= p~(h.p~y)
p~ (~o~h. p~)y = pa(h.p~)y
:
:
(w~a,~h.pap~)y
~ h " PaP~Y = w ,~ho pa(kop~) : ~- ~ w a,~
= ~,~h o w~,~k, pap~ • It follows
that
w~
is a homomorphlsm.
We now give another proof nificance
of the
X
of 4.1,
maps in that case.
228
T h e n so is
which shows When
X~
J
the sig-
C ~ ~ ,C
is
GRILLET also a right GS functor
coset congruence,
G'
(dual of
G)o
hence gives
Furthermore,
ments
Pa
: Ga
~ G'a ' defined by: g'Pa = p o ¢
yields
(see 1.6). ctor
G'
Maps
for each @~
a ~ A
:G~~
this yields LEMMA
G~
> G~
g
for all
are furnished e
the
X
4.2. For every
the ele-
ca
Cp~
g ~ Ga by the fun-
can be used to transfer
> G ~ • The following
precisely
choosing
an isomorphism
and the Isomorphlsms
them to maps
rise to a right
lemma says that
maps°
a,~ ~ A, the followi~g
diagram
commu te s :
G~
> O~
a~ Proof. pa(hop~)
For all
h ~ O~ •
= pa(p~.~h)
= P~tP~ o q~[~[5 h =
=
~ c~h =
=
Then
X~,~h
a~
~
a , ~ " P~P~
"
= c~*~
•-~ ~ c~h
¢~ h
follows
from (3"3)°"
This lemma immediately implies that in this case X~,~_ is a homomorphlsm, thereby giving another proof of 4.1. Furthermore it shows that X inherits the composition property from @ ; in view of (3.9~, this is stated as part of: COROLLARY and
identity
4.3. so
When
C = ~,
c a n be w r i t t e n
~
depends
only
Furthermo
l~pon
×== i s
the
on G om
However,
the main consequence
C = ~ , our structure
theorem
sion is essentially
self-dual.
G
X~
of 4.2 is that,
for the corresponding
in case exten-
If we consider the groups
as g i v e n and the homomorphlsms
229
q0 as unknown,
then the
GRiLLET theorem
becomes
playing
dual
X
as g i v e n
roles. as
that
the o
terms
of
a better
only
furthermore,
4.Z
also
of the
in fact g i v e s
a precise
when
the maps
where the
these
a,~
(i.e.
and
composion
A
construct
This
3.3
in
is u n d o u b -
in case
cohomologyo[We commutative
C ¢ ~ ; do not
case)
be-
for w h i c h
this
advantageous.]
our preference case
set
measure
in of,
rather
(3.8)
to,
chosen
a ,~ = i ); the
X
that
is r e l a t e d
lack
It also
can be c h o s e n
for
C ~ ~ ) it s h o w s
the
involved.
c a n be
Pa
and
be d i s c u s s e d ,
in the
factor
maps
to the
~
of c o h e r e n c e gives
coherently: so that
other
p~p~
us
two
one
is
= p~
is w h e n all
G
abelian. 3.
We n o w
~ ~ o The within
turn
first
any
A) a
implies
fixed
an
4.4.
£-class First
in case
behavior
is a l w a y s
for all
that
assume from
the
A,
gcG
lows
Xa Y,m
of
have
the
Y~ Z m
in
> Gya
good
then
which
X ~a ,
with
m,~
A.
there
that
By 1.7, that mC
~gomc
~ Cya
m = wa Y,~
230
(with
ex-
=
• We can • It fol-
isomorphism.
that
only
property°
where
see
ya £
depends
composition
such
~ = ym £ m , 6~ £ ~
(3.9)
extension,
isomorphisms
, c¢ Ca,
and
is an
coset
isomorphism,
• :G
c = pm , m = py
follows
is a n
assume
let
Now
extensions
that
In a left
X ya , a
isomorphism
= m(g.c)
shows
y~ ° F u r t h e r m o r e
Proof. ists
coset
one £-class.
and
in any
to left
result
PROPOSITION (in on
will
~ ,~ ,e
elements
f o r all are
the
in the
explains
(still
the a p p e a r a n c e
cases
~c A),
is not p a r t i c u l a r l y
m • Finally
(subject
theorem
for
~ ,X
if we c o n s i d e r
(3.7),(3.8)).
(save
extensions
presentation
between
to
with
from two f u n c t o r s
to e a c h
to p r e s e n t
follows
symmetric,
is true
we s t a r t
it is a must
cause g e n e r a l
Lemma
is,
(subject
way
it in w h a t
than
same
same group
tedly
use
The
left-right
well as the g r o u p s
tion property; assigning
entirely
Y, S c A). It
GRILLET X 8Y= ,ya (~h for all
o o y,¢ O Xy,~k) ¢
h ~ Gy , k ¢ G
isomorphism, y~
o Since
by the
first
69 £ ~ , X 6 , y ¢
part
a o X6y, ~k
~y,¢) = X
of the proof;
,9
~an
hence
~
X6,y~
Xy,~
= X6y,a
Next, induces g~ Ga
assume
that
k = = py ~ Oa
y¢ = ~
g ~ G
with
y ~ A)° T h e n
py(gop
k = o
= ~ y¢ , ~ g ' P y P ¢
Comparing
(with
; thus,
; in p a r t i c u l a r ,
py(g.p¢)
) = kg.p~
(let
we see
for all
g = I). But
= ~ y~ , ~ g ' ° y , ~ ' P y ¢
the above,
py c T
that
= °y,¢°Xy,¢g'P~ X~,ag
= g
~
for all
o Then
Xy,~ . ¥ but
assume
is the only
on
~
on
morphism
inverse
but only
on
x~
first
second
O
and
G
whenever
yields
o Hence
and
thus
We may
from
result
that
y~ = ~ ° T o g e t h e r
the
category
X
theorem
can be p r e s e n t e d group
presentation
maps
in terms
in the
do not
allow
to the
lack
general any
3.3
left
further
of c o h e r e n c e
use
of
between
23l
this
is the
is the
not d e p e n d
coset
isoon
denote
y
it by and
extension,
is the
(3.9')
identity
this
a group-valued as
left
that
coset
case,
We note the Leech ~
and
but
that
C ~ ~ , the
assigning a better we
cannot
our m e t h o d s
categories, X.
that
extensions
of functors this gives
means
on
functor
in the case
Again
for
and
on
the a b o v e . m
of a p a i r
situation.
then
for some
is then proved,
with
thus,
e c A.
~ = 69
if we wish
for w e l l - b e h a v e d
to each
9 = a,
not d e p e n d
Xy,~
does
X~,~_
to d e f i n e
]L(A);
of theorem
use it
then
of the s t a t e m e n t
the
on Leech's
the same
bT~ = ~
does
= X6y,a~
of a w e l l - b e h a v e d
we c a n use
structure
X~6,9 X y , ~
ye = 9 "
y ~ A)° If
hence
9 ~ a,
is now clear
case
still
and
X~, 9
half
half
4°4
that
of
(with
G
9 ° If
since
e
In the
on
and
seen
identity
• The
~ £ 9 : ¥~
identity
6¢ A ; we h a v e
the
= X 6Y¢, y ~ ( ~ h .
due
GRILLET 4.
We now give two more cases where left coset exten-
sions are well-behaved. THEOREM
8
implies
4.5.
periodic
combinatorial b_~y A
Let
A
be a semlgroup
in which
of perio d I (e.~_. let
semlgroup).
Then every
A
6~ = a
be a finite
left coset extension
is well-behaved. Proof.
= ~,~h.
It suffices
w~,6k
the formula
whenever
is trivial
further assume 2°8 that
to prove
h
that
a,~¢A,
when
and
66 = 9"
affected
by the choice pc
= (u.p6)p~
= ~6~u.p6p~
and
induces
C6, where
good behavior
6
is not
Pa ; by 2.4 we may
¢
implies
is, and then we also
o6, ~ = I
= qo u.p~ , where
with these remarks g e G~
(since
and
uop6 ¢ C 6
k = ~0~v
with
hop~ = induces
v ¢ Gc
we begin the proof
is induced
and
by
(t.py)p~
presentations
g,
t.py
¥6 = ~ , joe.
for
we have
for
hk, h
if
with
= q°Y~t'°¥,6_ ° P76 )" but we also need and
k ]o When
then
w ~a,6g • pap6 = pm(g.p~) = (w Y=,Yt • papy)p~
-~¥
"Y y t
: p=(t.py) p~ = = ~ay~-a¥wYa,Yt o p=pyp~
(
.ay , y yfi o w~ ,po¥,p = ~a¥13~a, which proves
proper, by
if and only
g = ~t. o oy,~
= (t.py)(1.p~)
use this presentation
the better
= i
o e,~ = I .
that
[We shall
~,61
and we may
of some
whenever
h = q0~u • Similarly,
g'P~ = (topy)p~ VP = ~
• Since
k = 1
of the elements
be idempotent
p6p~ : p~ • This
Armed
or
By 3°2,
know that
noting
h = i
=
h,k ~ 1 o In this case we know from
then let
h , and hence
m~,~(hk)
h,keO6
is induced by an element
is idempotent
c~ = ~
that
that
w ,~g = ~ a ~ a , y
232
o p=p~
,
,~Oy,~
o [Since
top¥
GRILLET g = ~ Y t . u y ,~' this sorts for ~ .]
is
Applying
to
tain
h, which
= qo~8u~6 ~ ~,6 u
• ~,~h
induced
this
a product-preserving
by
vope
(as
and hence
by
u.p 5 , we o b -
is induced
by
We then
k
; therefore
,ga6e,~
calculate:
=
c~8¢ (~8e 8¢ 8 ¢ v). cp~ ocL, 8 ¢ • cp~9 Xc~,8e(cpScu.o6,¢.Xs,¢ • ~o~8¢c~-I p 6,6~ ° ° ~ , 9 ° X ~ , 9 o ~ , ~
" o ~-I ,9
by
c~Se ~6 abe ~6 6 ~6e a 66e ¢ = q°c~ q°~Se°~, 6 ° cPo.~8 q°~8¢X6, 8 u " cP~9 ~8, ¢ ' q°6~ X~8,
a6¢0-I cc6
8¢9
c~8 8
"
u~8¢,9
It a l s o
follows
(3o7)
, hence
by 4o4 , s i n c e
~
v o
~6 ~o66~a~,6 o a~6,~ =
066,9
~8 -I 6 = cP6~q~,
(3.9)
implies
°6,9
~).×~s,~k
"
×i
:
(3.6)
(~6CO (16C C V = ~8,e " ~0 ~ Xas,e o a~Se, ~
233
°c~,~ , since
°
s
8~ = ~ • T h e r e f o r e
= o~6,e ~ • X~6 '
(3.6)
that
sp
slnee
v o
•
s
:
¢
by
(3.J)
(.3.7)
by
from
• o -±
68¢
06, 9 = 1 , X~ ~l = i • Also,
sp
q
~6e
o q(~,9 -i
= o ~,6~ ' X 8~, 6~06,~
= k
is
(u.ps)(v.pe)
ccSe L~Se 6 e = ~o(~9 (~,8¢ (~°8¢u ° 08,~ " X6,eV) ° ~
~,~(hk)
of
~ ,~i = I )o Similarly,
hk
6 ¢ v) ° PS~ = (cP6¢U ° °8,¢ ° X8,¢
0u~ ~(hk) ~,
is induced
property
i
and
yields
×~,pk :
GRILLET
~6 -I
• X~,~k
, by the
=8 -I ° Ma~q~,5
° a ,~ o X
above.
T h e re fo re
6 = M~6~ ~ ~,6 ° ~ X~8 ~,6u
~,~ (hk)
We n o t e
that
with
5~ = ~
shown
by e x a m p l e
2 .(However, C
have
that
the r e s u l t
for some 3,
the
no l o n g e r
has
period
in w h i c h
the same
classes
~
5
holds
more
,9
k o o -I a,~
if some
than
is p e r i o d i c
c a n be s a i d
~
but
of e x a m p l e
5e A
i : this
is
of p e r i o d
i , if we
let
¢ = [e] , 6 = [d} , ~ = [a,b} , ¥ = [c] , so
condition
in t h e o r e m
4.5
is c e r t a i n l y
not
neces-
sary o ) We a l s o
note
ly to p r e s e n t that m
a6,9
maps
presented.
that
h,k
the
hypothesis
in terms
= ce,~
preserve
splits,
that
of e l e m e n t s
= i • The products
rest
P~P~
is where
= Pa~
6,¢
elements
a left
the e l e m e n t s
for all
was of
of the proof
of g r o u p
If we now c o n s i d e r
that
of 4.5
coset Pa
~,~ ~ A ,
used A
such
shows
that
that
can be
extension
can be c h o s e n
then all
sole-
o~
so
which so
= ±
and
therefore : THEOREM
4.6.
Every
split
left
coset
extension
is
well-behavedoI 5. the
Our
case
last
of a left
PROPOSITION therefore,
4.7.
For
PaP~
all =
is of m i n o r
coset
extension
importance it f o l l o w s
In a c o m m u t a t i v e
commutative
Proof° w~,~g°
result
extensions
are
~,~ ~ A , g ¢ G
S
extension,
= ~ago
also
A)
is
234
commutatlve.ll
in
4.1o w = ~0 ;
,
p~(g-pl3) = ( g . p ~ ) p g
(and h e n c e
from
well-behaved.
= ~p~g • P~P~ , since
since
P~P~ =
G RI LLET
5o Equivalence
of extensions
and other topics
io We start with the following THEOREM
5.±. Two extensions
[G,A;o',X'] family
u
= S'
(5ol)
m
.
and
if there
is a
such that o u-i
× ,13u~
°s,l~
criterion :
= S
if and only
us ~ G s (a¢ A)
t°m~um°
:
[G,A;u,X]
are equivalent
of elements
equivalence
~F mi3
for all
s,~cA,
well-behaved Proof~ extensions. actions, hence
h e O~o If . .that . .is the case,
extension First, Since
we have
0(g,s)
if and only
let
0 : S'
if
~ S
0 preserves
S
is a
is.
be an e q u i v a l e n c e
projections
0(l,s) = (us,s)
= e(go(l,s))=
S'
then
to
for some
goG(l,s)
A
and group
ua g G
= (gum,a)
of
and
for all
g ¢ G s o N o w calculate: O((g,s)(h,13)) = ( ~ 1s3 g ' O(g's)O(h'13) Since and
8
s s = (~°s[3g ' q°~13us " °s,13
is an isomorphism,
h = i
(5.~).
we obtain
If conversely
then define
£
tion then shows further,
(5.1);
that
8
to
A
oX~
,13(hul3) , s13
these are equal; with
(5.1),(5.2)
by: £(g,s)
it is obvious
projections statement
u 's , ~ o X'13~h ~,p " um~ ' sl3 ) '
The previous of
S ;
is clear on (5.~).I
called equivalent. Ps
into
The last part of the
which satisfy
For example,
(5oi),
are always
equivalent:
choice
are which of ele-
for, the extensions
they give rise to must be equivalent
235
(5.Z)
two factor cosystems
arise from the same e x t e n s i o n by a different [G,A;u,X]
S'
u,
calcula-
is bijective and p r e s e r v e s
and group actions.
Two factor cosystems
ments
g = i we o b t ~ n
hold for some family
is a h o m o m o r p h i s m 0
with
g = ± and (5.1)
= (gum,s)o
that
) o
to the
GRILLET original
extension.
one changes
pa
This
into
X' , then comparing
can also
uaopa , thereby
the two values
ned from the two f o r m u l ~ same c a l c u l a t i o n proof
is that
(3.5))
as above.
in particular
o,X
arise from
(see the proof of 3.2),
from
for any choice so is
5.2.
of
If
u
cosystem
gG
formulae
(o',X') • If
in the same
from
is a factor
(mcA),
the
we obtain the following directly
(o,X)
of
u 's
(o,X)
(3.7)-(3°9)):
cosystem,
then
(5.1) , (5.2)
yield
is well-behaved,
then
(o',X') "| ~o
gated
We continue
to include
The first
that .....
X~,m~
tent and
with two results
5°3. Let
(o,X) on
G
tor cosystem
such that
(~',X') when
e
=
Proof. otherwise Then
~
Let
(a',X')
(o,X)
by 5.1~ X
= 1
u
'a is still the idenX¢,~ and ea = a, and further-
whenever
be arbitrary
¢
if
__is idempotent
and
~
is not idempotent,
o',X'
by (5.1),(5.~).
cosystem
and it is clear from
transfers
to
whenever
¢
Further
from a
to a fac-
o
is a factor
sis on
such
is idempo-
is equivalent
u¢ = o -I ¢,e • Define
let
¢
cosystem arising
(a,X)
is idempotent
~¢,~' = i ' °h,c = i
ca=~,t3c
whenever
(e.g. any factor Then
tity o__nn G
theory°
be a factor cosystem
left coset extension).
more
extension
obli-
normalization.
is the identity
em = a
which one feels
in any Schreier-like
concerns
PROPOSITION
o'
by the choice
will arise
; hence
can also be proved
PROPOSITION a factor
a)
cosys-
pm o Since
then for any
by (5.1),(5.2)
pm' = ( u
(which
two factor
[G,A;a,X]
defined
result
) by the
if they arise from the same
data
fashion
(obtai-
of the direct
of the elements
Pa = (I,~) o',X'
if
into o',
(5ol),(5.
the converse:
choices
o,X
(g.pm')(hop~)
yields
if and only
by different
changing
of
The advantage
it also shows
tems are equivalent extension
be proved directly:
assume
by 5.Z, equivalent (5.2)
X' • F u r t h e r m o r e
that the hypothe-
(5.1)
shows
that
is idempotent. ¢~ = m, ~¢ = ~ • Then 236
to
(3.7) yields
GRILLET
OC~EE i.e.
0 t
eg~
=
o'
and
o
whence
~,e
X~,EE e , ~ = ~ e e E u e , g
~ee . ×
We call
I
normalized.
5.3 essentially of
to an extension
e, C
G
by
of
G
In the case of a left follows
A
from
splits
by
A
2°4°
in case
in which
it is equi-
o ,~ = i
for
m,~¢A. PROPOSITION
following
5.4. For any extension of
G
by
A
the
are equivalent:
i) the extension
splits;
Ii)
ua~O a (acA)
there
exist
= ~°a~u a - l ' u ~ ' ~ , ~ u ~ ) -I for all ill) it is possible
iv) there
exists
is the
)
such that
the elements
for all
a homomorphism
identity
on
A
ca, ~ =
a,~ ¢ A (we__ say ~ is__trivlal);
to choose
o ,~ = i (i.e_o pap~ = p ~
fog
and
;
~E:E E;~e = ~o g e [ 3 , e
(o',X')
3. An extension all
=
'
= loi
coset extension,
valent
o'
eg,~
g:A
(where
Pa
m,~eA
so that
; > S
f :S
such that
) A
is the
projection) o Proof.
That
if conversely is e q u i v a l e n t
i)
i0
implies
holds,
it) is immediate
then by 5.2 the given e x t e n s i o n
to an extension
(5.1) and hence
a'
11) are equivalent° from the remarks Recognizing
= I
in which
for all
the structure
through good behavior), red by other means
of
that of
split extensions also
it implies
o'
is given by
a,~eAo
The equivalence
after 5.±;
cause it simplifies
from 5.1
Thus, i)
i)
and
iii) and
iv)
is important, theorem
is c]ea~I
first
be-
(in p a r t i c u l a r
because when splitting the existence 237
and
ill) follows
is pro-
of a splitting
GRI LLET homomorphism.
Hence it would be interesting
semigroups
A
have the property
(or, every
left coset extension,
with
C = ~ ) splits.
ties;
the literature
[3], the bieyellc
provides
[30].(In these examples by r e c t a n g u l a r
that every e x t e n s i o n
~
by
A
every left coset e x t e n s i o n
Free semigroups
semlgroup
to find which
have all three proper-
more examples:
[25],
semilattices
polycycllc
is the equality,
semlgroups but e x t e n s i o n s
bands need not split even in case
C = ~ .)
4. We n o w give the last part of the main theorem; this tells which extensions THEOREM and
5.5. Let
by
A,
S
> A. Then
valent
C
be_ the congruence C
a a X6,
and
Proofs
te that
S
that
and
C
C,
and
with
G
(g,a) C (h,~)
and
can d i f f e r only
6 g = ~0 k • a6, a o
if and only
C.
in the action of (FI), (F2)
if
and that
arising from
that when
if:
there are
is a left coset congruence S'
i_ssequi-
6~ = c~ ;
g ~ l,
6a = a
S
if and only
whenever
G
to the e x t e n s i o n S'
of
[No-
G
hold
on for
they also hold for any equivalent
(a',x'):
e.g.
when
6a = =
6
hence
g = ~p k . a6, ~
sion a r i s i n g
from
C-
(F2)
is e s t a b l i s h e d
g~ G
with
g ~ I
SI), say
implies
g = ~pj(ku~l)oa 6,a ' • Hence
S = S'
In this
case,
S
is the exten-
(FI) follows
as in the proof of 4°5,
(k, 6), in p a r t i c u l a r
Conversely,
i.e.
from 4.4;
i.e.
is induced by some element of
= (k, 6)(l,a) , which implies
implies
factor c o s y s t e m 6 and a~, a = ~au= o a6, =
then
we may as well assume that
just
on
g~ G
We see on (5.1),(5.2)
(a,x)
from
k e G 6 such that
is e q u i v a l e n t
Ca . ]
and
We know that
= ~ • Assume S
a~A
be an e x t e n s i o n
congruence,
arising
i_ss the identity
extensions.
induced by the p r o j e c t i o n
is a left coset
(F%) for every 6¢ A
are left coset
S = [G,A;a,X]
to the extension
(FI)
some
[G,A;a,X]
assume
6a = ~
that
238
(not
(g,a) = g.(l,=) and
(FI),(F2)
(k, 8)(g,a) = (~p~k.a6, = ~
every S
• g I a)
=
6 g = ~ a k . ab, a • holdo Note whenever
that 6a
(F]) a
@
G RI LL ET Take if
g,h~
g ~ h,
kEG6, sing
G a • If
then
g h -I ~ ~
6a = a the
(g,a)
(by
roles
of
g
for
that
some
and
(k,6)(h,a)
left
coset Let
= (h,a)
a e A. with
(F~), O~
and
that
Ta
pax = p a y
6a = ~ . Hence
on
Ca,
the
that
Pa
only
an a c t i o n - p r e s e r v i n g group
G(C a)
of
Ca , w h i c h
of
S
Ga
a,13¢A,
= i , so
,a
C
iE S I
is a
o a6, a
; then
already
acts
x (g,a)
= way
all
x ¢ Ta, (g,a) E Cao simply
= p x.(g,=)
is a h o m o m o r p h i s m
wax
and
Pa : Ta
for all
formula
Thus
we can use to
=
> Ga
= ~k
Ga
isomorphism
upon
relative
If
if
of
a mapping
Since
implies if and
that
• Therefore
consists
by : p~l = i , 0 ~ ( k , 6 )
transitively
Rever-
(k, 6) (g,a)
q0~k • o 6
hE G
with
(g,a)~
conclude
b= = a ,
for all
is surjeetive.
immediately
=
If f u r t h e r m o r e
x (g,a) = ( 0 x • g , a ) = pax • (g,a) By
£ (h,a).
by ~ o I -
We see
can be d e f i n e d
(g,a)
(k,6)(h,a)
we a g a i n
, then
congruence,
(k, 6) E S
h
clearly
g h -1 = ~0a6k. o6, a
Hence
0 ~ £.
g~ G
then
and
(F%)).
£ (h,a) o Thus
(g,a)
g = h,
and
• Therefore
that
there
is
of the Sch[~tzenberger
Ga
is a Sch[~tzenberger
ira d e f i n i n g
"the"
left
GS
group functor
C-
then
(g.(l,a))(1,[3)
= ( ~ 1 3 g " go.,13 ' ~13 ) =
= ~ aa ~ g o((l,a)(l,~)) ; hence the left GS functor of S relatlve to C is t h e n p r e c i s e l y G . Finally, it is c l e a r f r o m the above same
that
as a r i s e s We note
~0~6
is the
clear
from
the g i v e n from
that
the
some_
C
following
(F3)
for be A
composition on
condition
every and
In t h a t
~cA k~G5
case
bly s t r e n g h t h e n e d
of
in 5.5
property
whenever
G
the h a n d l i n g
the c o n g r u e n c e
structure
on
S
is the
C-I
the
identity
extension
£
a~ = a .
in the p r e v i o u s
is c o n t a i n e d also
and
we have
that seen
(see 4.1,
in
~
q0 i m p l i e s It is proof
if and
then that only
if
holds: g~ Ga
such
of
a6 = a that
4.3)°
239
with
(FI)
g ~ i, there and
g = o
are 6 ,6.X~,6k
can be c o n s i d e r a -
G Ri LL ET 5.
It follows
functor
on
(relative implies with
A
to some
that
Ga
~a = a
functor
C
by left
semigroup
cosystem
S
A
is no
directly).
More
Adding
(F3)
relative These
cause
functor
precisely,
G
on a semi-
is the
to some
if and only
values
in
G
left GS
left
coset
if there
exists
such that
(FI),
G
(F~)o
ral)
a characterization contained
characterizations
all f a c t o r
factor
yields
to congruences
we c a n n o t
functor
tell at one glance
(FI)
in (F2);
is s h o w n
stems
that
this
first
(or 4°4)
elements
Oa
~
of
thermore
then test
from
the appearance be improved
Let
C
has two elements.
of
Ga
and
of the (in gene-
by: i (continued)°
while
calculate
cannot
have
classes
C 6 = {d} , C a = [a,b] , Cy = [c] o We see that vial,
be-
whether a g r o u p - v a l u e d
cosystems
satisfying
~.
very satisfactory,
but must
T h e difficulty
set
are not
of left GS func-
in
is a left GS functor
EXAMPLE
ment
8 ¢ A
from 3 . 1 , 5°5:
relative
with
instance,(F2)
when there
divisibility)
S/C ~ A )
on
For
semigroup
holdom
tots
for
is obvious
(with
of some
congruence)°
5°6. A g r 0 u p - v a ! u e d
of some
a factor
not every g r o u p - v a l u e d
is also c l e a r
(preordered
congruence
(F2)
left coset
result
COROLLARY A
that
must be trivial
(this
the f o l l o w i n g
group
from 5.5
can be the left GS functor
A
such that
cannot
~
be w r i t t e n
the d i f f i c u l t y
here
Since
= a, as
C e = {e],
G c , G&
8,e
are tri-
are the
the n o n - t r i v i a l
~0~k
with
only ele-
~a = a .(Fur-
is not a l l e v i a t e d
by g o o d
behavior. )| More suitably
satisfactory strong
no factor
set
extension
by
tic condition. potents
c
of
characterizations
restrictions
will appear A
splits. There A
such
are p l a c e d
in (F2)
The next
we denote that
A.
For
in case every
Ea
if
instance
left
coset
a less
dras-
the set of all
idem-
result
by
ca = a .
240
c a n be a c h i e v e d on
gives
GRILLET PROPOSITION semigroup 6~ = a
A
5.7.
implies
Xa
O
=
U(
Proof.
Ea ~ ~ • If
there
is no n)
5~ A
element
with
and
G
at once
hE G c , k¢G
(FI)
Condition case
when
under finite
of
G
commutative
G
;
is trivial, or
can be w r i t t e n
(3.9)
on
A
and
the
b)
A,
that for
holds.
hold. set
In v i e w of
in
a)
¢¢ E
(FI)
is nor-
• Then
assume
¢ = 6n ~ E
5a =
for some
n,
implies
o When
= X6,ea(q°ca h " g ¢, ~) " X s ¢ , e k h = I, this
reads
that
this
q0
of
241
improved,
in the
has a least
semlgroup
Ga
and
from 5.6.I
=U(
element A
(for then
E a) o Then every
; hence
a)
proves
is the case when
combinatorial
of all elements
E
(by
which
follows
in 5.7 can be further
through
for any
implies
whenever
b)o To prove
2.8)o
8n = ¢ c E
b)
factor
as
from
q9¢ 1
as
Thus
o¢,~ = I
on
then
i¢ G
T h e n the converse b)
in
e g E
in turn followed
a)
E a ~ ~ , if we know
factors
G
(otherwise,
that
left divisibility;
the p r o d u c t ¢ ~E a
and
a ~ ak = k X6a 'ak = Xs,al o X¢,
holds.
values
of 4°3
write
that
Xs,¢a(qg¢~h • g ¢,a o X~,ak)
that
with
holds,
(this
from
the hypothesis
normallzatlon):
if:
a)
is trivial°
assume
5¢ = ¢ . T h e n
for all
is the
We saw in the proof
5a = a
in p a r t i c u l a r
By
G
whenever
E a = ~ , then the hypothesis
follows A.
G
= ~
A.
a~
we can by 5.3 assume
(F)
A
is a left GS functor, take
and hence
mallzed~
in
G
k¢G c , ¢ ~ E
Conversely, a)
on
E
then we can also
¢ cE
some
identity
io Then
on
on a
in which
if and only
cosystem
either
non-trivial
for some
functor
) o
• Next,
that every
If
E
¢ ¢
When
by 3 . 3 , 4 . 4
q0~k
~ ¢ A,
qo¢ G¢ ;
of p e r i o d
a factor
is the
b) for each
be a g r o u p - v a l u e d
of some s e m i g r o u p
exists
such that
G
by left divisibility)
6 periodic
left GS functor a) there
Let
(preordered
is a N
q0~ with
~a~o¢ ; ¢ E E e )
is
G RILLET can be replaced by: O
= ~O
, i.e. ~
is surJective.
This
will be further refined in the last section (see 7.6).
6o Reduced and left reduced semigroups
io In this section we investigate the extent and effectiveness to which our main theorem can be applied to construct semigroups
from groups.
The main theorem tells us that any semigroup be (at least groups
in theory) explicitly
and its
S
can
constructed in terms of
(presumably simpler)
quotient by any left
coset congruence. Nothing can be done if the equality on is the only left coset congruence; left reduced.
We call
the only congruence
S
in that case we call
reduced in case the equality
contained in
is
~.
Clearly all left reduced semigroups are reduced; the other hand,
S S
on
example I is reduced but not left reduced.
From section 2, more precisely
from 2.5 , 2. 9 , we also
obtain the following two results: PROPOSITION 6.1o A regular semigroup is left reduced if and only if it is reduced.I PROPOSITION 6.2° All finite are left reduced (and, dually,
combinatorial semigroups
right reduced also)o|
Reduced and left reduced semigroups are just the semigroups
than cannot be constructed from simpler semigroups
by extensions
arising from congruences under
set congruences°
~
or left co-
We now show that arbitrary semigroups
can
to a large extent be constructed in terms of groups and reduced or left reduced semigroups. 2.
We begin with reduced semigroups because the pro-
cess is then simple and satisfactory° LEMMA 6.3. Let
f :S
T 242
be a surJective homomor-
GRILLET phism.
When
ker f ~ £S ' then
Proof.
Assume
is surjeetive, and
ker f ~ ~S
there exist
f(b) = f(ya) • Since
b = vya
for some
f-l(£T)
~ £S
f(£s ) ~ fT
f-l(£T) and
f(a) £T f(b) • Since
x,y~ S I
such that
ker f ~ £S
f
f(a) = f(xb)
this implies
u , v ~ S I ; therefore
" The converse
= £S °
a = uxb,
a £S b , which proves
inclusion
is trivial as
always.ll
THEOREM congruence
6.4. O_n_nevery s e m i g r o u p CH
contained
in
Z.
S
there
Furthermore,
is a greatest S/C H
is
reduce d. Proof° xay ~ xby is indeed Let
One may define for all
~
S/C H = A
congruence
f-l(g)
= f(f-l(8)) This
is done
on
A
importance:
of groups
By 6.3,
contained
result
in
CH
in
~.
f-l(~ A) ZA
also
and thus
(see e.g.
can be explicitly
and reduced semigroups.
in one step by means
[ZI])
constructed
The construction
of a well-behaved
as examples all structure
it
extension,
theorems
mentioned
introduction°
general°
of reduced semigroups
They have been known u n d e r various names:
(Munn's
terminology),
antigroups
name for r e d u c e d
inverse semigroups),
(for c o m m u t a t i v e
semigroups)o
of this p a p e r
naturally
It is not,
result which shows
should be construetible
in terms
in
fuuda-
(the R u s s i a n School
alas,
ordered the purpose
to construct all reduced semigroups,
do have an easy
alone
that
contained
with our main theorem,
It also shows the importance meutal
S
~ CH = ker f
is an easy and w e l l - k n o w n
and includes in the
on
must be the equality.a
that all semigroups
in terms
g
z ~S ' f-l(g)
but it is of great implies
if and only if
be the projection.
= ZS ; hence any congruence satisfies
by: a C H b
x , y ~ S I , and it is immediate
the greatest
f :S
CH
that in g e n e r a l
of partially
but we they
ordered
sets
•
When
S
is any semigroup, 243
I = S/~
is a partially
or-
G RI LLET dered
set,
and a representation
of
of
(since
is a left
congruence).
~L
by o r d e r - p r e s e r v i r ~
g
sentation A = S/£ La.~LS
of
S
(now written = Las
• This
= (~R,~L)
of
S
I
~R
ing t r a n s f o r m a t i o n s
S
by order-preserv-
is w e l l - d e f i n e d
as right
Dually,
there
= Rsa
is a repre-
transformations
operators),
in turn yields by pairs
by: ~ R S . R a
of
well-defined
by:
a representation
of o r d e r - p r e s e r v i n g
transfor-
mations • PROPOSITION
6.5.
ker ~ = C H ~ thus
S
When
S = S I, or when
is reduced
if and only
S
is regular,
if
~
is faith-
ful o Proof.
Either
hypothesis
dually,
so that
of
, and dually.
URX
L x = Ly let
C
and
is the g r e a t e s t Thus
be any
congruence xa ~ ya
~ R x = ~Ry
on
S
and
; dually
that
x g xS
element
~x = ~y
x ~ y ; therefore
x C Y , then that
Rx
implies
and
of the
implies
range
Rx = Ry,
ker ~ g C H • Conversely, contained
Rxa = Ry a ~L x = ~Ly
in
~.
When
for all
ag S,
, and this
proves
so
C ~ k e r ~ .| This truct
is of course
reduced
characterization as
I
and
retrieve
A
of which
reduced
But
note
verse
semlgroups
will
unreasonable described 3. ces not
partially
ordered
from
result
to expect
in terms
[12];
deal
that a r b i t r a r y
of groups
and
We n o w turn to c o n s t r u c t i o n s contained
in
coset
can serve
of pairs
to do with
semigroups
ordered of in-
by the It is not
can be
ordered
in terms
of
the ex-
papers
semigroups.
partially
a
which
in the case
forthcoming
for regular
sets.
of c o n g r u e n -
M.
PROPOSITION 6.6. On every est left
the semigroup
[Zl] has a great
the same
sets
on such p a r t i a l l y
that the s i m i l a r
show
to cons-
it with
as well as conditions
semigroups
of a c o n s t r u c t i o n
result;
supplement
transformations
sets°
author
trivial
one must
in the above,
order-preserving
istence
a nearly
semigroups,
congruence
EL • 244
semigroup
there
is a great-
GRILLET Proof. coset
Let
CL
congruences
be the
on
their set-theoretical contained
in
£,
unlon~
and h e n c e
sitive
closure
exist
ao,al,..o,ane S
Cn
on
S
C L.
such
a n = b o Slnce
lows
from
thus, CL
and
All
the
transitive
left
coset
coset
is a l e f t
we e v e n t u a l l y
implies
obtain
coset
its
congruences
coset
xa o = a o C I a I
xa I = a I C 2 8~
congruences
xa = a C L b • T h e n
left
of are
tran-
there CI,..o ,
for all
i < n,
congruence,
it fol-
implies
xa I = a I o
xa 2 = a 2
• Continuing
xb = b • T h e n
congruence;
left
closure
t h e i r u n i o n and
a = a o , a i C i + I ai+ 1 CI
2.1 t h a t
is a l e f t
of the set of all
so are
Now assume
that
and
Similarly,
lou.b,
S ; that, ls,
~i
shows
it is e v i d e n t l y
that
the g r e a -
test.| The reduced
example and
hence
uo r e l a t i o n particular in
of r e c t a n g u l a r
to
left
reduced
the s i z e of
need
£
bands,
which
are
by 6.1,
shows
that
(other
not be the g r e a t e s t
than
(trivial~ CL
bears
C L ~ ~ ), and
congruence
in
contained
£. One m i g h t
expect
S/C L
next
example
shows
this
turn
implies
that,
unlike
coset
congruences
der appropriate EXAMPLE
to be
is not
left r e d u c e d ,
but
always
the case.
[This
congruences
contained
in
in g e n e r a l
d o not
pull back well
This
S
e u v w
a x
Y z
is the
semigroup
even
with table:
e u v w d x y z
a' a" ! b' b"
C
e u v w
a' a' a" a" b' b' b" b"
C C C C C C C C
u e w v
v w e u
w d x y z v z y x d u y z d x e d z y zy e u v w
dx x d z y wv ue y z d x v we u z y x d u e wv
a" a" a' a' b" b" b' b'
b' b" b" b' a' a" a" a'
b" b' b' b" a" a' a' a"
a"
a~ a"
a' a"
C
C
b' b"
b' b"
b' b"
0
¢
c
c
O
C
a !
245
in
~,
surjections.]
4.
the
C
left un-
GRILL~T It is built and had as
D4
from the dihedral
on the four vertices
is indeed
ate cases
of which
the £-classes
~a',a",b',b"} congruence
and
ou
ub" ~ b" , C [e,u,v,w]a' must also have
The
a left
S.
cannot
classes
the only
from the
fact
that
the a p p r o p r i -
We now find
CL •
C
(~ £ )
a'
be any
left
ua" = a"
or
a"
[d,x,y,z}a'
with
b'
= [b',b"]
from
coset
ub' ~ b' or
b" ; then
shows
that
d,x,y,z • C o n v e r s e l y
C
let
[e,u,v,w],[d,x,y,z],{a',a"],{b',b"],{c]
shows
coset
just
in its geome-
{e,u,v,w,d,x,y,z),
ua' = a'
e,u,v,w
He ~ D4
a',a",b',b"
by considering
are:
relate
= [a' a"]
follows
trivial)o
[c} o Let Since
separate
table
are
on
of a square
Associatlvlty
a group action,
(most
First,
C
D 4 : we took
it act by left m u l t i p l i c a t i o n acts
tric p r e s e n t a t i o n ° this
group
that
C
congruence
pairs
(p,q)
is a congruence. is readily
such
that
That
C
seen once we note
pq = q
•
is in fact that
are the pairs
(e,q) , (p,c) , (u,a') , (u,a") , (w,b') , (w,b") • T h e r e f o r e C = C L • Now
the table
old f r i e n d E x a m p l e 4. E x a m p l e
4 shows
be reached
left coset
congruence).
Define
for each
quotient
follows.
First,
ned,
many
ordinal
test left
coset
congruence
U
C~L o We call
PROPOSITION
nal
k,
series
is not left
reduced
it can always
number
a
on
A
of
family
S.
If
e
It is clear that
CL
on
is defi-
is the grea-
is a l i m i t
ordinal
of c o n g r u e n c e s
the
S.
It terminates
Proof.
a CL
~ S/C~ = A s , as
CL(A ~)
o If
this
i_ss !eft
be r e a c h e d
a congruence
Schutzenberger
Ak
can-
by one
steps.
left
which
reduced.|
semigroup
c h a i n of congruences.
for
6.7° The
ubiquitous
(= by quotient
= f I- (CL(A)) , where
left S c h u t z e n b e r g e r
ascending
seen
is our
and p r o j e c t i o n fa : S 0 CL is the e q u a l i t y on
c~+l
C~ =
step
However,
then
then
S/C L
that a left
in one
sufficiently
S , with
that
i , who we have
not always
if we a l l o w
shows
series at some
is an ordi-
reduced° eL ~ ~L
246
always.
We cannot
GRILLET (~ ~+i C L c ~L
have
ceed that dinal
k
shows
for all
~ , lest
the cardinal
of
S xS
ex-
of every ordinal; hence there exlsts a least orr,X+l • The d e f i n i t i o n of c_X+I with C LI = ~L L then
that
C L ( A i)
is the equality,
i.e. A k
is left
redu-
k
ted;
then an easy The
first
first
ordinal
induction
ordinal
~
such that
shows
such Ak
left S c h [ t z e n b e r g e r
rank of by
migroups £
have
CL
on
CL ~ £
= 0 , transfers for
ker f
= f-l(CL(k~))a { f~l(£A~) CL
4 has
for all
by r e c t a n g u l a r
dual c o n g r u e n c e
CR
PROPOSITION Proof.
C LNC R ~ £ N~
the c o r r e s p o n se-
rank 2.
and,
is trivial and,
for
when
_~+i
by 6.J,
" In particular,
bands.
we have
just
; hence both is a left
in
COROLLARY
it the
CL
true
:
~ L m £ " But
congruence
contained
If we also
consider
in
the
, we have:
CH
so is c o n t a i n e d
the
6~ 8. C L 6 C R = C L N C R = C H o
First
C H o Conversely,
we call
a : this
n e e d not be the g r e a t e s t
£ , as shown
is also
left r e d u c e d
ordinals,
= CL { £ = £S
a _~ k.|
I has rank ± (as in that ease
at once to limit
(~, Implies
again
and denote
S/CL) , example
that
reduced;
For example,
rank O, example
is trivial We note
S,
CL
for all
X = ~L+I CL ~L
that
is left
ding
congruence
CL = CL
CL
reduced
that
congruences coset
and dually
6.9. Every
~oroduct of a left
observed
reduced
CL N CR
are c o n t a i n e d
congruence also
in
in
2.Z)
and
CR.|
semigroup
semi~roup
(by
is a s u b d l r e c t
and a_ right
reduced
s e-
mlgroup .I
.5.
If
S
ger series
of
the f o l l o w i n g is a left
is any semigroup, S
extension llm A~
nal,
then
used
to r e c o n s t r u c t
through
semigroups
properties:
coset As
yields
the d i r e c t e d
S
then the left S c h u t z e n b e r A
A 0 = S ; AX by
Aa+ l ; if
(0 < ~ _~ k)
~
• If
X
from
A 1 , as there
colimit.
is a limit
is infinite,
this
is no way
A
ordl-
cannot
If on the other h a n d
247
with
is left reduced;
be
to back the
GRILLET series
is finite,
then the nain theorem tells us how to
construct
Am
from
back from
Ak
to
Aa+l; by repeated applications
S.
Thus any semigroup of finite
Sch{Jtze~berger rank can be explicitly of groups
divide
If
S
result can be obtained for finite
is finite,
in terms
duced semigroup
T.
quotients
But
T
until we reach a left re-
is also finite,
so taking suc-
by right coset congruences
right reduced semigroup. until it stops,
semi-
then we can as above successively
by left coset congruences
cessive
left
and a left reduced semigroup.
A stronger groups.
constructed
we can go
The procedure
will yield a
can be continued
which must happen after finitely many stelm,
at which point a semigroup has been reached which is both left reduced and right reduced. can be explicitly -and-right
constructed
reduced semigroups.
While the construction step,
it involves
tensions,
Thus every finite semigroup
in terms of groups and left-
is completely
explicit at each
repeated left coset and right coset ex-
which are not necessarily
well-behaved,
and so is
unmanageable
as a construction,
at least until more
known about
iterated extensions°
It should however
ful as a technique elements).
The only comparable
tion of minimal applying
congruences,
to arbitrary
reconstruction Again,
finite semigroups
description
of
but does not allow
semigroups,
is no loss of structure ought
we see that
global picture but again at each step
(in par-
through semigroup di-
to result in new structural
the group in Z. lO'~homld play a role in these . the class of left and right reduced semigroups
larger than that of combinatorial strictly
descrip-
is compared with the
of explicitness
vision). This advantage However,
of finite
to give a satisfactory
ticular there
is Rhodes'
from the quotient°
it has the advantage
insights;
technique
the number of
which has the advantage
when this construction
Krohn-Rhodes it fails
of proof by induction(on
is be use-
larger
is shown by: 248
semigroupso
That it is
is
GRILLET EXAMPLE
5. This
is the semigroup S
edabc
e d
edabc debac a a a a c b b b b c
a
b !C
it is anti isomorphic gue [ 8].(It we a zero
C
C O C Ci
to s e m i g r o u p ~ 7 5 6
is also semigroup ~ 9 9
adjoined;
with example
we leave
are
ned in
R
and
da = b )
and hence
is right
reduced°
Because
congruence tained
in
reduced.
relate
However, S
Without
This
class
of left
of the next
and
at least occurs
right
order
at small The
orders,
where
orders
situation
coset
tion are do with This
commutative. cohomology
and hence
must
thus as
reduced
that,
at small orders,
semigroups
is only
semigroups: by trivial
remark well may
processes,
Because
to the notation
inertia
249
Pe-
at h i g h e r
occur.
under
considera-
m o s t of this s e c t i o n to the additive
of group
or
cohomology
that all semigroups
we t h e n s w i t c h
slightly
for sem~groups
is due to Mario
naturally
the
or inflation,
be different
case:
of
but not c o m b i n a t o -
or identity,
(this
is left
d E H e .m
many of them do that some
could
be con-
S
5 is the only s e m i g r o u p
reduced
larger groups
We n o w assume
extends
b,
right
7 o The commutative
i.
it relates
aa = a , ab ~ b , a left and
of a zero
and contai-
thus
tend to be built
sufficiently
trich)o
(otherwise
of combinatorial
such as a d J u n c t i o n
congruence
be the equality;
to indicate
that
d
°A
must be the equality;
example
is left and
tends
and
a
catalo-
catalogue
[e,d],(a,b},{c},
is not combinatorial,
its zero,
rialo
than
e
and as above
order 4 which
larger
relate
cannot ~
are
{e,d},{a},{b],[c}
ea = a S
cannot
of Tamura's
of Forsythe's
the zero to show s i m i l a r i t y
io) The £-classes
the ~-classes
with table:
actions:
g.c
has
to
notation° becomes
GRILLET g+c
(again
ther
+
means
When S
it will always
S
with
Also
p ~ C , no l o n g e r seen d i r e c t l y p
only
T(C)
and
follows
: G (%
in
~,
~ G'(%
for each
PROPOSITION M, the
Proof. functors
= a+Y
(%~ A
ral e q u i v a l e n c e )
= ~
extension.
A
then trivial other hand,
there
a
alone.
the
is a c o n g r u e n c e isomorphism
contained
isomorphic.
commutative, Next
the
0
(% > ~
two
we have
is an isomorphism
• When
c = (~(%)(%gA
in
c (% : (i.e.
A
is a formal
Just
identity)
is an isomorphism
(natu-
G' oI
this
implies
holds
X = q0 (i.e.
in any
also
of
condition
S
(3.8') property
a(%,~ = q~,(%
commutativityo
250
w = ~
extension
can be c o n s t r u c t e d Furthermore,
X~
commutative
implies
in the g r o u p
by the c o m p o s i t i o n
to insure
of
that
= ~ (%1~' (% = ~ (% ¢ ~(% , whence
= ~
'~
are
category.
on the kernel
extensions
set
is also
= ~'
[Commutativlty
trivially
these
factor
(3.7)
(%,~
of 4° 7 ), which
acts
that
of 4.2,
for all
(%+9
proof
G
choice
to any congruence
y g A° - where
then ¢~ q 0(%~ m = e ~ (% (% ¢~ q0~ = ~ c(% ° Thus
In v i e w
Relative
e ~
for some
C
a canonical
the same d o m a i n
with
can be
(%c A •
A = S/C
for each
~- G'
by the
and
is surjective,
When
right GS functors
Since
have
seen that : Ga
7.1.
left and
C
but it is c l e a r ~
p,qe C.
we then have
~ G' (C)
p : this
it is affected
ep ~ = ~' • since for all
and so are
Cp : G(C)
of
on
~ (= £ )~ f u ~
S1
, for any coset
on the choice
(we remember
= T' (C)
contained
in
= g+p
in
divide
isomorphism
depends
congruences
contained
groups
up to inner automorphisms),
ep = eq
¢
the
p+epg
whe-
action).
the c o n g r u e n c e s
in 1.6 by
from the c o n t e x t
the left coset
all Sch[~tzenberger
a b e l i a n groups. defined
or g r o u p
is commutative,
coincide
thermore
addition
be clear
left
coset
(see also the
case
means
that
~ A o] It follows by means and of must
Consequently
of the
(3.9')
are
qo. O n the be a d d e d we call
to
GRILLET o = (oa,~)~,~¢ A A
with
(a ,~ ~ G +~)
values
(7.1)
in
~,~,YcA
Theorem
3.3
(where
has
7-2-
THEOREM
(7o2)
addition
(g,~)
Let
G
sion of
of G
be an a b e l i a n - g r o u p - v a l u e d
functor
A
G
by
(3o7),
(preordered
factor
set
on
union
A
by d i v i s i b i l i t y ) with
values
a~A ( G a x {a])
in
G.
with
c~ + o t ~ , l3 + ~ + 1 3 h , c~+13 ) = (~a+t3g
, together
actions
extension
just
by:
+(h,13)
[G,A;o]
and g r o u p
X = ~P )o
be the d i s j o i n t
defined
(7°3)
is
+ wa~+13+Y o l3+Y cr13, ¥ with
semigroup
be a c o m m u t a t i v e
[G,A;o]
Then
on
become:
on a c o m m u t a t i v e o
set
it s a t i s f i e s
~=+t3 cr~,I3 + crc~+13,Y = °~,13+Y w~+!3+v
for all
and
when
factor
• ,13 = al3, oa
(7.Z)
Let
G
a commutative
the
g + (h,~)
by A
with =
projection
(g+h,~)
A o Conversely,
(g,a)
, ~
, is a c o m m u t a t i v e
every
can be c o n s t r u c t e d
@
commutative
in that
fashion
eztenup to
e q u i v a l e n c e .| To which
this
must
follows THEOREM
[G,Aja] exists
be added
at once 7-3.
With
, [G,A;c'] a family
(7.~)
fo___r all
~'
criterion
for e q u i v a l e n c e ,
G,A
as above,
are e q u i v a l e n t
if and only
u = (ua)~ A
(u s c G )
~
u
+
13
two e x t e n s i o n s
such
if there
that
~,~ ~ A oli
There not e x t e n d will h o l d
the
from 5.I:
is an
immediate
corollary
to the general if we define
COROLLAF~ a commutative
7°4.
With
extension
case°
of '7-2 w h i c h
We note
that
we
could
(7ol),(7.2)
a ,~ = 0 ~ Ga+ ~ ; hence: G,A of
G
as above, by
251
A .II
there
always
exists
GRILLET 2o
When the theory is applied
semigroups,
we now fall in the more favorable
6°4 ; furthermore ence).
Thus
to the construction
CH = M
(since
reduced commutative
zed by the fact that their divisibility
~
~ = £ = g
PROPOSITION
is a congru-
equivalently
is an order relation
partially ordered sets appear agaln). another criterion
case given by
semigroups are characteri-
is the equality,
preorder
of
that
(as i n & 5
In the finite
case,
is: 7.5. A finite
commutative
semlgroup
is
reduced if and only if it is combinatorial. Proof.
This follows at once from 2.7 (in the same way
as ~.9).~(The
result can also be shown directly
We also want to characterize is new in the commutative 5.7° Condition finite,
a)
idempotent with
When applied directly
of
qo~ (~)
¢(=) + ~
7.6° Let
the Schutzenberger if and only
if
A
is trivial or
G
e(~)
is
E
~
(= relative
to
~ ),
(which again can be proved
be an abelian-group-valued semigroup
ones;
is reduced and, q0~ [~)
for each
is
semlgroup
a ¢ A,
either
functors
semigroups,
i) to construct
satisfying
in 7°6; 3) for each functor,
by induction
G
is surjective.m
2) for each combinatorial
abeilan-group-valued classes of factor
A o Then
funotor of some commutative
in view of what precedes:
condition
of
is
can be
= ~ , in case
To construct all finite commutative natorial
A
[13]):
PROPOSITION
needs,
b)
, where
functors
result
functor on a finite commutative
Gm
by 7.4 ; if
5.7 shows that
to Sch~tzenberger the following
What
case is a further sharpening
the remark following
this yields
[left] GS functors.
in 5.7 disappears,
replaced by the surjectivity the least
[13]o)
one
all combi-
semlgroup,
all
the surjectivity
all equivalence
sets. This has been done by John [13],
on the order of the reduced semigroup.
3. We now let
A
be any commutative 252
semlgroup and
G
GRILLET be any a b e l l a n - g r o u p - v a l u e d mutative
factor sets on
cr+T
by:
(u+T)~,~
that
~+ T
addition, values
functor
A
with values
is again a commutative G
factor sets
of
7°2,
Z2(A,G), 7°3
correspondance
with the elements
of
of
C = ~
semlgroups,
group to consider.
G
by
A
A does
Ext(A,G)
is finite
of
G.
4. Our goal
case,
on
is to present
copies
is therefore in 7.6 do
in the cohomology.
Ext(A,G)
as a second coho-
a resolution
for
~
A.
its
of all a b e l l a n - g r o u p - v a -
A o An
denote
of the commutative
the set of all unordered pairs builds c o m m u t a t l v i t y
(~,~)
the cartesian
semigroup with
into the resolution].
= (~i' •.o ,~n ) E A n , we denote F o r each
provided
to the Bar resolution but takes
n > 0 , n ~ 2 , let n
is known),
as the conditions
in the abelian category
lued functors
(F)
in the main
we want,
Ext(A,G)
We begin by constructing
It is quite analogous
satisfying
(as set forth in 7.6) are pla-
not seem to create any s i m p l i f i c a t i o n
mology group.
come
might not be the right
Ext(A,G) ; and,
The study of
(in the general
which
in c o n s t r u c t i n g
(and a construction
restrictions
classes
Ext(A,G) •
the extensions
consist of all extensions
that suitable and
B2(A,G) •
of the quotient g r o u p
the latter by
However,
ced on
duct of
with
are in a o n e - t o - o n e
are of interest
might not y i e l d a subgroup
For
A
that the trivial
only those left coset extensions
from the choice
values
show
which we denote by
show that the equivalence
ZZ(A,G) / B2(A,G) ; we denote
A
we can define
which we denote by
extensions
case where
are com-
factor sets on
we see on (7.1),(7.2)
of c o m m u t a t i v e
commutative
~,T
(constructed by: aa,~ = q°a+~u~ + ~0+ ~ u~ - u +~ )
Then theorems
Ext(A,G)
in G ,
becomes an abelian group,
form a subgroup
Since
If
factor set. Under this
the set of all commutative
in
A.
= u , ~ + m ,~ , and (7.1),(7.2)
Z2(A,G) o Furthermore
pursued
on
n > i, let
the sum
Kn(A) = 253
pro-
A ; Az
is
~,~ ~ A [this When
~i +''~ + a n
~ = ~ A
by
Ew.
be the free a b e l i a n group
G RI LLE T generated w , then
by all
me A n
with
Kn(A) ~ = 0 ). When
tor of
Kn(A) a
> Kn(A) ~
obtain for each
n > 0
A boundary
a functor
When
A,
"~
is no such
then every genera-
of
Kn(A) ~ ; we let
be the inclusion.
~ • Kn(A)
as follows.
(if there
in
is also a g e n e r a t o r
Kn(A) ~ : Kn(A) ~
n > i
Zw _~ m
~ ~ ~
In this
way we
Kn(A) ~ C .
Kn_I(A)
is defined
for each
w = (al,...,m n) c A n , put n-I
(7°5)
~w = (c~2,.o.,~ n) + i=lZ ( - 1 ) i ( = l ,.,,.,=i + = i + l ' ' ' ° '
e~n)
+ (-i) n(al, .- ,(~n_l ) (as c a l c u l a t e d ~
when
•
is a generator
the mapping
rators
of
Kn(A) a
extends
(as this ~
A n-I )o When
boundary
formula
of
a > ~
G
groups.
n = i
we let
6 : HomG(KI(A),G) 5. First
of
we have
> Kn_I(A)
topology;
K(A)
hence
in
of G o
the proof that
~
K2(A) <
~roups
use
in
functor
comple~ of
We C.
functors
in the general
groups
Hn(A,G)
HI(A,G)
...
are the homology
The homology
case. on
A
Homc(K(A),G)
of this new complex A
with values
be the kernel
in
of
~ HomG(K~(A),G)).
We now describe
Homc(K(A),G)
on generators
is any abe lian-group-valued
are the cohomology (for
~OKn(A) ~ =
holds
(i.e. G ~ C), we can form the cochain of abelian
~ :
in a n u m b e r of books such as [19].
A ; they are of little
If now
on gene-
is the same as the familiar
K(A) : KI(A)
objects
then
Kn(A)
(7°5)
in algebraic
now have a complex
~WCKn_I(A) m ;
(in particular)
obviously
formula
is available
The homology
then
~w ; i.e.
to a homomorphism
is a morphism
Our boundary
Hn(A)
Kn(A) a
> Kn_I(A) ~ • If
Kn(A) ~) , and
= 0
of
~) just defined
Kn(A) ~
= Kn_I(A) ~ o ~
~
on
>_ ~ , then the same is true of each term of
hence
:
in the free a b e l i a n group
the first
to give an alternate •
254
two cohomology description
of
groups.
G
GRILLET LEMMA
7o7 • Homc(Kn(A),G)
=- ~ O • The ,~A n Zw
isomorphisms
are g i v e n by:
(gu) w : ~0EWum , ®e = (cEw(w))¢A n
u ~
' c : Kn(A)
~ n GEw ~eA Proof.
Let
c~ : Kn(A) ~
in p a r t i c u l a r
'> G
is defined
cEw(w)
Thus we can define
be a morphism
on
~¢ A n
is always
in
whenever
~.
Then
Zw _> ~ ;
defined and lies
in
GE~.
~ n Gzw as in the statement, esw~A by the coordinate condition (®c)~ = cEw(w) o In
sentially
the a b e l i a n wise
'> G o
c : Kn(A) Ga
.... :~
for all
(ioe.
~c ~
functor category (c+d)
= c
+d
8
morphisms
are added point-
) and so it is clear that
®
is a homomorphism. Conversely, generator ~
Zw
of
u~E G
u = (Uw)wcA n e w~A n oT'~
; there
is a unique homomorphism
by
(~u)a, such that
Ew >__ ~ o If furthermore
(~u)~ Kn(A) ~ ~ whenever
=
•
is a
> G
(~u) w =
= ~ C~a Z w u~
=
9
is
is a morphism
~ ~ (~u)~ w
a
Kn(A)
"> G •
homomorphism.
We now calculate (G~u)~
which shows
®~
=
(~ u)z®(w)
: ~ zZ~ ®uw
is the identity;
= u®
'
and
z®(cz~(®)) = o,~(Kn(A)~(®)) (using n a t u r a l i t y (~®c)
= c The
of
for all
isomorphlsms
c) ~
whenever and
~
Ew _> ~ ; this
= c(®) implies
too is the identlty.I
given by 7o? carry the c o b o u n d a r y 255
,
when-
(gu)~ Kn(A)~ = ~p~ (gu)~ , which
9u = ((9u)~)~¢ A
Again it is clear that
Kn(A) ~
~ > 9, then
(~u)~w = ~ E w u
£~J _> ~ ; hence
shows that
• If
Kn(A) = , ice. E~ >_ = , we can consider
which we denote ever
let
GRILLET 6 = Homc(~,O) to maps late
6' = g6~
8'
when
but we are which
: Homc(Kn_I(A),G) :
~An_iOzm
n = ~-,3
interested
> Homc(Kn(A),O) ..>... A n
GEm
(the result
only
in these).
,c
> c~,
• We now calcu-
would extend
to all
We remember
(7-5),
n
implies :
~(~,~,¥) a(~,~) LEMMA (7.6)
7.8. For
(6'
whenever (7"7)
u = (u)~
-
u +~
A , us ~ O
Let
E(~,~)
(~,~),
l__ss~iven by: +
~+~u
;
+¥ s ~,y - s~+~,y + s , ~ + y - W -~~++~f +l y s a,~ = ~ ,~+~+y
s = (sa,~)(~,~)~ ~
Proof.
(6'u)
n = 2,3, 6'
~ u~ u)~,~ = ~a+~
(6's)~,~,y
whenever
that
= (~,¥) -(e+~,v) + (~,~+¥)= (~)-(=+~) +(e) .
u
, sa, ~ ~ O +~ •
be as in (7.6)
= a +~
and take
(m,~) ~ A ~
(so
)o Then
,~ : (S6~u)(=,~)
= (6~u)=+~(a,~)
=((~u)~)a+~(~,~)
=
= (~u)~+~(~(a,~)) = (~u)~+~(~)
(~u)~+~(~+~)
u ~+~ = ~=+~ ~ - ~+~u which proves
(7.6)° Next,
~ u +~ + ~ + ~ ~ , let
s
be as in (7°7) and take
(~,~,¥) e A 3 ; put
E(~,~,Y)
= ~+~+¥
= ~n ~ ~+Y~o~,y
- ~s=+~,y
+ ~s
which proves 6. These
+ (~u)~+~ (a)
= ~ • As above,
,~+y - ~ + ~ s
,~ ,
(7,77.| two lemmas
provide 256
us with a complex
which
is
GHILLET isomorphic
to
gy groups.
We use this to calculate
of
A
Homc(K(A),G)
with values
in
We start with
u = (u)
the cohomology
in dimensions
HI(A,G),
of all l-cocycles, chains
G,
and hence has the same homoloI and
which by definition
i.e., up to isomorphism,
¢A, u ¢ G
, such that
G
by
A,
l-cochains
namely
A l-cochaln
u
mapping
g :A
identity Clearly
the split extension
and 1-cocycles
on
A
there
of all l-co-
> S, ~ : >
abellan groups
f :S
> A
g :~ : >
only
if
for all
Therefore
+ e~ 13 U~ , ,~+~
~,[~A~
is the
is the projection) °
correspondance
g
between
1-coof
Joe.
then
is a homomorphism
a+~)
these splitting
S.
the addition of cross-sections
: ~ : > (u + v , e ) • N o w
¢+[)
fog
(u ,~) , h : e : > (v ,e),
g+h
~" (cp~+~u
in
((u~,~))a¢ A , i.e.
it becomes an isomorphism
if we define
if
S = [ G , A ; O ] . The
(u ,e) , such that
is a bljectlve
chains and cross-sectlons; as follows:
extension
have a ready i n t e r p r e t a t i o n
yields a cross-sectlon (where
consists
6'u = 0 o We remem-
ber from 7.4 that there is always a commutative of
groups
2°
if and
= (u ,~)+(u~ ~ ~) = (u +~
if and only if
homomorphisms
u
,
is a cocycle.
form a group under
the induced addition and we obtain: PBOPOSITION of splitting
by
7.9. HI(A,G)
homomorphisms
is isomorphic to the group
of the split extension
G
A ol We n o w t u r n to
tisfies paring
s a ~ = s~, e (7o71
coincide
H2(A,G). (since
First,
(a,~) = (~,e)
with the commutative
for some
tor sets.
any 2-cochain
sa-
A z )o Com-
factor sets. On (7.6)
l-cochain
u)
(cochalns
coincide
we
of the form
with the trivial fac-
Therefore:
THEOREM
7o10o
H2(A,G)
=" E x t ( A , O ) ° I
One use of theorem 7.i0 is to give Ext(A,G)
in
s
and (7°2) we then see that the 2-cocycles
also see that the ~_-coboundaries 6'u
of
depends
on
G:
we see that 257
insight
Ext(A,G)
in the way is a cova-
GRILLET riant functor of
G~
a short exact sequence of functors
yields a long e~act sequence
connecting the groups
G
Ext(A,G)
with the corresponding groups of splitting homomorphisms. There
A
has to remain the same throughout.
depends on
A
How
is a more complicated question,
ge of semigroup
A
requires a change of functor
finite combinatorial
A,
Ext(A,G)
since a chanG.
For
this is investigated in [13]~
7" One might hope for a universal coefficients theorem whereby the cohomology groups could be calculated from and the homology functors of
A.
The difficulty
neral situation is that the abelian category high a homological dimension in general, subobjects
of projectives
in
G
G
G
in the gehas too
in particular
need not be projective.
The
author obtained results of that nature predicated on unnatural assumptions projective
that suitable subfunctors of
(note that
K (A) be n itself is projective, e.g.
Kn(A)
by 7"7)* The difficulty will not appear if we only wish to consider functors group:
for then
G
which essentially depend on just one G
can be replaced by a functor category
which is isomorphic to the category of abelian groups°
The-
re are specific cases where this happens. Call a commutative extension of
G
by
equisected in case all
A
(and the corresponding functor ~
maps are isomorphisms ; in the
case of an extnnsion with a zero element (written to additive notation),
q0~ is an isomorphism when
G co = 0 ; in particular all
size. For example, tive semigroups
C
(~ ~ go)
finite subdirectly
irreducible commuta-
(with zero) [8].
For equisected extensions
there is a very satisfactory
universal coefficients theorem;
since a zero element can
always be adjoined if necessary,
we present it in the case
of equisected extensions with zero. Note that and
~ # co,
are the same
(other than groups and nilsemigroups) yield
equisected extensions
a zero,
go, due
we say that the extension is equi-
sected with zero in case and
G)
m ~_ co for all
m~ A
258
A
then has
(i.eo go really ought to
GRI LLET be -Go) o F o r each that
be d e f i n e d abelian e.go
as
on
happens
that
property. <
Its h o m o l o g y
of Now
ctorial
be the set of all of a b e l i a n
For each (with
n >_ i , Kn(A)
Kn(A)
if
A
= 0
if
w e A n such
groups
can
is the
then free
A n = ~ , which
is nilpotent).
When
n > 1,
In this K~(A)
groups
way
<
are
we o b t a i n
o.. the
a complex
of free a b e l i a n
(equisected)
groups.
homology
groups
A. let
G
be an e q u i s e c t e d
isomorphisms
isomorphism), ~ oo
An
eventually
: K~(A)
H~(A)
~
An
K'(A)
~ K' (A) is d e f i n e d as before on g e n e r a t o r s , n-i note that when Em > oo all terms of (7°5) inhe-
by (?.5);
K'(A)
follows°
group
: Kn(A)
rit
n > 1 , let
Ew ~ co. A complex
are
with
affect
we may as well
the
same group,
a > ~
LEMMA
do not
>
7.11.
are
(~
When
G
functor
the
A.
Since
the c o h o m o l o g y
assume
also
on
that
all
denoted
identity
by
on
is e q u i s e c t e d ,
G
G,
fun-
(up to with
and
that all
G o Hom~(K(A),G)
--- Hom(K' (A) ,G) • Proof. When
Let
~ ~ co,
: Kn(A) a
f : K'(A) > G be a [group] h o m o m o r p h i s m . n Kn(A) a ~ Kn(A) and we let (~f)a :
~ Ga = G
(~f)co: Kn(A)co ~f : Kn(A)
) Geo=
"~ G
Conversely, naturality ca(w)
here
= ca(u)
let
. There
(®c)(~) ~
restriction
is trivial. and
that
c : Kn(A)
means
such that
Now
0
is n a t u r a l
clear
that
be the
that,
'~ G
when
of
to
It is clear ~
Kn(A) a that
is a h o m o m o r p h i s m .
be a m o r p h i s m
in
C;
Zw >_ ~ > ~ > 06 , t h e n
is one h o m o m o r p h i s m
= Czw(W)
f
for all
®c : Kn(A)
we A n • Again
> G it is
is a h o m o m o r p h i s m .
calculate,
(~ec)~(®) by n a t u r a l l t y ;
this
for
a ~ co,
= (®c)(~) also holds
259
Z~ > a
:
= czw(®) (trivially)
= o (w) if
~ = Go,
and
GRILLET hence
~®
is the identity.
Similarly,
for all
w g A n
(@~f) (w) = (~f)zw(w) It remains
= f(w) , so that
to show that
complexes,
i.e. preserve
complex, 5
is composition
on all
w e A n ; hence,
~
and
Hence
by ~ , and
if
is an isomorphism
Since (when of
A
G
Hom~(K(A),G)
is a complex
of
orem 4.1
of [19],P.77):
accordingly)} splitting,
THEOREM in
namely
when
furthermore,
7.12.
in
When
Hn(A,G)
We note
groups
K(A) o Now
K'(A)
in this situation
coefficients
theorem
(the-
Hn(K'(A),G) (with
K'(A)
n < 0 , and the homology the isomorphism,
and direct
back to
is equlsected,
exdefined sum
A, we obtain:
there
is an
z Hom(H~(A),G) @ E x t ( H ~ _ I ( A ) , G )
~ Hom(H~(A),O) @ E x t ( H ~ ( A ) , G )
that the equisected
are also rather easier
gy functors
of
interesting
interpretation.
Since
B~(A)
is the subgroup
= ]~(K'(A))
= ( ~ ) - (~+~) + ( ~ )
with
f(e) , we see that
f
HI(A)
a+~ class
while (a)
(= modulo
is a partial
H~(A) ; that is, f ( ~ + ~ )
generated
~ ~,
K$(A)
= 0
, where
by all K~(A)
with
~(a,~) is the
a ~ ~ ° If
B~(A))
of
(~)
homomorphlsm
of
A \~
= f(a) +f(~) 260
H~(A) ,
than the homolo-
we had to take
by all
.I
also has an
HI'(A) = K~(A)/B~(A)
group generated
the homology
groups
to calculate
we have
free a b e l l a n
homology
A o The first group
for the theorem,
into
~ = ~ •
, we may as well
the cohomology than
G o Coming
G
if
G • In particular, Ext(A,G)
we denote
(trivially)
• Ext(Hn_I(K'(A)),G) = 0
are natural
isomorphism
H~(A)
calculate
K'(A) , rather
Hom(Hu(K'(A)),G)
= (6f)w = (O6f) w
of complexes.|
simple universal
tended by: K3(A)
natural
: f(~)
of free abelian groups;
there is a very
of
6 o In either
~) has the same action
~ Hom(K'(A),G)
is equlsected)
by means
are isomorphlsms
a ~ oo :
Zw ~ e ; this also holds
~
~
~
is the identity.
the coboundaries
6((Of)e(~)) = (~f)a(~w) whenever
@~
whenever
by
=
GRILLET
~+~
~ ~. PROPOSITION
H I'(A) grgup_
7.13- With the partial homomorphism
f,
is the universal abelian group of the partial
semi-
A \ c o : that is, for every partial homomorphism
g : A\co
) O , there is a homomorphism
unique such that Proof.
First
k : K{(A)
g = hof g
extends uniquely
~ G ; since
see that
g
tors through
Thus
These
commutative A
G
by
semigroups
H{(A)
H~(A) : 0
factors
and hence
g
fac-
follows from the fact that
determines, of
from group homoA \ co into abelian
all ideal extensions
A ; for instance,
of
finite archimedean
are constructed by that process
a nilsemigroup).
A
k
we
.I
in turn determine
PROPOSITION r__o) by
~ HI(A)
all partial homomorphisms
abelian groups (with
K[(A)
H{(A)
we see that
morphisms, groups.
is a partial homomorphism,
f o The uniqueness
generates
~ G
to a homomorphism
B[(A) ~ Ker k, and it follows that
through the projection
f(A\oo)
h : HI(A)
o
Another application
is:
7.14. All equisected extensions
split if and only
if
H~(A)
(with ze-
is free and
.
Proof.
This
is clear by 7. I~.|
(Inasmuch as the equl-
sected homology groups are fairly easy to calculate, is an explicit
this
criterion.)
REFE BEN CE S [i] ANDERSON, L.W. and HUNTER, R.P. Groups, homomorphisms and Green's relations, Proc. Intern~ Conf. Theory of G-~up-s ~-Ca~be-~ 19-~6~I, I-5. Gordon and Breach, 1967. [Z] CARRUTH, J.H. Building S from S/Z, Second Florida Symposium on Automata and Semigroups, 18pp. Gainesville Flao 1971o [3] CLIFFORD, A.Ho Semi~rouRs admitting relative Ann. of Math. 42 (1941) 1037-1049.
261
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°
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Tulane University New Orleans,
Louisiana
70118
Received January 18, 1973 Revised August 14, 1973
263
