Combinatorial Algebra: Syntax and Semantics

Combinatorial Algebra: Syntax and Semantics provides comprehensive account of many areas of combinatorial algebra. It contains self-contained proofs of  more than 20 fundamental results, both classical and modern. This includes Golod–Shafarevich and Olshanskii's solutions of Burnside problems, Shirshov's solution of Kurosh's problem for PI rings, Belov's solution of Specht's problem for varieties of rings, Grigorchuk's solution of Milnor's problem, Bass–Guivarc'h theorem about growth of nilpotent groups, Kleiman's solution of Hanna Neumann's problem for varieties of groups, Adian's solution of von Neumann-Day's problem, Trahtman's solution of the road coloring problem of Adler, Goodwyn and Weiss. The book emphasize several ``universal" tools, such as trees, subshifts, uniformly recurrent words, diagrams and automata. With over 350 exercises at various levels of difficulty and with hints for the more difficult problems, this book can be used as a textbook, and aims to reach a wide and diversified audience.  No prerequisites beyond standard courses in linear and abstract algebra are required. The broad appeal of this textbook extends to a variety of student levels: from advanced high-schoolers to undergraduates and graduate students, including those in search of a Ph.D. thesis who will benefit from the  “Further reading and open problems” sections at the end of Chapters 2 –5. The book can also be used for self-study, engaging those beyond the classroom setting: researchers, instructors, students, virtually anyone who wishes to learn and better understand this important area of mathematics.

“This book concentrates on the following three topics: the Burnside problem, the finite basis property and growth of algebras. … this book is a very valuable addition to the mathematical literature. … shows that there are ‘universal’ methods which can be used to investigate many algebraic problems from different parts of algebra … . it should also be noted that the book presents self-contained proofs of important recent results which were available only in research articles till now.” (Konrad P. Pióro, Mathematical Reviews, July, 2015)“The book explores connections and interactions between ‘semantics’ (that is, ‘objects’ like graphs, topological spaces, etc.) and ‘syntactics’ (i.e. the means to speak about objects like words, laws, etc.) which occur in the analysis of algebraic structures. … The book … is addressed to active readers of a certain level of mathematical maturity. Included are more than 350 exercises and a collection of open problems.” (K. Auinger, Monatshefte für Mathematik, 2015)“The book should be viewed as an encyclopedia of algebraic facts which have strong combinatorial flavor both in their statement and in proofs. … The book should be of interest to different audiences who want to be introduced to the fascinating world of combinatorial algebra, ranging from undergraduate students to senior scientists.” (Vadim Gorin, zbMATH 1319.05001, 2015)

Mark V. Sapir is a Centennial Professor of Mathematics at Vanderbilt University.