Combinatorial Algebra: Syntax and Semantics

Combinatorial Algebra: Syntax and Semantics explores the foundations of the subject and many important applications. With over 350 exercises at various levels of difficulty and with hints for the more difficult problems, this textbook aims to reach a wide and diversified audience; no prerequisites beyond standard courses in linear and abstract algebra are required, although a certain mathematical maturity (the ability to understand and produce complicated proofs) will help the reader.§The five chapters cover classical results that formed the core of modern combinatorial algebra, including Burnside-type problems, growth and identities of groups, semigroups and associative algebras as well as relatively recent breakthroughs, including the road coloring problem. In this self-contained exposition, the author delves into the deep connections between syntactic and semantic methods in combinatorial algebra. Many proofs emphasize several ``universal" tools, such as trees, sub shifts, uniformly recurrent words, and so on.§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. Additionally, this text opens a path to 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.§