##
**The algebraic theory of semigroups. Vol. I.**
*(English)*
Zbl 0111.03403

Mathematical Surveys. 7. Providence, R.I.: American Mathematical Society (AMS). xv, 224 p. (1961).

Although we have a book “Semigroups” written in Russian by E. S. Lyapin (Moscow: FizMatGiz) (1960; Zbl 0100.02301), the book by Clifford and Preston is the first one on semigroups written in English. Since about 1950, the theory of semigroups has been vigorously studied by many mathematicians. Sometimes, however, different persons have discussed the same fundamental results accidentally and also the same concepts are expressed in many different terminologies, so that the beginners, who are interested in this field may not know how to begin a study. At this time the publication of this book systematizes the theory of semigroups to some extent and provides a guiding light for interested mathematicians. This book is the first of two volumes; the first volume consists of five chapters.

Chapter 1. Elementary concepts.

The most important fundamental concepts which are introduced in this chapter are relations or congruences (§1.4,§1.5), translations (§1.3), and lattices (§1.8). By a right translation \(\rho\) we mean a mapping \(\rho\) of a semigroup \(S\) into itself such that \((ab)\rho=a(b\rho)\) for every \(a,b\in S\); a left translation \(\lambda\) is defined to be a mapping \(\lambda\) satisfying \((ab)\lambda=(a\lambda)b\). Translations play an important role in then theory of extensions, for example, the translational hull (§1.3). In §1.4, it is interesting to define a composition on relations; and the authors discuss in §1.4 how an equivalence is generated by any relation and in §1.5 how a congruence is generated by any relation. Further, in §1.5, they give a few fundamental theorems on homomorphisms and congruences, especially the following important principle due to Kimura and the reviewer: there is the maximal homomorphic image of a groupoid \(G\) of a given type. The reviewer thinks that this principle can be stated more briefly or nicely. The concept of congruences is very important in the theory of semigroups, so that a more precise discussion might be expected in Vol. 2.

Semilattices are valuable in the decomposition of any semigroup. It is natural to consider the relationship between groups and semigroups, that is, there are two: groups included in a semigroup, semigroups included in a group. As to the former, the authors discuss existence of maximal subgroups in a semigroup (§1.7) and, as to the latter, embedding of semigroups in a group (§1.10). In §1.10 Ore’s theorem (1931) that a right reversible cancellative semigroup can be embedded in a group is proved in Rees’ way (1948), and furthermore it is shown that, according to Dubreil, right reversibility is a necessary and sufficient condition in order that a cancellative semigroup \(S\) can be embedded in a group of left quotients of \(S\).

As semigroups which resemble groups, inverse semigroups (§1.9) and right groups (§1.11) are discussed.

Chapter 2. Ideals and Related Concepts.

Five kinds of equivalences are defined on any semigroup \(S\) as follows (§2.1): Let \(a,b\in S\).

\[ aLb \quad\text{iff}\quad a\cup Sa=b\cup Sb;\qquad aRb\quad\text{iff} \quad a\cup aS=b\cup bS; \]

\[ aDb\quad\text{iff}\quad aLb \text{ or }aRb;\qquad aHb\quad\text{iff} \quad aLb\text{ and }aRb; \]

\[ aJb\quad\text{iff}\quad S^1aS^1=S^1bS^1 \] where \(S^1\) is \(S\) with adjoint identity.

These were first introduced by Green (1951) and are used in the study of semigroups throughout Chapters 2 and 3. It can be said that these equivalences make it easy to resystematize the theory since Rees.

The Schützenberger group (§2.4) on an \(H\)-class is in close connection with the Schützenberger representation.

In §2.5, the fundamental properties of 0-simple (left 0-, right 0-simple) semigroups and the structure of 0-minimal ideals are nicely developed by the results given by Munn and Schwarz. For example, a 0-minimal ideal \(M\) is either a 0-simple subsemigroup or a zero-semigroup; and if a 0-minimal ideal \(M\) of \(S\) contains at least one 0-minimal left ideal of \(S\), then \(M\) is the set union of all 0-minimal left ideals of \(S\) contained in \(M\).

In §2.6, principal series, factors, relative ideal series, composition series with respect to ideals are defined so that Jordan-Hölder-Schreier’s theorem holds. According to Munn, there is no distinction between a principal series and a composition series if \(S\) is semisimple. In §2.7, the authors study 0-minimal left ideals of a completely 0-simple semigroup and the structure of \(D\)-classes of \(S\) based on Clifford, Green and Munn’s results.

Chapter 3. Representation by matrices over a group with zero.

In this chapter, the authors treat the two representation theorems: one by Rees, the other by Schützenberger. The Rees matrix semigroup and the regular Rees matrix semigroup over a group with 0 are defined in §3.1, in §3.2, the following Rees’ theorem is proved: A semigroup\(S\) is completely 0-simple if and only if \(S\) is isomorphic to a regular Rees matrix semigroup over a group with zero. Since this theorem is proved by the method of Miller and Clifford, that is, it is argued from the general standpoint of \(H\)-classes in \(D\)-classes, Rees’ original complicated proof is improved and simplified. Rees’ representation treats only completely 0-simple semigroups, while Schützenberger’s representation is concerned with any semigroups.

In §3.5, the authors introduce the Schützenberger representation, i.e. a representation of any semigroup by matrices over the Schützenberger group on an \(H\)-class. Also the direct sum of all the Schützenberger representations of \(S\) or of all the dual Schützenberger representations of \(S\) are defined and a necessary and sufficient condition for these direct sums to be faithful (§3.6). The results in this chapter were given by Schützenberger and were improved by Preston.

Chapter 4. Decomposition and Extensions.

By a decomposition of a semigroup we mean classifying elements of a semigroup into the set union of disjoint subsemigroups. In this chapter, necessary and sufficient conditions on a semigroup \(S\) in order that the following conditions are satisfied are given: (1) \(S\) is the set union of disjoint left simple subsemigroups, (2) \(S\) is the set union of disjoint groups, (3) \(S\) is the set union of disjoint simple subsemigroups.

In §4.1, Croisot’s result is introduced, that is, the required conditions are expressed by the notions of semiprimeness of ideals and of left (right) regularity of \(S\). In §4.2, the condition for \(S\) to be a union of groups is given by Clifford such that the structure of such \(S\) is clarified. In §4.3, a decomposition of any commutative semigroup \(S\) into a semilattice is discussed. First the following theorem is stated: a commutative semigroup \(S\) is a semilattice of disjoint archimedean semigroups and this gives us the greatest semilattice decomposition of \(S\). This was given by Kimura and the reviewer. Next, the notion of separativity of \(S\) is defined by the condition that \(ab=a^2=b^2\) implies \(a=b\), and it is shown that the above mentioned decomposition of \(S\) is not greater than the greatest separative decomposition of \(S\). This was given by Hewitt and Zuckerman.

in §4.4 the extension theory is treated. Let \(S\) be a semigroup and \(T\) be a semigroup with zero. By an ideal extension \(\Sigma\) of \(S\) by \(T\) we mean a semigroup \(\Sigma\) containing \(S\) as an ideal such that the Rees quotient semigroup \(\Sigma/S\) is isomorphic with \(T\). The problem of finding \(\Sigma\) for given \(S\) and \(T\) is not solved in general. In this chapter only the special cases are treated: the case where \(S\) is weakly reductive is given by Clifford, and the case where \(S\) is a group and \(T\) is a completely 0-simple semigroup is given by Munn. In this book, a semilattice decomposition of a non-commutative semigroup is not discussed. The reviewer mentions just one important thing: a semigroup is a semilattice of semilattice-indecomposable semigroups and this gives us the greatest semilattice decomposition.

Chapter 5. Representation by Matrices over a Field.

In this chapter, analogous to group theory, a semigroup is represented by finite matrices over a field. In §5.2, Munn and Ponizovsky give us a necessary and sufficient condition for a matrix algebra over a finite semigroup to be semisimple. Now let \(S\) be a (not necessarily finite) semigroup and suppose that \(S\) satisfies minimal conditions with respect to principal ideals. In §5.3, it is shown that all irreducible representations of \(S\) are determined by principal irreducible representations of \(S\). This was given by Hewitt and Zuckerman and is proved by the method of Munn. In §5.4, Suschkewitch and Clifford’s result is given: irreducible representations of a completely 0-simple semigroup are constructed by representations of the ground group; consequently, it follows if \(S\) satisfies minimal conditions with respect to right principal ideals and left principal ideals, then irreducible representations of \(S\) are determined by those subsemigroups of \(S\).

Finally in §5.5, the theory of characters of commutative semigroups is stated. This nice theory was obtained by Hewitt and Zuckerman and independently by Schwarz. By a semicharacter \(\chi\) we mean a homomorphism of a commutative semigroup \(S\) with identity into the field of complex numbers and it is called a character if it satisfies \(|\chi(a)|=0\) or \(1\) for all \(a\in S\). All characters form a commutative semigroup \(S^*\) if a binary operation \(\chi\psi\) is defined by \((\chi\psi)(a)=\chi(a)\psi(a)\). The following things are proved: Separativity of \(S\) is equivalent to that of \(S^*\). Characters of a commutative semigroup \(S\) with identity are the extensions of characters of subsemigroups if the maximal semilattice homomorphic image satisfies minimal conditions.

Thus the explanation of the main outline has been finished.

The reviewer believes that the main part of the book consists of Chapter 2 and Chapters 3, 5, in which a beautiful mathematical procession is developed. The reviewer feels that some materials on semilattices and ordered semigroups should have been added, but he is also aware of the inevitable circumstances mentioned in the preface. The reviewer knows what a great task it is to arrange a lot of papers and to systematize this undeveloped field. In fact the authors refer to many of the main papers up to 1958, and have incorporated many of the results not only in the book but also in the exercises. It goes without saying that Clifford and Preston have devoted much to this book; the characteristics of the English School are quite evident. Munn’s valuable contributions are accented. The foursome, Rees-Green-Preston-Munn, is representative of England.

The authors promise that Vol. 2 will include materials of the active French School. Also the reviewer has heard that books on semigroups by Dubreil and Croisot for France and by Schwarz for Czechoslovakia may be published soon. An English translation of Lyapin’s book is being published by the American Mathematical Society. It will be interesting and valuable to have books each of which has a different characteristic and color.

At any rate this book is very nice and indispensable for graduate students and mathematicians interested in this field. Finally the reviewer wishes to thank the authors for their great task.

Chapter 1. Elementary concepts.

The most important fundamental concepts which are introduced in this chapter are relations or congruences (§1.4,§1.5), translations (§1.3), and lattices (§1.8). By a right translation \(\rho\) we mean a mapping \(\rho\) of a semigroup \(S\) into itself such that \((ab)\rho=a(b\rho)\) for every \(a,b\in S\); a left translation \(\lambda\) is defined to be a mapping \(\lambda\) satisfying \((ab)\lambda=(a\lambda)b\). Translations play an important role in then theory of extensions, for example, the translational hull (§1.3). In §1.4, it is interesting to define a composition on relations; and the authors discuss in §1.4 how an equivalence is generated by any relation and in §1.5 how a congruence is generated by any relation. Further, in §1.5, they give a few fundamental theorems on homomorphisms and congruences, especially the following important principle due to Kimura and the reviewer: there is the maximal homomorphic image of a groupoid \(G\) of a given type. The reviewer thinks that this principle can be stated more briefly or nicely. The concept of congruences is very important in the theory of semigroups, so that a more precise discussion might be expected in Vol. 2.

Semilattices are valuable in the decomposition of any semigroup. It is natural to consider the relationship between groups and semigroups, that is, there are two: groups included in a semigroup, semigroups included in a group. As to the former, the authors discuss existence of maximal subgroups in a semigroup (§1.7) and, as to the latter, embedding of semigroups in a group (§1.10). In §1.10 Ore’s theorem (1931) that a right reversible cancellative semigroup can be embedded in a group is proved in Rees’ way (1948), and furthermore it is shown that, according to Dubreil, right reversibility is a necessary and sufficient condition in order that a cancellative semigroup \(S\) can be embedded in a group of left quotients of \(S\).

As semigroups which resemble groups, inverse semigroups (§1.9) and right groups (§1.11) are discussed.

Chapter 2. Ideals and Related Concepts.

Five kinds of equivalences are defined on any semigroup \(S\) as follows (§2.1): Let \(a,b\in S\).

\[ aLb \quad\text{iff}\quad a\cup Sa=b\cup Sb;\qquad aRb\quad\text{iff} \quad a\cup aS=b\cup bS; \]

\[ aDb\quad\text{iff}\quad aLb \text{ or }aRb;\qquad aHb\quad\text{iff} \quad aLb\text{ and }aRb; \]

\[ aJb\quad\text{iff}\quad S^1aS^1=S^1bS^1 \] where \(S^1\) is \(S\) with adjoint identity.

These were first introduced by Green (1951) and are used in the study of semigroups throughout Chapters 2 and 3. It can be said that these equivalences make it easy to resystematize the theory since Rees.

The Schützenberger group (§2.4) on an \(H\)-class is in close connection with the Schützenberger representation.

In §2.5, the fundamental properties of 0-simple (left 0-, right 0-simple) semigroups and the structure of 0-minimal ideals are nicely developed by the results given by Munn and Schwarz. For example, a 0-minimal ideal \(M\) is either a 0-simple subsemigroup or a zero-semigroup; and if a 0-minimal ideal \(M\) of \(S\) contains at least one 0-minimal left ideal of \(S\), then \(M\) is the set union of all 0-minimal left ideals of \(S\) contained in \(M\).

In §2.6, principal series, factors, relative ideal series, composition series with respect to ideals are defined so that Jordan-Hölder-Schreier’s theorem holds. According to Munn, there is no distinction between a principal series and a composition series if \(S\) is semisimple. In §2.7, the authors study 0-minimal left ideals of a completely 0-simple semigroup and the structure of \(D\)-classes of \(S\) based on Clifford, Green and Munn’s results.

Chapter 3. Representation by matrices over a group with zero.

In this chapter, the authors treat the two representation theorems: one by Rees, the other by Schützenberger. The Rees matrix semigroup and the regular Rees matrix semigroup over a group with 0 are defined in §3.1, in §3.2, the following Rees’ theorem is proved: A semigroup\(S\) is completely 0-simple if and only if \(S\) is isomorphic to a regular Rees matrix semigroup over a group with zero. Since this theorem is proved by the method of Miller and Clifford, that is, it is argued from the general standpoint of \(H\)-classes in \(D\)-classes, Rees’ original complicated proof is improved and simplified. Rees’ representation treats only completely 0-simple semigroups, while Schützenberger’s representation is concerned with any semigroups.

In §3.5, the authors introduce the Schützenberger representation, i.e. a representation of any semigroup by matrices over the Schützenberger group on an \(H\)-class. Also the direct sum of all the Schützenberger representations of \(S\) or of all the dual Schützenberger representations of \(S\) are defined and a necessary and sufficient condition for these direct sums to be faithful (§3.6). The results in this chapter were given by Schützenberger and were improved by Preston.

Chapter 4. Decomposition and Extensions.

By a decomposition of a semigroup we mean classifying elements of a semigroup into the set union of disjoint subsemigroups. In this chapter, necessary and sufficient conditions on a semigroup \(S\) in order that the following conditions are satisfied are given: (1) \(S\) is the set union of disjoint left simple subsemigroups, (2) \(S\) is the set union of disjoint groups, (3) \(S\) is the set union of disjoint simple subsemigroups.

In §4.1, Croisot’s result is introduced, that is, the required conditions are expressed by the notions of semiprimeness of ideals and of left (right) regularity of \(S\). In §4.2, the condition for \(S\) to be a union of groups is given by Clifford such that the structure of such \(S\) is clarified. In §4.3, a decomposition of any commutative semigroup \(S\) into a semilattice is discussed. First the following theorem is stated: a commutative semigroup \(S\) is a semilattice of disjoint archimedean semigroups and this gives us the greatest semilattice decomposition of \(S\). This was given by Kimura and the reviewer. Next, the notion of separativity of \(S\) is defined by the condition that \(ab=a^2=b^2\) implies \(a=b\), and it is shown that the above mentioned decomposition of \(S\) is not greater than the greatest separative decomposition of \(S\). This was given by Hewitt and Zuckerman.

in §4.4 the extension theory is treated. Let \(S\) be a semigroup and \(T\) be a semigroup with zero. By an ideal extension \(\Sigma\) of \(S\) by \(T\) we mean a semigroup \(\Sigma\) containing \(S\) as an ideal such that the Rees quotient semigroup \(\Sigma/S\) is isomorphic with \(T\). The problem of finding \(\Sigma\) for given \(S\) and \(T\) is not solved in general. In this chapter only the special cases are treated: the case where \(S\) is weakly reductive is given by Clifford, and the case where \(S\) is a group and \(T\) is a completely 0-simple semigroup is given by Munn. In this book, a semilattice decomposition of a non-commutative semigroup is not discussed. The reviewer mentions just one important thing: a semigroup is a semilattice of semilattice-indecomposable semigroups and this gives us the greatest semilattice decomposition.

Chapter 5. Representation by Matrices over a Field.

In this chapter, analogous to group theory, a semigroup is represented by finite matrices over a field. In §5.2, Munn and Ponizovsky give us a necessary and sufficient condition for a matrix algebra over a finite semigroup to be semisimple. Now let \(S\) be a (not necessarily finite) semigroup and suppose that \(S\) satisfies minimal conditions with respect to principal ideals. In §5.3, it is shown that all irreducible representations of \(S\) are determined by principal irreducible representations of \(S\). This was given by Hewitt and Zuckerman and is proved by the method of Munn. In §5.4, Suschkewitch and Clifford’s result is given: irreducible representations of a completely 0-simple semigroup are constructed by representations of the ground group; consequently, it follows if \(S\) satisfies minimal conditions with respect to right principal ideals and left principal ideals, then irreducible representations of \(S\) are determined by those subsemigroups of \(S\).

Finally in §5.5, the theory of characters of commutative semigroups is stated. This nice theory was obtained by Hewitt and Zuckerman and independently by Schwarz. By a semicharacter \(\chi\) we mean a homomorphism of a commutative semigroup \(S\) with identity into the field of complex numbers and it is called a character if it satisfies \(|\chi(a)|=0\) or \(1\) for all \(a\in S\). All characters form a commutative semigroup \(S^*\) if a binary operation \(\chi\psi\) is defined by \((\chi\psi)(a)=\chi(a)\psi(a)\). The following things are proved: Separativity of \(S\) is equivalent to that of \(S^*\). Characters of a commutative semigroup \(S\) with identity are the extensions of characters of subsemigroups if the maximal semilattice homomorphic image satisfies minimal conditions.

Thus the explanation of the main outline has been finished.

The reviewer believes that the main part of the book consists of Chapter 2 and Chapters 3, 5, in which a beautiful mathematical procession is developed. The reviewer feels that some materials on semilattices and ordered semigroups should have been added, but he is also aware of the inevitable circumstances mentioned in the preface. The reviewer knows what a great task it is to arrange a lot of papers and to systematize this undeveloped field. In fact the authors refer to many of the main papers up to 1958, and have incorporated many of the results not only in the book but also in the exercises. It goes without saying that Clifford and Preston have devoted much to this book; the characteristics of the English School are quite evident. Munn’s valuable contributions are accented. The foursome, Rees-Green-Preston-Munn, is representative of England.

The authors promise that Vol. 2 will include materials of the active French School. Also the reviewer has heard that books on semigroups by Dubreil and Croisot for France and by Schwarz for Czechoslovakia may be published soon. An English translation of Lyapin’s book is being published by the American Mathematical Society. It will be interesting and valuable to have books each of which has a different characteristic and color.

At any rate this book is very nice and indispensable for graduate students and mathematicians interested in this field. Finally the reviewer wishes to thank the authors for their great task.

Reviewer: Takayuki Tamura

### MSC:

20Mxx | Semigroups |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |