Combinatorial Algebra

Syntax and Semantics

Mark V. Sapir, Victor S. Guba, Mikhail V. Volkov

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English

Springer International Publishing AG
22 September 2016

Mathematical foundations; Mathematical logic; Discrete mathematics; Groups & group theory

Series: Springer Monographs in Mathematics

"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 t

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

By: Mark V. Sapir
Contributions by: Victor S. Guba, Mikhail V. Volkov
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: Softcover reprint of the original 1st ed. 2014
Dimensions: Height: 235mm, Width: 155mm, Spine: 20mm
Weight: 5.621kg
ISBN: 9783319375908
ISBN 10: 3319375903
Series: Springer Monographs in Mathematics
Pages: 355
Publication Date: 22 September 2016
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Introduction.- 1. Main definitions and basic fact.- 2. Words that can be avoided.- 3. Semigroups.- 4. Rings.- 5. Groups.- Bibliography.- Index.

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

Reviews for Combinatorial Algebra: Syntax and Semantics

“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)