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Probability and Its Applications
Published in association with the Applied Probability Trust

Editors: J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz

For other titles published in this series, go to
www.springer.com/series/1560


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About the Authors:
Akademi
Găoran Hăognăas is Professor of Applied Mathematics at his alma mater Abo
University, from which he received his Ph.D. in Mathematics in 1974. Professor
Hăognăas is the co-author of a text on applied mathematics as well as co-editor
of the Proceedings of the Third Finnish-Soviet Syposium on Probability Theory
and Mathematical Statistics. He serves on the editorial board of the Journal of
Theoretical Probability. Professor Hăognăas was the director of the Finnish Graduate
School in Stochastics and Statistics (1998–2009).
Arunava Mukherjea is Professor of Mathematics at the University of Texas–Pan
American, and prior to 2007, was Professor at the University of South Florida.
He received his Ph.D. in Mathematics in 1967 from Wayne State University. As
visiting professor, he spent sojourns at various academic institutions including Tata
Institute of Fundamental Research in Mumbai, India and Universite of Paul Sabatier
in Toulouse, France. He was a Fulbright scholar. He is an associate editor (2005–
present) of the Journal of Theoretical Probability, and was its editor-in-chief prior
to 2005.

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Găoran Hăognăas Arunava Mukherjea

Probability Measures
on Semigroups
Convolution Products, Random Walks,
and Random Matrices

ABC


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Găoran Hăognăas
Akademi University
Abo
Făanriksgatan 3B

20500 Abo
Finland


Arunava Mukherjea
University of Texas Pan American
Department of Mathematics
1201 West University Drive
Edinburg, Texas 78539

USA


Series Editors:
Søren Asmussen
Department of Mathematical Sciences
Aarhus University
Ny Munkegade
8000 Aarhus C
Denmark

Joe Gani
Centre for Mathematics
and its Applications
Mathematical Sciences Institute
Australian National University
Canberra, ACT 0200
Australia


ISSN 1431-7028
ISBN 978-0-387-77547-0
DOI 10.1007/978-0-387-77548-7

Peter Jagers
Mathematical Statistics
Chalmers University of Technology
and Găoteborg (Gothenburg) University
412 96 Găoteborg
Sweden

Thomas G. Kurtz
Department of Mathematics
University of Wisconsin - Madison
480 Lincoln Drive
Madison, WI 53706-1388
USA


e-ISBN 978-0-387-77548-7

Library of Congress Control Number: 2010938438
c Springer Science+Business Media, LLC 2011
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Springer is part of Springer Science+Business Media (www.springer.com)


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From the Preface to the First Edition

“A Scientific American article on chaos, see Crutchfield et al. (1986), illustrates a
very persuasive example of recurrence. A painting of Henri Poincar´e, or rather a
digitized version of it, is stretched and cut to produce a mildly distorted image of
Poincar´e. The same procedure is applied to the distorted image and the process is repeated over and over again on the successively more and more blurred images. After
a dozen repetitions nothing seems to be left of the original portrait. Miraculously,
structured images appear briefly as we continue to apply the distortion procedure to
successive images. After 241 iterations the original picture reappears, unchanged!
Apparently the pixels of the Poincar´e portrait were moving about in accordance
with a strictly deterministic rule. More importantly, the set of all pixels, the whole
portrait, was transformed by the distortion mechanism. In this example the transformation seems to have been a reversible one since the original was faithfully
recreated.
It is not very farfetched to introduce a certain amount of randomness and irreversibility in the above example. Think of a random miscoloring of some pixels or
of inadvertently giving a pixel the color of its neighbor.
The methods in this book are geared towards being applicable to the asymptotics
of such transformation processes. The transformations form a semigroup in a natural way; we want to investigate the long-term behavior of random elements of this
semigroup.
To be more specific, let us consider a sequence of independent and identically
distributed random variables X0 ; X1 ; X2 ; : : : taking values in a set of affine maps
from Rd into Rd , that is, maps of the form f .x/ D Ax C B, where B and x are
d 1 column vectors and Â
A is a Ã
d d real matrix. Since f can be identified with the
AB
.d C 1/ .d C 1/ matrix
, the random variables Xi s can also be regarded as
0 1
.d C 1/ .d C 1/ random matrices; thus, , the distribution of X
Âi , is aÃprobability
AB
measure on the set of .d C 1/ .d C 1/ matrices of the form

. Let S be
0 1
the closed (with usual topology) multiplicative semigroup generated by the support
of . Then the study of the random walks Yn ; Yn D X0 X1 : : : Xn with values in S
and distribution n (the nth convolution power of ), and the set of recurrent states

v


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vi

From the Preface to the First Edition

of .Yn / become relevant in the context of the so-called iterated function systems
introduced by Barnsley and his colleagues [see Barnsley (1988)].
Let us briefly discuss another example.
Suppose that we are monitoring a random system with two states denoted 0 and 1.
Let
010001101001100010011001111000
and
101110010110011101111001111000
be observed time series of the successive states of the system. The observations
seem rather like a record of independent coin tosses, with 0 for heads and 1 for tails,
say. Viewed as a Markov chain on the two-state state space X D f0; 1g our process
would have the transition probability matrix
Â
Ã
1=2 1=2
P D

:
1=2 1=2
Let us assume, however, that the above time series are concurrent. Then another
interpretation imposes itself: the state space is subjected to a succession of random
transformations. (The first two transformation are transpositions, 0 and 1 just trade
places. At the third and fourth steps the identity map is at work. A sequence of transpositions and identities then follows, but at step 19 everything is mapped onto the
state 1. From then on the two paths are identical.) The transformations are the four
possible mappings of X into itself, the identity Ã, the transposition , and the two
constant mappings 0 and 1. The transition matrix P is then a convex combination
of matrices representing those transformations:
Â
Ã
Â
Ã
Â
Ã
Â
Ã
1 0
0 1
1 0
0 1
P Da
Cb
Cc
Cd
;
0 1
1 0
1 0

0 1
where a, b, c, d are nonnegative numbers with a C b C c C d D 1.
Thus, a natural way to analyze our observed time series is to think of them as
emanating from an independent, identically distributed sequence of mappings of
the state space into itself, or, in other words, a random walk on the transformations
of X .
A Markov chain on a finite state space can always be regarded in this way. Its
transition matrix P is a convex combination of 0-1 matrices representing mappings
of the state space X into itself. (If P is doubly stochastic we can write it uniquely
as a convex combination of permutation matrices, this is the celebrated Birkhoff
theorem.) The corresponding result is true even for a large class of Markov chains
on a topological space X , see Kifer (1986), Chapter 1.
To consider an example in the context of particle systems, let V be an arbitrary
countably infinite set (with discrete topology), and let € denote the semigroup of


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From the Preface to the First Edition

vii

functions f W V ! V under composition. We can then identify each f in € with
an infinite 0-1 stochastic matrix Af such that
.Af /ij D 1

if and only if

f .i / D j:

By a configuration Á of V , we mean a nonnegative integer valued function on V

such that
X
Á.x/ < 1:
x2V

The idea is that Á.j / is the number of particles that occupy the site j 2 V , and that
when we apply the mapping f W V ! V , all these particles move to the site f .j /,
and the configuration changes to Á Af , where
.Á Af /.x/ D

X

Á.y/ ıf .y/ .x/:

y

The new configuration has at site x all the particles that the map f has sent to
the site x from the sites of the original configuration. Thus, to study the random
motions of finite systems of particles on V , without births or deaths, where each site
may be occupied by a finite number of particles, and all particles at a particular site
move together, one needs to study the random transformations F (that is, the infinite
random stochastic matrices AF ). Instead of studying the different configurations, we
study a sequence of independent identically distributed countably infinite stochastic
matrices, and among other things, will be interested in gaining some insights in the
limiting laws of products of these matrices.
To mention yet another context where probability measures on countable semigroups have been found useful, we mention the paper of Hansel and Perrin (1983),
where the authors utilized the structure of an idempotent probability measure on a
semigroup in order to have some insights in certain problems in coding theory.
It is also relevant to mention that Ruzsa (1994) utilized his results on weak convergence of the sequence 1
2 :::

n , where the i s are probability measures
on a countable semigroup, in proving a generalization of a result in number theory
due to Davenport and Erdăos (1936). This last mentioned result simply says that every multiplicative ideal A of the set N of positive integers has a logarithmic density,
that is,
X 1
1
.A/ D lim
;
n!1 log n
a
a2A
aÄn
exists. Note that for a set A
N , its logarithmic density
asymptotic density d , given by
1 X
1;
n!1 n
a2A
aÄn

d.A/ D lim

may exist while its


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viii

From the Preface to the First Edition

may not exist. [It is well known, however, that .A/ exists whenever d.A/ does,
and then .A/ D d.A/.] Ruzsa’s result says the following: if f is a homomorphism
from the multiplicative semigroup of integers to a commutative semigroup H , then
for every h 2 H , the set fn 2 N W f .n/ D hg has a logarithmic density.
Let us finally mention, before we go to the text proper, that abstract semigroup
theory was of crucial importance in developing the methods used in Hăognăas and
Mukherjea (1980) to study the set of recurrent states of a random walk taking values
in n n real matrices.”

References
Barnsley, M. F., Fractals Everywhere, Academic Press, Orlando (1988).
Crutchfield, J. P., J. D. Farmer, N. H. Packard, and R. S. Shaw, “Chaos,” Scientific American 255,
No. 6, 38–49 (1986).
Gelbaum, B. R. and G. K. Kalisch, “Measures in Semigroups,” Canadian J. Math. 4, 396–406
(1952).
Hansel, G. and D. Perrin, “Codes and Bernoulli partitions,” Math Systems Theory 16, 133–157
(1983).
Hennion, H., “Limit theorems for products of positive random matrices, The Annals of Probability
25, No. 4, 15451587 (1997).
Hăognăas, G. and A. Mukherjea, “Recurrent random walks and invariant measures on semigroups of
n n matrices,” Math Zeitschrift 173, 69–94 (1980).
Kifer, Y., Ergodic Theory of Random Transformations, Birkhăauser, BostonBaselStuttgart
(1986).
Raugi, A., Une demonstration du th´eoreme de Choquet-Deny par les martingales,” Ann. Inst.
Henri Poincar´e Statist. Probab. 19, 101–109 (1983).
Ruzsa, I. Z., Logarithmic density and measures on semigroups, Manuscripta Mathematica 89, 307–
317 (1996).

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Preface to the Second Edition

In this new edition, we have corrected all errors (mathematical and otherwise) that
we could find or were brought to our attention from the book’s first edition by our
colleagues and students. We have also updated the references, all notes and comments at the end of each chapter, and more importantly, added exercises at the end
of each section in each chapter. We have also added new results often in the main
text and in the appendices.
The book, we feel, can be a useful reference for courses in the area of probability
on algebraic structures. It can also serve as a text for graduate or senior undergraduate students for an one semester course on “Probability measures on semigroups”
designed around the following four core topics:
(i) Completely simple semigroups: their structure and the Rees-Suschkewitsch
theorem (see Chapter 1)
(ii) Convolution products of probability measures on locally compact semigroups,
their weak convergence, and their weak limits which are idempotent probability measures, with supports always closed completely simple subsemigroups,
and structure theorem for idempotent probability measures (see Chapter 2)
(iii) Recurrent random walks on semigroups: characterization of the set of recurrent
states as closed and completely simple subsemigroups (see Chapter 3)
(iv) Tightness and convergence in distribution of products of d by d i.i.d. random
nonnegative matrices: The Kesten-Spitzer theory and various generalizations
to random real matrices (see Chapter 4)
The first chapter of the book can also serve as a text for an one semester course
on Semigroups for senior undergraduate or graduate students. Sufficiently many
exercises of varying levels of difficulty have been included to help the instructor of
such a course.
Semigroups are very general structures and we often come across them in various
contexts in science and engineering. (See the preface to the first edition.) The results
that we have presented here on weak convergence or random walks or random matrices using semigroup ideas are for the most part complete and best possible. Still

ix


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x

Preface to the Second Edition

there are other results that remain to be completed. These are all mentioned in the
notes and comments at the end of each chapter, and this, we hope, will keep the
readership of this book enthusiastic and interested for some time to come.
˚ Akademi University, Abo,
˚
Abo
Finland
UTPA, Edinburg, Texas 78539, USA
August 30, 2010

Găoran Hăognăas
Arunava Mukherjea

Acknowledgements

We want to thank Abo
Akademi University student Susanna Nyberg for technical
assistance and help with the exercises in Chap. 1. Also we are much indebted to
student Andreas Anckar who made the LaTeX compilations for us in a very competent way. Let us also thank many of our colleagues and friends for assisting us in
various ways in the completion of the present edition. We are specially indebted to
Greg Budzban, Edgardo Cureg, S.G. Dani, Phil Feinsilver, Yves Guivarc’h, Herbert
Heyer, Karl Hofmann, B.V. Rao, Ricardo Restrepo, Todd Retzlaff, M. Rosenblatt,

Imre Ruzsa, T.C. Sun and Nicolas Tserpes. Last but not least, we must thank Vaishali
Damle (the Springer editor) who very patiently and efficiently helped us through
different stages of the production of this book.


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Contents

1

Semigroups . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.2 Homomorphisms, Quotients, and Products .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.3 Semigroups with Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.4 The Rees–Suschkewitsch Representation Theorem .. . .. . . . . . . . . . . . . . . . .
1.5 Topological Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.6 Semigroups of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.7 Semigroups of Infinite Dimensional Matrices . . . . . . . . . .. . . . . . . . . . . . . . . . .
1.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

1
1
6
10
12
22
33
53
60

2 Probability Measures on Topological Semigroups . . . . . . . .. . . . . . . . . . . . . . . . . 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 63
2.2 Invariant and Idempotent Probability Measures .. . . . . . .. . . . . . . . . . . . . . . . . 64
2.3 Weak Convergence of Convolution Products of Probability Measures 83
2.4 Weak Convergence of Convolution Products
of Nonidentical Probability Measures . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .137
2.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168
3 Random Walks on Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .171
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .171
3.2 Discrete Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .180
3.3 Locally Compact Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .198
3.4 Compact Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .219
3.5 Completely Simple Semigroups . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .241
3.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .247
4 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253
4.2 Recurrent Random Walks in Nonnegative Matrices . . .. . . . . . . . . . . . . . . . .254
4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence 284
4.4 Invariant Measures for Random Walks in Nonnegative
Matrices: Laws of Large Numbers .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .333
4.5 Notes and Commments .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .355
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xii

Contents

A Products of I.I.D. Random Stochastic Matrices: Their
Skeletons and Convergence in Distribution . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .359
B An Example Due to Chamayou and Letac . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .365
C Asymptotic Behavior of kXn Xn 1 : : : X0 uk for I.I.D.
Random Nonnegative Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .369
D Continuous Singularity of the Limit Distribution of
Products of I.I.D. Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .383
E Rate of Decay of Concentration Functions on Discrete
Groups, by T.M. Retzlaff .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .397
E.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .397
E.2 Irreducible Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .404
E.3 Adapted Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408
References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .413
Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .423


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Chapter 1

Semigroups

1.1 Introduction
Chapter 1 contains the basics of semigroups: definitions, elementary concepts,
and fundamental examples. We assume some familiarity with standard notions of
point-set topology (see [120, 163]); the algebraic portions of Chap. 1 are, however,
completely self-contained. Without going into any details whatsoever, it is perhaps
prudent to remark at this point that our main interest centers around asymptotics,
invariance questions, etc. Our treatment is a reflection of this. We concentrate on
algebraic concepts corresponding to such phenomena as absorption, stability, and

invariance: zeros, simple semigroups, minimal ideals, maximal subgroups, and so
on. We strive to keep the digressions at a minimum. Clifford and Preston [43] offers a wealth of information on all aspects of algebraic semigroups, and this text is
recommended to any reader interested in a much more elaborate treatment of this
fascinating subject.
Sections 1.1–1.5 contain basic material necessary for the development of all
subsequent chapters, while Sects. 1.6 and 1.7, which deal with more specific applications, can be skipped at first reading.
Arguably, the most important notion in mathematics is that of a mapping (or
function or transformation). The ultimate goal of the research presented in this
book is to describe the long-term behavior of random transformations of some set.
Transformations of a set form a semigroup in a natural way. Indeed, we see that any
semigroup is (algebraically) a transformation semigroup in a canonical way. Linear
transformations of a vector space form another family of fundamental examples.
We devote considerable effort to those semigroups, which incidentally may just as
well be viewed as semigroups of matrices.
Let S be a set. If S is endowed with an associative binary operation [which
we call multiplication and denote by a dot, . / or simply by juxtaposition], then the
S is called a semigroup. Strictly speaking, the semigroup is the pair .S; /, but the
intended operation is usually quite clear from the context. When we are dealing with
a specific application, we of course use the established notation.
If s is an element of a semigroup S and A and B are subsets of S , then we denote by sA the set fsa 2 S ja 2 Ag and by AB the set fab 2 S ja 2 A; b 2 Bg.

G. Hăognăas and A. Mukherjea, Probability Measures on Semigroups: Convolution
Products, Random Walks and Random Matrices, Probability and Its Applications,
DOI 10.1007/978-0-387-77548-7 1, c Springer Science+Business Media, LLC 2011

1


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2

1 Semigroups

(As is, of course, defined analogously.) Note that products with more than two
factors, such as abc and aBcS , are well defined due to the associativity of the
multiplication. aa; aaa; : : : are usually written a2 ; a3 ; : : :
A nonempty subset T of S is called a subsemigroup if it is stable under multiplication, i.e., if T T
T . If T is also a group, we call it a subgroup of S . As we
see later, it is important in many applications to identify the subgroups of a given
semigroup S .
A subsemigroup L of S is called a left ideal, if SL L; right ideals are similarly
defined. A nonempty subsemigroup I is a bilateral or two-sided ideal or just ideal
of S if it is both a right and a left ideal: SI I; IS I . S is said to be left (right)
simple if it contains no proper left (right) ideal. Similarly, S is simple if the only
ideal of S is S itself. A left (right) ideal is a principal left (right) ideal if it is of the
form fag [ Sa (fag [ aS ) for some a 2 S .
Note that S is left simple if and only if for any given a; b 2 S , the equation
xa D b is soluble. (Sa is a left ideal of S for all a 2 S . On the other hand, any left
ideal L contains a subset of the form Sa.)
An element e 2 S is called a left (right) identity element of S if es D s .se D s/
for every s 2 S . e is a two-sided identity element of S , or simply identity of S , if it
is both a right and a left identity of S . It is easy to see that the identity is unique if it
exists.
An element z of S is called a left (right) zero element of S if zs D z .sz D z/ for
all s 2 S . If z is both a left zero and a right zero of S , we simply call it a zero of S .
A semigroup has at most one zero.
The semigroup S is said be left (right) cancellative if for any a 2 S , the equation
ax D ay .xa D ya/ in S implies x D y.
An element a 2 S is idempotent if a2 D a. Zeros and identities are idempotent.
An idempotent is in a trivial fashion the identity element of a subgroup of S .

Elements a and b of S are said to commute if ab D ba. If all elements of S commute with each other, S is called a commutative or abelian semigroup. In abelian
semigroups, the operation is often called addition and denoted by C, the identity
element by 0 and the inverse of a by a.
To put the preceding concepts into perspective, we now investigate some of their
relationships to a group structure. Most textbooks define a group as a nonempty set
G with an associative binary operation with identity element e and inverses; i.e., for
all a 2 G, there is a b 2 G such that ab D ba D e, (see, for example, [62] or
[217]). There is, however, a multitude of alternative, seemingly weaker, but in fact
equivalent definitions, see [43], Chap. 1.
In our context, Proposition 1.1 is a convenient characterization of a group.
Proposition 1.1. A semigroup is a group if and only if it is both left and right simple.
Proof. A group is clearly both left and right simple. Conversely, let S be a semigroup which is both left and right simple. For any a 2 S , the equation ae D a has
a solution. On the other hand, Sa is all of S ; hence, e is a right identity of S . In
the same way, we can produce a left identity f which turns out to be equal to e.
(We have f D f e D e.)


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1.1 Introduction

3

It is just as straightforward to obtain the two-sided inverse for an element a 2 S .
Let b and c be the solutions of the equations ab D e and ca D e, respectively. Then
b D eb D .ca/b D c.ab/ D ce D c: Thus, S is a group.
t
u
If A is a subsemigroup of S we denote by hAi all the elements in S that can
be expressed as finite products of the elements in A. In the case hAi D S , we call
A the generator of S . If a semigroup is generated by a single element, we call it

monothetic or cyclic.
Remark 1.1. In a finite semigroup S , hxi contains a subgroup for every x 2 S , cf.
Exercise 1.6.
Example 1.1 (The semigroup of transformations of a set).
Let X be any finite or infinite nonempty set. If f and g are mappings from X
to itself, we define as usual the composition f ı g of f and g by .f ı g/.x/ D
f .g.x//; x 2 X . (The domain of the mappings is always understood to be all of X .)
The composition of mappings is an associative operation; hence, the set of all mappings from X into X forms a semigroup with composition as multiplication. This
semigroup is called the full transformation semigroup on X and denoted by TX .
A more complete treatment of TX is given in [43], Chap. 2.2.
TX has an identity element, the identity mapping Ã: Ã.x/ D x; x 2 X . The
constant mappings are left zeros, since c ı f D c for all f if c.x/ D x0 ; x 2 X
(where x0 is a particular element of X ). On the other hand, TX does not admit any
right zeros unless of course X is a singleton.
Define the range R.f / of an f 2 TX to be the set f .X / Á ff .x/ j x 2 X g
(where Á means equal by definition). Clearly, R.f ı g/ R.f /. If R0 is a subset
of X , then mappings with range inside R0 , ff 2 TX jR.f /
R0 g, form a right
ideal of TX .
The partition .f / of X generated by an element f 2 TX is the equivalence
relation on X defined by x .f /y ” f .x/ D f .y/; x; y 2 X . In other words,
elements are .f / equivalent if and only if they have the same image under f . The
.g/-equivalence implies .f ı g/- equivalence, .f ı g/
.g/. Consequently,
if 0 is a given equivalence relation on X , then those f 2 TX with .f /
0
form a left ideal of TX .
Define the rank of f to be the cardinality of R.f /, which we denote by jR.f /j.
Note that the cardinality of R.f / is the same as that of the quotient space X= .f /
[the number of .f /-equivalence classes]. Mappings with rank no larger than a

given cardinality r, ff 2 TX j jR.f /j Ä rg, form a two-sided ideal of the semigroup TX .
The constant mappings, i.e., the mappings of rank 1, form the minimal two-sided
ideal of TX . This subsemigroup is a left zero semigroup, since all its elements are
left zeros.
Let us now identify the subgroups of TX . Since any subgroup has an idempotent
as its identity element, our first task is to determine idempotents in TX . Let e be
idempotent with range R and partition , whence e.x/ D x; x 2 R. As before
jRj D jX= j. This is possible if and only if R is a complete set of representatives of


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4

1 Semigroups

the -equivalence classes, i.e., every equivalence class contains exactly one element
of R. In the terminology of Clifford and Preston [43] R is a cross section of .
Suppose f belongs to a subgroup G of TX and the identity element of G is the
idempotent e previously discussed. We can immediately conclude from the relation
e ı f D f ı e D f that the range of f is R and the partition corresponding to
f is . Furthermore, f is a one-to-one mapping from R to R precisely because
R is a cross section of . We can then construct a g belonging to TX with the
following properties: g has range R and partition , and the restriction of g to
R is the inverse of the restriction of f to R. It is then clear that g ı f D f ı
g D e, in other words, the inverse of f in the subgroup G is g. Our result is
thus the following, see [43], Theorem 2.10: f belongs to a subgroup of TX if and
only if R.f / is a complete set of representatives of .f /. The sets ff jR.f / D
R; .f / D g where jRj D jX= j are groups if and only if R is a complete set of
representatives of .
If there is an f whose range is not a cross section of .f /, then f ı f has a

smaller range than f : R.f ı f / is a proper subset of R.f /. Clearly, such an f
cannot belong to a group.
The preceding discussion holds almost verbatim for any transformation semigroup on X ; i.e., any subsemigroup S of TX . The conditions are necessary only in
the general case. For example, f 2 S can belong to a subgroup of S only if R.f /
is a cross section of .f /. For a transformation semigroup S on a finite set X we
have the converse: If elements of a subsemigroup of S have common range R and
partition , where R is a cross section of , then it is a group. (Each element f is
of finite order, i.e., some power of f equals the identity mapping e on R with the
given partition .)
Let X be countably infinite and G a subgroup of infinite rank of TX . If e is the
identity element of G, then we can construct a mapping ˛ with the properties that ˛
restricted to the range of e is a bijection onto all of X (practically the definition of
an infinite subset of X ) and ˛ has the same partition as e. If we denote the inverse
of ˛ restricted to the range of e by ˇ, then ˇ is injective, and has the same range
as e. Hence, ˛ ı ˇ D Ã (the identity mapping on X ) and ˇ ı ˛ D e. For any g 2 G,
˛ ı g ı ˇ is a bijection on X . Conversely, the elements of G can be written ˇ ı h ı ˛,
where the h’s are bijections on X .
The semigroup BX of relations on the set X consists of all subsets of the Cartesian product X X equipped with the composition operation ı: For any two and
in X X define .x; y/ 2 ı if and only if there is a z 2 X such that .x; z/ 2
and .z; y/ 2 .

Section 1.1 Exercises
Exercise 1.1. Show that a semigroup with more than one element contains a subsemigroup S 0 Ô S .


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1.1 Introduction

5

Exercise 1.2. Prove that a finite semigroup that is cancellative is in fact a group.
Exercise 1.3. Show that if a semigroup S contains a left identity element e, such
that there for any x 2 S exists a y 2 S such that yx D e, then S is a group.
Exercise 1.4. Prove that the following are groups:
(i) A left simple semigroup that contains a left identity.
(ii) A cancellative and simple semigroup containing an idempotent
Exercise 1.5. Let S be a semigroup so that for every x 2 S there exists an a 2 S
such that x D xax. Show that the following are equivalent:
(a) S has exactly one idempotent
(b) S is cancellative
(c) S is a group
Exercise 1.6. Let S be a finite semigroup generated by a single element x. Prove
that there exists m; r 2 N, such that x mCr D x r and
hxi D fx; x 2 ; : : : ; x mCr
Also show that the set fx r ; x rC1 ; : : : ; x mCr

1

1

g:

g is a subgroup of S .

Exercise 1.7. Let S be a finite commutative semigroup. Show that S can be partitioned into maximal subsemigroups fP .e.i //ji D 1; 2; : : : rg, where fe.1/; e.2/; : : :
e.r/g is the set of all idempotents of S , and x is in P .e.i // iff x k D e.i / for some
positive integer k. Further, show that there are smallest positive integers K and D
such that for all x in S , x K D x KCD .
Exercise 1.8. Verify that the definition of the operation ı defined above on the semigroup BX of relations on the set X coincides with the composition on TX defined
in Example 1.1 in case the relations are transformations of X . (f 2 TX can be

viewed as a relation f 2 BX when we identify f with its graph, a subset of X X :
.x; y/ 2 f if and only if y D f .x/.)
Show that if a relation
ı D .

is reflexive and transitive, then it is idempotent, i.e.,

Exercise 1.9. A d d real matrix is called circulant and denoted by circ(x.0/;
x.1/; : : : ; x.d 1/) if its first row is x.0/; x.1/; : : : ; x.d 1/, its second
row x.d 1/; x.0/; x.1/; : : : x.d 2/, the third row x.d 2/; x.d 1/; x.0/; : : : ;
x.d 3/ and so on:
0

x.0/
Bx.d 1/
B
@ :::
x.1/

1
x.1/ x.2/ : : : x.d 1/
x.0/ x.1/ : : : x.d 2/C
C:
::: ::: :::
::: A
x.2/ x.3/ : : : x.0/

Prove that the circulant matrices form a commutative semigroup with respect to
matrix multiplication.

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6

1 Semigroups

Exercise 1.10. Let x be a d d real matrix. Show that the following are equivalent:
(i)
(ii)
(iii)
(iv)

x is circulant;
x T , the transpose of x, is circulant;
xP D P x, where P is the permutation matrix circ(0; 1; 0; : : : ; 0);
x D f .P /, where f is a polynomial of order less than d .

1.2 Homomorphisms, Quotients, and Products
A mapping
between two semigroups .S; / and .T; / is called a semigroup
homomorphism (antihomomorphism) if
.a b/ D .a/

.b/ . .a b/ D .b/

.a//; a; b 2 S:

The is said to be a semigroup isomorphism (anti-isomorphism) if it is bijective
(i.e., onto and one-to-one) as well. We however usually suppress the explicit reference to semigroups when it is clear from the context that we are dealing with a
semigroup structure. The S and T are isomorphic (as semigroups) if there exists a

semigroup isomorphism between them. Using the antiisomorphism a b 7! b a
we can always convert an antihomomorphism to a homomorphism, if need be.
As an example, consider the set of left translations on a given semigroup S :
a ; a 2 S , where a is defined by a .x/ D ax; x 2 S . Clearly, ab D a ı b . The
is thus a homomorphism from S to the full transformation semigroup on S , TS .
Right translations define in a similar way a semigroup antihomomorphism from S
to TS . If in addition ax D bx for all x 2 S implies a D b (the left translations act
effectively on S), then is injective and S is isomorphic to a subsemigroup of TS .
In particular this is the case when S has a right identity element.
If we extend the idea of left translations slightly, we obtain the useful result that
any semigroup S is isomorphic to some transformation semigroup. We need only
take X D S [ 1 and define a1 D a; a 2 S . Left translations a ; a 2 S on X define
an injective homomorphism from S into TX .
An equivalence relation on a semigroup S compatible with the multiplication
is called a congruence on S . More formally, is a congruence if for all a; b; s 2 S ,
a b implies as bs and sa sb. If is a congruence on S , then the multiplication of
equivalence classes in the natural way will be a well-defined operation on S= . (For
a; b 2 S , we take ŒaŒb D Œab, where Œa is the equivalence class containing a).
The semigroup thus obtained is called the quotient or factor semigroup of S mod .
The discussion in the preceding paragraphs shows that if is the congruence on
S defined by
a b ” ax D bx for all x 2 S;
then S= is isomorphic to a subsemigroup of TS .
Definition 1.1 presents another useful congruence in semigroup theory.


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1.2 Homomorphisms, Quotients, and Products

7

Definition 1.1. Let I be an ideal of S . If we define a relation

by

a b ” a D b or else both a and b 2 I;
then is a congruence on S . The corresponding factor semigroup is usually written
S=I , and it is called the Rees factor (or quotient) semigroup of S mod I . The
intuitive idea behind the Rees factor semigroup is to lump all elements of I together
into a single zero.
Take a semigroup S and view it as transformation semigroup on X . For f 2 S ,
define a matrix Bf .x; y/ indexed by X according to the following prescription:
Bf .x; y/ D ıf .y/ x, i.e., Bf .x; y/ D 1 if x D f .y/ and 0 otherwise, x; y 2 X .
Multiplying the matrices according to the usual rules of matrix multiplication, we
see that indeed Bf ıg D Bf Bg , [see [53] where the matrices Af have the antihomomorphism property instead]. Hence, any semigroup can be described as a
semigroup of 0–1 matrices, transformation matrices, with exactly one 1 in each
column. If there is exactly one 1 in each row as well, we obtain the familiar permutation matrices corresponding to bijections on the set X .
For semigroups .S; / and .T; /, we obtain a new structure on their cartesian
product by the rule
.s; t/ ? .s 0 ; t 0 / D .s s 0 ; t

t 0 /:

The resulting semigroup .S T; ?/ is called the direct product of .S; / and .T; /.
Direct products with several factors are defined analogously.
Let G be a group and E be a right zero semigroup (where ee 0 D e 0 ; e; e 0 2
E). Consider the direct product of G and E. Multiplication in G E is given by
.g; e/.g 0 ; e 0 / D .gg 0 ; e 0 /.
The usefulness of this structure is due to the fact that any right group has this
representation.

Definition 1.2. A right group (left group) is a semigroup that is right simple (left
simple) and left cancellative (right cancellative).
Alternative characterizations are given in Proposition 1.2
Proposition 1.2. For a semigroup S , the following statements are equivalent:
(i)
(ii)
(iii)
(iv)

S is a right group;
For any a; b 2 S , the equation ax D b has one and only one solution;
S is right simple and contains an idempotent;
S is isomorphic to the direct product of a group G and a right zero semigroup E.

Proof. The equivalence of (i) and (ii) follows immediately from the definitions and
the characterization of right simplicity in Sect. 1.1.
Let S satisfy (ii). Then for any a 2 S the equation ax D a has a solution e, say.
We have aee D ae; by left cancellativity, ee D e: Thus (ii) implies (iii).


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8

1 Semigroups

Assume (iii) and let ax D ay for some a; x; y 2 S . There is an idempotent e in
S , e is a left identity of S D eS . There is a b 2 S such that ab D e. .ba/.ba/ D
b.ab/a D bea D ba, so ba is an idempotent too and consequently a left identity
of S . We finally obtain ax D ay H) bax D bay H) x D y; i.e., (i) holds. The
equation .g; e/.x; y/ D .g 0 ; e 0 / in the direct product G E has the unique solution

.x; y/ D .g 1 g 0 ; e 0 /. This shows that (iv) implies (ii).
Suppose now that S satisfies (iii), and, equivalently, (i) and (ii). Let E be the set
of idempotents in S . We saw above that any idempotent is a left identity, so ee 0 D
e 0 ; e; e 0 2 E. In other words, E is a right zero (sub)semigroup (of S ). Take an e 2 E.
We show next that Se is a subgroup of S . Se is right simple: seSe D Se. Se is also
left simple. To see this, take an element se 2 Se and let t 2 S be such that set D e.
Then se.te/ D .set/e D ee D e and se.tese/ D .sete/se D ese D see (since
e is a left identity of S ), so that tese D e by left cancellability. For an arbitrary
ue 2 Se; the equation .xe/.se/ D ue can be solved, namely by x D uet 2 S . By
Proposition 1.1, Se is a group. Take a particular idempotent e0 and let Se0 D G.
Consider the map .g; e/ 7! ge; g 2 G; e 2 E: Call it . We prove that is the
desired isomorphism.
The map is a homomorphism: For g; h 2 G and e; f 2 E, gehf D ghf
because each idempotent is a left identity.
is injective. To see this, let ge D hf . Then g D ge0 D g.ee0 / D .ge/e0 D
.hf /e0 D h.f e0 / D he0 D h. Left cancellativity then yields e D f .
is surjective, since any a 2 S can be written in the form ae for some
idempotent e [see the proof of (ii) H) (iii) above]. e0 is a left identity of S , so
a D a.e0 e/ D .ae0 /e.
t
u
To understand right groups concretely, let us look at the full transformation semigroup TX and its subsemigroups. The possible idempotents and subgroups were
characterized in Sect. 1.1. Note that the subgroups consist of bijections (permutations) of the set R. It is not difficult to see that subgroups are isomorphic to a
permutation group on R (i.e., a subgroup of the symmetric group GR on R consisting of all the bijections from R to R).
It is evident that a right group S
TX consists of mappings with a common
range R0 ; otherwise, it could not possibly be right simple. It follows from the assumption that S is closed under multiplication that the common range R0 is a cross
section of all partitions generated by elements of S .
Indeed, we obtain Proposition 1.3 in the case of a finite X :
Proposition 1.3. A subsemigroup S of TX is a right group if and only if the elements of S have common range.

Proof. The only if statement is immediate, so let us concentrate on the sufficiency,
the if statement. Any f 2 S is bijective on the common range, so left cancellativity
follows. Since X is finite, the powers f n ; n D 1; 2; 3 : : : form a group whose identity is a left identity of S . The equation f ı g D h thus has a solution g D f r 1 ı h
(where r is the order of the group generated by f ). In other words, S is right simple
and hence a right group.
t
u


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1.2 Homomorphisms, Quotients, and Products

9

To see how the concepts work for countably infinite X , let us return to the last
paragraph of Example 1.1. If H is a subgroup of the symmetric group GX , then
ˇ ı H ı ˛ is an infinite-rank subgroup of TX . Conversely, any such subgroup is
isomorphic to ˇ ı H ı ˛ for some H; ˛, and ˇ. Right groups are then obtained by
varying the partition in the construction of ˛: Any right group of infinite rank is
isomorphic to ˇ ı H ı E, where E is some set of mappings constructed exactly as
˛ but with the partition varying; of course, the fundamental property of the range
as a cross section of the partition must be maintained. In particular for an ˛ 0 2 E,
˛ 0 ı ˇ D Ã and ˇ ı ˛ 0 is the identity of some group of mappings with the same
range as e.
Definition 1.3. We conclude this section by introducing another important product
structure, the Rees product. Let E be a left zero semigroup, F a right zero semigroup, G a group, and a function from F E to G. Define a multiplication on
E G F by
.e; g; f /.e 0 ; g 0 ; f 0 / D .e; g .f; e 0 /g 0 ; f 0 /:
Note that this product is direct if the sandwich function maps everything onto the
identity element of the group G. Such a is termed trivial.

We emphasize at this point that is a completely arbitrary function from F E
to G. Different choices of may produce isomorphic semigroups. If, for example,
maps everything onto a constant c 2 G, then the resulting Rees product is isomorphic to the direct product of E; G; and F . We will return briefly to this question
in Sect. 1.4 (Proposition 1.10).
Remark 1.2. The cylinder subsets of the form feg G ff g , called cells, are all
groups isomorphic to G. The identity of such a group is the element .e; . .f; e// 1 ;
f /. Any subsemigroup of the form feg G B (where B F ) of the Rees product
is a right group. This fact is an immediate consequence of Definition 1.2.

Section 1.2 Exercises
Exercise 1.11. Let W S ! T be a homomorphism, where S and T are semigroups. Show that maps subsemigroups of S to subsemigroups of T , and also
idempotents of S to idempotents of T . Show further that if is onto, then right ideals of S are mapped to right ideals of T , and the identity of S to the identity of T .
Exercise 1.12. Let S be a semigroup and T a subsemigroup of S , such that xT D
T x for every x 2 S . Prove that , defined by a b , aT D bT , is a congruence.
Exercise 1.13. Prove that every equivalence relation on a semigroup of left zeros is
a congruence.


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1 Semigroups

Exercise 1.14. Let S be a semigroup and let be a congruence on S . Prove that if
e is an idempotent, e 2 S , then its equivalence class e= is a subsemigroup of S ,
and is an idempotent in the quotient S= . Also, prove that if S is finite and x= is
an idempotent of S= , then x= contains an idempotent.
Exercise 1.15. Prove that the following are equivalent:
(i) S is a right group
(ii) There exists a right identity element e 2 S such that e 2 aS for every a 2 S .

(iii) For each element a 2 S , Sa contains a left identity element of S .
Exercise 1.16. Show that if X D f1; 2; 3; 4g and R D f1; 2g, then the semigroup S
of all maps from X into X with range R is a right group, but not a group. Find all
the eight elements of S and the right group decomposition as the direct product of a
group and a right zero semigroup.

1.3 Semigroups with Zero
Recall that an element z of a semigroup S is a zero if sz D zs D z for all s 2 S .
A zero is unique if it exists. We henceforth adhere to the common convention and
denote a zero by 0.
The notions of (right, left) simplicity are trivial in the presence of a zero 0. For
instance, S is simple if and only if S D f0g: For semigroups with zero, it is therefore
useful for many purposes to restrict some of the definitions to nonzero elements of S .
It is often practical to have a special notation for these elements; A is the generic
notation for the nonzero elements of the set A
A Á A n f0g Á fa 2 Aja Ô 0g:
An ideal I Ô f0g of a semigroup S is said to be 0-minimal if f0g is the only ideal
of S properly contained in I . A 0-minimal left (right) ideal is defined analogously.
A semigroup S is called right (left) 0-simple if S 2 Ô f0g and its only right (left)
ideals are 0 and S itself. S is called 0-simple if S 2 Ô f0g and 0 is the only proper
two-sided ideal of S .
A semigroup S with the property that all products are 0, S 2 D f0g, is called a
null semigroup. Such a semigroup obviously satisfies the second condition of the
preceding definitions.
Elements (nonzero elements) a and b of a semigroup with 0 are called divisors
of zero (proper divisors of zero) if ab D 0.
Propositions 1.1 and 1.2 have their counterparts for semigroups with 0, which
follow immediately from the original Propositions and Lemma 1.1.
Proposition 1.4. A semigroup with 0 is a group with 0 if and only if it is both left
and right 0-simple.

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1.3 Semigroups with Zero

11

Proposition 1.5. For a semigroup S with 0, the following statements are equivalent:
(i)
(ii)
(iii)
(iv)

S is a right group;
For any a 2 S ; b 2 S the equation ax D b has one and only one solution;
S is right 0-simple and contains a nonzero idempotent;
S is isomorphic to the Rees quotient of the direct product of a group with zero
G 0 and a right zero semigroup E modulo E f0g.

Lemma 1.1. A right (left) 0-simple semigroup has no proper divisors of zero.
Proof. Let S be a right 0-simple semigroup and assume that a and b are proper
divisors of zero. For a given a, solutions of the equation ax D 0 form a right ideal
of S . Consequently by our assumptions on S , the existence of one nonzero solution
b implies that all elements of S are solutions to ax D 0; i.e., aS D f0g. Those as
with this property also form a right ideal of S . Again that right ideal has to be all of
S , implying S S D f0g, which contradicts the assumption of right 0-simplicity. u
t
Definition 1.4. A product structure analogous to the Rees product in Sect. 1.2 is the
Rees product over a group with zero which is defined as follows: Let G 0 be a group
with a zero 0 adjoined. Let E and F be as in the definition in Sect. 1.2. denotes a

map from F E to G 0 . Form the product E G 0 F with the same multiplication
rule as before
.e; g; f /.e 0 ; g 0 ; f 0 / D .e; g .f; e 0 /g 0 ; f 0 /; e; e 0 2 E; g; g 0 2 G 0 ; f; f 0 2 F:
This defines a semigroup. The set I Á E f0g F consisting of triples with zero
middle factor is a two-sided ideal of the semigroup. The Rees product over G 0 is
then obtained by collapsing all of I into a zero. More precisely, the Rees product
over the group with zero is the Rees quotient
E

G0

F mod I:

The cells feg G ff g are again groups with identity .e; . .f; e// 1 ; f / provided the middle term exists, i.e., .f; e/ Ô 0. These cells are called group cells.
Null cells are characterized by .f; e/ D 0. The null cells (with 0 adjoined) are null
subsemigroups of the Rees product over G 0 .
The sandwich function is said to be regular if the mappings .f; / and . ; e/
are not identically 0 for any f 2 F or e 2 E. In other words, for each f 2 F there
is an e 2 E (and for each e 2 E there is an f 2 F ) such that .f; e/ Ô 0. If is
pictured as a matrix indexed by F E, then it is regular if and only if there are no
zero rows or zero columns.
Remark 1.3. The condition that be regular is necessary and sufficient for the Rees
product over G 0 to be 0-simple: Take elements .e; g; f /; .e 0 ; g 0 ; f 0 / 2 E G F .
The equation
.x; y; z/.e; g; f /.x 0 ; y 0 ; z0 / D .e 0 ; g 0 ; f 0 /


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12

1 Semigroups

can be solved if and only if .z; e/ and .f; x 0 / can be chosen to be nonzero. In that
case, a solution is provided by x D e 0 ; y D . .z; e/g .f; x 0 // 1 g; z0 D f 0 ; and y 0
the identity of G.
In contrast to the case without 0, sets of the form feg G 0 B (where B F )
are not necessarily right 0-simple unless B is all of F .
In general semigroup theory, a semigroup S is regular if a 2 aSa for all a 2 S .
This notion and the regularity of the sandwich function just discussed are consistent in the following sense: The Rees product over a group with 0 is regular as a
semigroup if and only if its sandwich function is regular. To see this, consider the
equation .e; g; f / D .e; g; f /.x; y; z/.e; g; f /, which can be solved provided the
functions .f; / and . ; e/ are not identically 0.

Section 1.3 Exercises
Exercise 1.17. Let S be a semigroup with zero 0, and let S Ô 0. Prove that S is
0-simple if and only if S xS D S for every x Ô 0 of S .
Exercise 1.18. Let S be a 0-simple semigroup, and let for every x Ô 0 in S , x n Ô 0
where n D 1; 2 : : :. Show that S contains no proper divisors of zero.
Exercise 1.19. Let M be a 0-minimal ideal of a semigroup S with zero. Prove that
either M D 0 or M is a 0-simple subsemigroup of S .
Exercise 1.20. Define Jk as the set of elements of rank at most k for a transformation semigroup on a finite set X . Let S be a subsemigroup of Tn . Identify the
divisors of zero in S=Jk . Prove that Jn =Jn 1 does not contain any divisors of zero.
If S D Tn , show that Jk =Jk 1 has divisors of zero for 2 Ä k < n.

1.4 The Rees–Suschkewitsch Representation Theorem
The Rees–Suschkewitsch representation theorem comes in two versions or even four
if we take into account the topological considerations to be presented in Sect. 1.5.
In Sect. 1.4, we discuss the algebraic case only for semigroups with and without 0.
Needless to say, our strategy is dictated by applications where both versions appear
naturally.

We will begin by studying the case without 0 in some detail. As seen in Chap. 4,
this case is more prevalent in the applications. We only outline proofs of results in
the latter part of Sect. 1.4, where the semigroups with 0 are treated. However, we
point out the main pitfalls in going from one theory to the other.
Definition 1.5. An idempotent in S is said to be a primitive idempotent if it is
minimal with respect to the partial order Ä on the set E.S / of idempotents of S
defined by
e Ä f ” ef D f e D e

.e; f 2 E.S //: