A SOURCE BOOK IN MATHEMATICS
CONTENTS CHAPTER 1 ALGEBRA AT THE BEGINNING OF THE TWENTIETH CENTURY 1 1 A CHANGING DISCIPLINE 1 2 THE TERM ‘SEMIGROUP’ 1 3 AN OVERVIEW OF THE DEVELOPMENT OF SEMIGROUP THEORY CHAPTER 2 COMMUNICATION BETWEEN EAST AND WEST 2 1 COMMUNICATION DOWN THROUGH THE DECADES 2 2 ACCESS TO PUBLICATIONS CHAPTER 3 ANTON KAZIMIROVICH SUSHKEVICH 3 1 BIOGRAPHY 3 2 THE THEORY OF OPERATIONS AS THE GENERAL THEORY OF GROUPS 3 3 GENERALISED GROUPS 3 4 SUSHKEVICH’S IMPACT CHAPTER 4 UNIQUE FACTORISATION IN SEMIGROUPS 4 1 POSTULATIONAL ANALYSIS 4 2 E T BELL AND THE ARITHMETISATION OF ALGEBRA 4 3 MORGAN WARD AND THE FOUNDATIONS OF GENERAL ARITHMETIC 4 4 ALFRED H CLI?ORD 4 5 ARITHMETIC OF OVA 4 6 SUBSEQUENT DEVELOPMENTS CHAPTER 5 EMBEDDING SEMIGROUPS IN GROUPS 5 1 THE THEOREMS OF STEINITZ AND ORE 5 2 EMBEDDING ACCORDING TO SUSHKEVICH 5 3 FURTHER SU?CIENT CONDITIONS 5 4 MALTSEV’S IMMERSIBILITY CONDITIONS 5 5 OTHER EMBEDDING PROBLEMS CHAPTER 6 THE REES THEOREM 6 1 COMPLETELY (0-)SIMPLE SEMIGROUPS 6 2 BRANDT GROUPOIDS 6 3 SUSHKEVICH’S ‘KERNGRUPPEN’ 6 4 CLI?ORD’S ‘MULTIPLE GROUPS’ 6 5 THE REES THEOREM 6 6 UNIONS OF GROUPS AND SEMIGROUPS CONTENTS CHAPTER 7 THE FRENCH SCHOOL OF ‘DEMI-GROUPES’ 7 1 PAUL DUBREIL AND MARIE-LOUISE DUBREIL-JACOTIN 7 2 EQUIVALENCE RELATIONS 7 3 PRINCIPAL EQUIVALENCES AND RELATED CONCEPTS 7 4 SUBSEQUENT WORK CHAPTER 8 THE EXPANSION OF THE THEORY IN THE 1940S AND 1950S 8 1 THE GROWTH OF NATIONAL SCHOOLS 8 2 THE SLOVAK SCHOOL 8 3 THE AMERICAN SCHOOL 8 4 THE JAPANESE SCHOOL 8 5 THE HUNGARIAN SCHOOL 8 6 BRITISH AUTHORS CHAPTER 9 THE POST-SUSHKEVICH SOVIET SCHOOL 9 1 EVGENII SERGEEVICH LYAPIN 9 2 LYAPIN’S MATHEMATICAL WORK 9 3 LAZAR MATVEEVICH GLUSKIN 9 4 GLUSKIN’S MATHEMATICAL WORK 9 5 OTHER AUTHORS CHAPTER 10 THE DEVELOPMENT OF INVERSE SEMIGROUPS 10 1 A LITTLE THEORY 10 2 PSEUDOGROUPS AND CONCEPTUAL DI?CULTIES 10 3 VIKTOR VLADIMIROVICH WAGNER 10 4 WAGNER AND GENERALISED GROUPS 10 5 GORDON B PRESTON 10 6 PRESTON AND INVERSE SEMIGROUPS CHAPTER 11 MATRIX REPRESENTATIONS OF SEMIGROUPS 11 1 SUSHKEVICH ON MATRIX SEMIGROUPS 11 2 CLI?ORD ON MATRIX SEMIGROUPS 11 3 W DOUGLAS MUNN 11 4 THE WORK OF W D MUNN 11 5 THE WORK OF J S PONIZOVSKII CHAPTER 12 BOOKS, SEMINARS, CONFERENCES, AND JOURNALS 12 1 MONOGRAPHS 12 2 SEMINARS ON SEMIGROUPS 12 3 CZECHOSLOVAKIA, 1968, AND SEMIGROUP FORUM CHAPTER ALGEBRA AT THE BEGINNING OF THE TWENTIETH CENTURY SINCE THE THEORY OF SEMIGROUPS IS CLEARLY A BRANCH OF ABSTRACT ALGEBRA, IT IS AP- PROPRIATE TO SET THE SCENE FOR LATER CHAPTERS BY BEGINNING THIS BOOK WITH A VERY SHORT ACCOUNT OF THE DEVELOPMENT OF ABSTRACT ALGEBRA UP TO THE START OF THE TWENTIETH CEN- TURY IN PARTICULAR, I DISCUSS ALGEBRA’S TRANSITION, OVER THE COURSE OF THE NINETEENTH CENTURY, FROM A SUBJECT CONCERNED ENTIRELY WITH THE SOLUTION OF POLYNOMIAL EQUA- TIONS TO A DISCIPLINE THAT DEALS WITH GENERAL ‘STRUCTURES’ WITHIN MATHEMATICS, WHERE ‘STRUCTURE’ MAY BE UNDERSTOOD LOOSELY TO MEAN A SET TOGETHER WITH ONE OR MORE OPERATIONS WHICH ARE SUBJECT TO CERTAIN CONDITIONS THIS TRANSITION IS THE SUBJECT OF SECTION 1 1 MY ACCOUNT OF ALGEBRA’S PROGRESSION IS NOT OF COURSE A COMPREHENSIVE ONE — I REFER THE INTERESTED READER INSTEAD TO CORRY (1996) THE CONSOLIDATION OF GROUP THEORY AT THE BEGINNING OF THE TWENTIETH CENTURY, TOGETHER WITH SOME POSTULATIONAL CONSIDERATIONS, LED, IN 1904, TO THE COINING OF THE FRENCH TERM ‘SEMIGROUPE’, WHICH WAS TAKEN OVER INTO ENGLISH NATURALLY, AND ALMOST IMMEDIATELY, AS ‘SEMIGROUP’ THE ORIGINAL NOTION OF A ‘SEMIGROUP’, HOWEVER, DID NOT CORRESPOND TO THE MODERN DE?NITION MOREOVER, THE TERM WAS COINED IN THIS IS OF COURSE AKIN TO THE WAY IN WHICH THE TERM ‘ALGEBRA’ IS USED IN SCHOOLS TO THIS DAY, ALTHOUGH IT WAS A LONG TIME BEFORE SUCH ‘ALGEBRA’ CAME TO BE WRITTEN SYMBOLICALLY (SEE, FOR EXAMPLE, KATZ 2009, §12 4)