WHY STUDY SEMIGROUPS?
John M. Howie
Lecture given to the New Zealand Mathematical Colloquium
(Received June 1986)
1.
Introduction
Before tackling the question in my title I should perhaps begin by
saying what a semigroup is.
operation
.
A non-empty set
(,xy)z
If in addition there exists
1
S
-
in
lx
we say that
S
is called a semigroup if, for all
=
endowed with a single binary
x ,
y ,
z
in
S ,
x(yz) .
S
Xl
such that, for all
=
x
in
S ,
X
is a semigroup with identity or (more usually) a monoid.
I shall be confining myself today to semigroups that have no additional
structure.
Thus, though semigroups feature quite prominently in parts of
functional analysis, the algebraic structure of those semigroups is usually
very straightforward and so they scarely rate a mention in any algebraic
theory.
Equally, although they are often of greater algebraic interest, I
shall say nothing about topological semigroups.
Let me begin by answering a slightly different question:
Who studies semigroups?
Section 20
in Mathematical Reviews is entitled
'Groups and generalizations' and has two. 'leper colonies' at the end, called
20M
Semigroups and
20N
Other generalizations.
In the Introduction to
their book The algebraic theory of semigroups in 1961 Clifford and Preston
[2 ]
remarked that about thirty papers on semigroups per year were currently
appearing.
A brief look at recent annual index volumes of Mathematical
Reviews shows that the current figures are
1982
321
1983
310
1984
311 .
Math. Chronicle 16(1987), 1 -14 .
1
Incidentally, comparable figures for 'other generalizations' are about
one third of these.
So it is clear that of all generalizations of the group
concept the semigroup is the one that has attracted the most interest by
far.
I shall in due course hazard a guess as to why this is so.
Mathematicians are rightly a bit suspicious of theories whose only
motive seems to be to generalize existing theories - and if the only motiva
tion for semigroup theory were to examine group-theoretical results with a
view to generalization, then I would have no very convincing answer to the
question of my title.
The test proposed by Michael Atiyah for a generaliza
tion - that it should have at least two distinct and interesting special
cases - is a reasonable one provided it is not applied too dogmatically;
and the semigroup concept passes the test, since from the outset semigroup
theory drew its ideas partly from group theory and partly from ring theory.
For clearly every ring
the operation
+ .
are semigroups
operation
+
(#,+,.)
is a semigroup if we simply neglect
The converse is certainly not true:
(S,.)
so as to create a ring
(5,+,.) .
this is to recall the known result that a ring
that
x2 = x
for all
tive
- i.e.
satisfies
S
= (4 xB) U {0} ,
and make
S
that is, there
with zero on which it is not possible to define an
x
in
R
xy
where
with the property
(a Boolean ring) is necessarily commuta
= yx
A ,
The easiest way to see
(fl,+,.)
B
for all
x
in
R .
Now let
are non-empty sets and
0 £ A * B ,
into a semigroup with zero by defining
(a,i)
0
=
0
(a,£)
(aj.fcj) (a2,&2)
00
=
=
(altb2) .
for all
0
x
,
Then
S
has the property that
hand
S
is certainly not commutative and so cannot be made into a ring.
2.
x1 = x
=
in
S .
On the other
Pure mathematical reasons
Let me now turn to my question 'Why study semigroups?'
I am a pure
mathematician by instinct and so I begin by offering some pure mathematical
reasons.
I shall come later to what might be called 'applied mathematical'
reasons.
2
My first point, not a fashionable one in these utilitarian times, is
that they are fun, that they provide an elegant theory, with arguments that
any mathematician can actively enjoy.
Let me give an example.
The concept
of regularity, introduced for rings by no less a person than von Neumann,
plays a much more central role in semigroup
theory.
We say that
(5,.)
(V a
theory than it does in ring
is regular if
€ S)(] i
(. S')
axa
= a
As in ring theory, idempotent elements - elements
are very important.
xa
From equation
e
such that
e1 = e
we readily see that both
ax
-
and
are idempotent.
Next, note that a semigroup
(V a
It is clear that
a'
(1)
(1 )
= xax ;
The element
€ S) (3 a' € S )
(2) —= (1) .
then from
a'
(5,.)
(1 )
aa'a
is regular if and only if
= a ,
To show that
a'aa'
(1)
= a' .
(2 )
(2)
simply take
we have
is usually called an inverse of a ,
but it should be noted
that this is a weaker concept of inverse than the one used in group theory:
for example in the four element semigroup with Cayley table
it is easy to check that every element is an inverse of every other element.
Theorem 1.
(1)
The following ooniditons on a regular semigroup
Idempotents cormrute;
3
S are equivalent
(2)
Inverses are unique
Proof.
(1) ==* (2) .
Suppose that idempotents commute.
inverses of
a .
a'
=
a'aa'
=
a*aa'aa*aa'
=
a*aa*
(2) =» (1) .
of
=
a'aa*aa'
=
Let
=
=
a*
be
by
by commuting idempotents
a*
e ,
(2 )
by
f
be idempotents and let
(2 ) .
x
be the unique inverse
ef :
fxe
=
ef ,
ef
xefx
=
x .
is idempotent, since
is an inverse of
But an idempotent
so
a'aa*cua*aa'
a*aa'aa'aa*
(fxe)2
and
a' ,
Then
efxef
Then
Let
ef = fxe ,
fxe ;
=
f(xefx)e
=
fxe ,
(ef) (fxe) (ef)
=
efxef
=
ef .
is its own unique inverse
The unique inverse of
ef ,
ef
=
(fxe)(ef) Cfxe)
i
(ef)(fe)(ef)
f(xefx)e
fxe :
an idempotent.
is an inverse of
It follows that
=
=
=
ef
Similarly
is thus
ef
fe
(iii = i ,
H i = i)
and
is idempotent.
itself.
On the other hand
fe
since
(ef)2
fe ,
=
ef ,
(fe) (ef) (fe)
=
(fe)2
=
fe .
as required.
That argument goes back to the early 1950s, to some fundamental work by
Vagner [16 , 17]
and Preston [12 , 13 , 14] .
A regular semigroup
satisfying either one (and hence both) of the conditions in Theorem 1
4
is
called an inverse semigroup.
A very impressive theory has been created for
such semigroups, as is evidenced by the publication in 1984 of a
monograph by Petrich [11]
But do semigroups of this kind occur 'in nature'?
very well known example due to Schein [15]
be a group and let
includes
G
K{G)
itself and also the cosets of the subgroup
G .
Ha * Kb
This is a natural definition:
Define an operation
=
=
Conversely, suppose that
6
Pa
and so
HaKb
1 ,
*
Let
G
This
which are
on
by
it is not hard to check that
HaKb .
(HaKa~l)ab £
Pc = Pab .
G .
(H vaKa~l)ab .
smallest coset containing the product
ab
Let me give a not
and McAlister [10] .
be the set of all right cosets of
effectively the elements of
HaKb
674-page
devoted entirely to semigroups of this kind.
£ Pa
Ha * Kb
is the
[Certainly
[H vaKa~l)ab .
(€ K(G)) .
Then in particular
Now
Hab £ HaKb c Pab
and so
H c P ;
also
(iaKa~1)ab
and so
aKa"1 £ P .
Thus
=
aKb £ HaKb £ Pab
H v aKa~l £ P
(fl \iaKa"^)ab c Pab
It is a routine matter to check that
and that
(a"lHa)a~1
Now suppose that
Ha
is an inverse of
= Pa .]
*
is an associative operation
in the semigroup
(K(G) , *) .
is idempotent:
Ha
"hen in particular
Ha
and so
=
a 2 = la2
Ha * Ha
Ha ;
=
(fl v a & M )a 2 .
i.e.
5
a 2 = ha
for some
h
in
H .
Hence
a
= h € H
and so
Ha = H .
are precisely the subgroups of
H- * K
=
N
of
G
are such that, for all
(ffvff)a
=
(NH)a .
Ha * N
=
(# vaNa~l)a
=
(ff vtf)a
=
N * la
=
(K(G) , *) .
a
in
(K(G) , *)
, K
of
G
is an inverse semigroup.
=
Na
H
K * H .
N * Ha
central idempotent then for all
N
=
(K(G) , *)
and so are central idempotents in
and so
For any two subgroups
Svif
Thus idempotents commute and so
The normal subgroups
In fact the idempotents of
G .
=
Ha
in
K(G) ,
(M)a ,
Conversely, if
N
is a
G
la * N
=
(1 vatfa
=
aN
is normal.
That is a slightly quaint example, but I mention it because I have the
feeling that it has not yet been adequately exploited.
The main reason that
semigroups turn up in mathematics is that one is very often interested in
self-maps of a set of one kind or another, and whenever
f ,
g ,
h
are
such maps it is automatically the case that
(/ O g) O h
=
/ O {g O h) .
If the maps are bijections then the appropriate abstract idea is that of a
group; if not then inevitably we must consider a semigroup.
It is this
connection with maps (arising from the associative axiom) that is the
strongest reason why semigroups are more important both theoretically and
in applications than the various non-associative generalizations of groups.
There is another pure mathematical reason for being interested in
semigroups.
It is possible to take a very general standpoint in algebra
and to discuss a so-called
of operations, where
n-algebra
: Ani — *■ A
A
having a family
ft = {uk
is an n^-ary operation.
6
: i € 1}
[For example,
in a group one can take
I
= {1 ,2 },
[(o1(a1,a2)
If
: A — +■ 3
0
=
a :a2 ,
a is map between
morphism if (for all
i
in
n1 =
J
,
2
« 2
^(aj)
=
=
,
1
a"1 .]
ft-algebras then we say that
and for all
a
cu,
i
1
<(>(aii (a1 , ... , an ))
i
If we regard
with
<j> as applying to
=
<f> is a
in
, ••• ,
A)
)) .
i
£ 1'1
in an obvious way then we can express
this property succinctly as a commuting condition
<(> O OK
A congruence on
that (for all
A
is
=
Consider the quotient set
an equivalence relation
“
i4/~
[a]
,
with the property
=
W
whose elements are equivalence classes
{x € A
: x ~ a} .
A/~
inherits the
ft-algebra structure
: we simply define
w.CCaj] , ... , [a
])
=
[u»i (a1 ,
i
and the compatibility condition
sense.
~
..... an ) ~ “i K ..... •
^
t
The congruence property means that
A
(3)
i)
a l ~ 2 I .....’ an . ~ an.
v
^
from
U). O $ .
Lj(a)
and the definition
(5)
ensures that the definition makes
: A —
=
)]
£
(4)
There is a natural map
,a
/5/~
[a]
(a
defined by
6
A)
can be interpreted as saying that
id• ° H
7'
1
=
□
H
7
;
0
*
(5)
hence, comparing with
(3) ,
Now suppose that
$
is a morphia image of
is a morphism.
is a morphism from
A .
a ~ a'
Define
~
on
if and only if
It is easy to verify that
for
we see that
~
A
A
onto
B ;
we say that
B
by the rule that
$(a)
=
is a congruence.
.
The first isomorphism theorem
ft-algebras is then as follows:
Theorem 2.
Let
im $ - B ,
A ,
B
be
and let
~
be the congruence on
Then there is an isomorphism
n-algebras, let
a
: A/--- - B
$
—*■B
: A
A
be a morphism with
defined by
(6 ) .
such that the diagram
A ./~
commutes.
This is, in one form or another, one the cornerstones of abstract
algebra.
It says in effect that an
n-algebra
A
carries its morphic
images 'within itself' and that to reveal them we need only consider the
quotients of
A
by its various congruences.
The result applies to groups, of course, but it is not usually stated
in quite this way.
This is because for a group
correspondence between congruences
N
a ~ b
The quotient
-4/~
=
{a € A
~
A
there is a one-one
and normal subgroups
: a ~ 1} ,
if and only if
is always denoted by
I
given by
8
given by
or
ab~l_ € N .
A/N .
Similarly, for a ring
there is a one-one correspondence between congruences
ideals
N
~
and two-sided
A
I
=
a ~ b
and the quotient
i4/~
{a € A
: a ~ 0} ,
if and only if
is always written
or
a - b i I ,
A/I .
In semigroups no such device is available and we
as such.
must study congruences
So semigroups consititute the simplest, most manageable and most
natural class of algebras to which the methods of universal algebra must be
applied.
Applied Mathematical reasons
3.
Let me now turn to less exalted reasons for studying semigroups.
One of the striking aspects of semigroup conferences these days is that
many of the participants, between a third and a half, at a guess, come from
departments of computer science.
The reason is that semigroups have found
significant applications in the theory of automata, languages and codes.
If
A
is a non-empty set (an alphabet, as we often want to call it)
then the set of all finite words
in the alphabet
position.
word
1
A
is a semigroup if we define multiplication by juxta
Denote the length of
(with
jl| =
0
)
u
by
This is the free monoid, generated by
A*
is usually denoted by
Now let
f
j
If we include the empty
4+ .
A .
The set of non-empty words in
A subset of
4*
is called a language .
we normally write
Q .
The function
f(.q,a)
f
simply as
qa
can be extended to
and think of
Q * A*
A
q1
=
q(q € Q)
q(wa)
=
(qw)a
= (Q,f)
is an
{q € Q ,
w € A* ,
A*-automation.
9
A
by defining
(inductively)
We say that
A* .
be a finite non-empty set and suppose that we have a map
: Q * A —+ Q ;
as 'acting' on
ju| .
then we obtain a monoid, which we denote by
a € A) .
(In the terminology of
Eilenberg [4]
this is a complete deterministic automaton.)
We may think
of it as a very rudimentary machine whose states (the elements of
be altered by various input (the elements of
Suppose now that among the elements of
Q
there is an element
we call the initial state and that there is a subset
.set of terminal states.
A language
L
recognizing
Example.
T
of. Q
=
{u € i4* ;
which
iw € T} .
is the language recognized by the automaton
A .
is called recognizable if there exists an automaton
We can picture an automaton via its state graph.
lb = 1 ,
i
called the
A
L .
A = {a b] ,
Let
L
can
Let
L
Then we say that
Q)
A) .
Q = {0,1,2,3} ,
2b = 3 ,
i = 1 ,
Qa = Ob = 0 ,
= 1 ,
T = {1,2,3} .
la = 2a = 2 ,
If
3a = 0 ,
then we can draw the picture
It is easy to see that
0
is a 'sink' state
from which no escape is possible, and that
1 aba
In fact
L ,
words in
A*
If
L
,
by
L .
Let
2 aba
=
0
.
the language recognized by this automaton, consists of all
not containing
wz (. L2 } .
=
L2 c A*
If
Lei*
F
then
then
aba
as a segment.
.L2
< L>
is defined as
^w xw z
'
■ w\ ^
denotes the submonoid of
be the set of all finite
10
subsets of
A* .
A*
>
generated
Then the set
Rat A*
from
of rational subsets of
F
A*
is the set of subsets of
by means of the operations of
U
(finite union),
A*
.
obtained
and
< >.
This leads to an important characterization of recognizable subsets of
Theorem 3.
(Kleene [7]).
A language
L
A* :
is recognizable if and only if it
is rational.
Another characterization of recognizable languages is more algebraic in
character.
of
L
If
L
is any subset of
is the relation
~
on
w 1 ~ w2
A*
A*
then the syntactic congruence
~
defined by
iff
v
€ L
iff uu2 y
€ L]
.
(7)
(I have not gone into the mathematical theory of grammar at all, but one can
perhaps dimly see that this is saying that
and
w2
are mutually
interchangeable in the set of 'meaningful' sentences that constitutes
Thus (very roughly speaking)
cat ~ dog
but
if
cat ^ black .
u = (the),
A* .
Then
is usually denoted by
Theorem 4.
L
L .
then
So 'cat' and 'dog' are syntactically equiva
lent but 'cat' and 'black' are not.)
congruence on
u = (sat on the mat)
4*/~
It is easy to verify that
~
is a
is called the syntactic monoid of
L
and
M(L) .
is recognizable if and only if its syntactic monoid
M(L)
is
finite.
This is by no means as deep as the Kleene theorem.
round it is virtually obvious.
of A on
M{L)
A
= (M{L) ,f)
=
L = {u
recognizes
4
z € L ,
w ~ z .
i.e.
(<;
*)
w a
into an
initial state and
z
M(L)
Indeed one way
is finite we can define an action
by
f(w,a)
making
If
=
wa
[w € M(L)
A*-automation.
: w € L)
But then
if and only if
lul € L
A
and hence
= (M{L) ,f)
If we then take
1
as
as the set of terminal states then
lz f I ,
if and only if there exists
that the automation
= 4*/~)
w
in
A*
z = lzl € L
recognizes
11
i.e.
L .
A
if and only if
such that
by
(7) .
w € L
and
We conclude
I mention this proof (or rather half-proof) because it emphasised the
very close links between automata and monoids.
A further connection is
provided by the theory of codes.
It is not the case that every submonoid of a free monoid is free.
example, in
{a,b}*
consider the submonoid
14* = {a 1 :
i > 2} .
The base of
indecomposable elements of
freely generated by
M)
M
(i.e.
are codes.
= M* 'J {1} ,
the set
is {a2 ,a3} ,
{a2 ,a3} .
but
M
= [a2 ,aba , ab 2 ,b},
C2
M .
For
where
Af*-\ [M*)2
of
is certainly not
We say that a subset
if it is the base of a free submonoid of
Cj
M
C
of
A*
is a code
For example,
= {a4 , b ,ba 2 ,ab ,aba2 }
This means, for example, that any word in four letters - say
x 3 x 1 x 1+x 2 x 3 - can be encoded unambiguously (in
Cx ,
say) as
ab2a2baba2b2 .
is in fact an example of a prefix code:
any word in
be decoded without hesitation by reading from the left.
C 1 fl C 1 A+ = 0 .
<C-1> = C*
can
This is because
By contrast if one tries to decode
aba^ba2
using
C2
the answer (unique since
might first have tried
x^xj ?
or
C2
is a code) is
x 5XjX 2 ?
,
but one
before reaching the correct
solution.
That the theory of codes is intimately bound up with the theory of
monoids is illustrated by.
Theorem 5.
base.
Then
Let
C
M
be a submonoid of a free monoid
is a prefix code if and only if
u ,
{M
M
ux € M =» x € M . .
is called left unitary.)
12
A*
and let
satisfies
C
be its
This has been a very sketchy introduction to a rich and rapidly growing
area.
Those whose appetites have been whetted should turn to Eilenberg [4,5]
Lallement [8 ] ,
Berstel and Perrin [l] ,
and Lothaire [9] .
REFERENCES
1.
J. Berstel and D. Perrin, Theory of codes, Academic Press, 1984 .
2.
A.H. Clifford and G.B. Preston, The algebraic theory of semigroups,
American Math. Soc., 1961 .
3.
P.M. Cohn, Universal Algebra, Harper and Row, 1965 .
4.
S. Eilenberg, Automata, Languages and Machines, vol. A , Academic Press
1974 .
5.
S. Eilenberg, Automata, Languages and Machines, vol. B , Academic Press
1976 .
6
.
7.
J. M. Howie, An introduction to semigroup theory, Academic Press, 1976 .
S.C. Kleene, Representation of events in nerve nets and finite auotmata,
Automata Studies, pp. 3-42 (Princeton University Press, 1956) .
8.
G. Lallement, Semigroups and combinatorial applications, Wiley, 1979 .
9.
M. Lothaire, Combinatorics on Words, Addison-Wesley, 1983 .
10.
D.B. McAlister, Embedding inverse semigroups in coset semigroups,
Semigroup Forum, 20 (1980), 255-267 .
11.
M. Petrich, Inverse semigroups, Wiley, 1984 .
12.
G.B. Preston, Inverse semi-groups, J. London Math. Soc. 29 (1954),
396-403 .
13.
G.B. Preston, Inverse semi-groups with minimal right ideals,
J. London
Math. Soc. 29 (1954), 404-411 .
14.
G.B. Preston, Representations of inverse semi-groups, J. London Math.
Soc. 29 (1954), 411-419 .
15.
B.M. Schein, Semigroups of strong subsets, Volzskii Matem. Sbomik
4 (1966), 180-186 .
(Russian).
13
16.
V.V. Vagner, Generalised groups, Doklady Akad. Nauk SSSR 84 (1952),
1119-1122 .
17.
(Russian).
V.V. Vagner, Theory of generalised heaps and generalised groups,
Matem. Sobomik (N.S.) 32 (1953), 545-632 .
University of St. Andrews,
SCOTLAND.
14
(Russian).