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 Olshanskiis solutions of Burnside problems, Shirshovs solution of Kuroshs problem for PI rings, Belovs solution of Spechts problem for varieties of rings, Grigorchuks solution of Milnors problem, Bass-Guivarch theorem about growth of nilpotent groups, Kleimans solution of Hanna Neumanns problem for varieties of groups, Adians solution of von Neumann-Days problem, Trahtmans 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 t

he classroom setting: researchers, instructors, students, virtually anyone who wishes to learn and better understand this important area of mathematics.