Special classes of semigroups (Q2723165)

In the theory of semigroups, certain kinds of band decompositions are very useful in the study of the structure of semigroups. In the preface the author says: There is a number of special semigroup classes in which these decompositions can be used. The semigroups are decomposed into special bands of left Archimedean or right Archimedean or t-Archimedean or Archimedean semigroups. In this book the author focuses his attention on such classes of semigroups and provides a systematic review on this subject for more than 30 years up to date. This book consists of 16 chapters.NEWLINENEWLINENEWLINEWe will look over all chapters, first we review the first two or three chapters. Chapter 1 presents the basic notions, the lemmas and the theorems which are used in this book, including results of high level, some of them with proofs, some without proofs leaving them to the references. This chapter includes Archimedean semigroups, band or semilattice decompositions, orthodox bands of groups, subdirectly irreducible semigroups, \(\Delta\)-semigroups. A semigroup \(S\) is called a \(\Delta\)-semigroup if the lattice of all congruences on \(S\) forms a chain with respect to inclusion. Chapter 2. Mohan S. Putcha (1973) obtained the following theorem: A semigroup \(S\) is a semilattice of Archimedean semigroups if and only if \(S\) satisfies the condition that, for every \(a,b\in S\), \(a\in S^1bS^1\) implies \(a^n\in S^1b^2S^1\) for some positive integer \(n\). A semigroup which satisfies this condition is called a Putcha semigroup. In this chapter a stronger notion, left (right) Putcha, is introduced. A semigroup \(S\) is called a left (right) Putcha semigroup if, \(\forall x,y\in S\) NEWLINE\[NEWLINE[y\in xS^1]\Rightarrow[\exists m\in\mathbb{Z}_+,\;y^m\in x^2S^1],\qquad ([y\in S^1x]\Rightarrow[\exists m\in\mathbb{Z}_+,\;y^m\in S^1x^2]).NEWLINE\]NEWLINE A left (right) Putcha semigroup is a Putcha semigroup. The chapter includes the following: The proof of Putcha's theorem, the determination of the least semilattice congruence on a Putcha semigroup, simple left and right Putcha semigroups, the structure of an Archimedean (left and right Putcha) semigroup containing at least one idempotent element in terms of (retract) ideal extension of a (completely) simple semigroup by a nil semigroup.NEWLINENEWLINENEWLINEChapter 3. Commutative semigroups. This chapter consists of three parts. The first part includes: The semilattice decomposition of commutative semigroups, the structure of a commutative Archimedean semigroup containing at least one idempotent element, existence of a non-trivial group homomorphism of a commutative Archimedean semigroup without idempotent element, separativity and cancellation. The second part of the chapter is to determine subdirectly irreducible commutative semigroups in which the following three cases are considered: The case containing a globally idempotent core, the case containing zero and a non-trivial annihilator, the case containing a nilpotent core and a trivial annihilator. The third part of this chapter treats commutative \(\Delta\)-semigroups.NEWLINENEWLINENEWLINEReviewing over the other chapters, Chapter 4 through Chapter 16 are devoted to other special semigroup classes, furthermore, Chapter 2 and Chapter 3 play important parts as the goal and the start of the backbone of the book as we will explain later. The following is the list of the semigroup classes. The numbers are those of the chapters and the classes at the same time. If there are cases `left' or `right' then the lower suffices denoting `l' or `r' respectively.NEWLINENEWLINENEWLINE2. Putcha semigroups (\(2_l\) left Putcha semigroup, \(2_r\) right Putcha semigroup), 3. Commutative semigroups, 4. Weakly commutative semigroups (\(4_l\) left weakly commutative, \(4_r\) right weakly commutative), \(5_R\). \(R\)-commutative semigroups, \(5_L\). \(L\)-commutative semigroups, \(5_H\). \(H\)-commutative semigroups, 6. Conditionally commutative semigroups, 7. \(RC\)-commutative semigroups (\(R\)-commutative and conditionally commutative semigroups), 8. Quasi commutative semigroups (\(8_l\) left quasi commutative, \(8_r\) right quasi commutative, \(8_\sigma\) \(\sigma\)-reflexive), 9. Medial semigroups, 10. Right commutative semigroups, 11. Externally commutative semigroups, 12. \(E\)-\(m\) semigroups (\(12_{ex}\) exponential semigroups), 13. \(WE\)-\(m\) semigroups, 14. Weakly exponential semigroups, 15. \((m,n)\)-commutative semigroups, 16. \(n_{(2)}\)-permutable semigroups.NEWLINENEWLINENEWLINEFor each class of the semigroups the following topics are considered: (1) The greatest semilattice decomposition of a semigroup, in particular observation of the Archimedean components (if it exists). When does it exist? (2) The determination of (0-)simple semigroups in each class. (3) The structure of the Archimedean component containing at least one idempotent element, it is (retract) ideal extension by a nil semigroup. (4) Existence of a non-trivial group homomorphic image of the Archimedean component without idempotent element. (5) Embedding into a group or union of groups. (6) Separativity and cancellation. (7) The least congruence with some condition on each semigroup. (8) \(\Delta\)-semigroup in each class, etc.NEWLINENEWLINENEWLINEThe reviewer examines most precisely (1). For simplicity we denote each class of semigroups in the above list by \(X\)-semigroup or \(Y\)-semigroup where \(X\) or \(Y\) denotes its name, for example, `weakly commutative' or `\(WE\)-\(m\)' etc., at the same time the No. of the class, in other words, in the greatest semilattice decomposition of \(X\)-semigroup, the component is an Archimedean \(Y\)-semigroup. The following four cases are possible for \(X\). Case 1. \(Y=X\), Case 2. \(Y\neq X\), Case 3. Any component is Archimedean but no \(Y\) exists, Case 4. Not necessarily Archimedean.NEWLINENEWLINENEWLINEReviewing chapter 2 through chapter 16, \(X\) are classified as follows: Case 1: 3, 4, 9, 10, 11, \(12_{ex}\), 12, 13, 14, 15, 16. Case 2: 7, 8, Case 3: \(2_l\), \(2_r\), 2, \(5_R\), Case 4: 6. A diagram expresses the mutual inclusion relations among the classes considered.NEWLINENEWLINENEWLINEThe reviewer respects the author for his achievement collecting and organizing the massive material and references. This book is recommended to researchers in this field.