Synopses & Reviews
This major volume presents an authoritative overview of developments in algorithmic and combinatorial algebra which have been achieved during the last forty years or so, with an emphasis on the results obtained by the Novosibirsk school of A.I. Mal'tchev and A.I. Shirshov and followers. The book has nine chapters. These deal with Applications of the Composition (or Diamond) Lemma to associative and Lie algebras (Chapters 1 and 3), to subalgebras of free Lie algebras and free products of Lie algebras (Chapters 2 and 4), to word problems and embedding theorems in varieties of Lie algebras and groups (Chapters 5--7) and to the constructive theory of HNN-extensions and its use in analysing the word and conjugacy problems in the Novikov--Boone groups (Chapters 8 and 9). Many results described here appear for the first time in a monograph. The volume concludes with a discussion of three applications. For graduate students and researchers whose work involves algorithmic and combinatorial algebra and its applications.
Synopsis
Even three decades ago, the words 'combinatorial algebra' contrasting, for in stance, the words 'combinatorial topology, ' were not a common designation for some branch of mathematics. The collocation 'combinatorial group theory' seems to ap pear first as the title of the book by A. Karras, W. Magnus, and D. Solitar 182] and, later on, it served as the title of the book by R. C. Lyndon and P. Schupp 247]. Nowadays, specialists do not question the existence of 'combinatorial algebra' as a special algebraic activity. The activity is distinguished not only by its objects of research (that are effectively given to some extent) but also by its methods (ef fective to some extent). To be more exact, we could approximately define the term 'combinatorial algebra' for the purposes of this book, as follows: So we call a part of algebra dealing with groups, semi groups, associative algebras, Lie algebras, and other algebraic systems which are given by generators and defining relations {in the first and particular place, free groups, semigroups, algebras, etc. )j a part in which we study universal constructions, viz. free products, lINN-extensions, etc. j and, finally, a part where specific methods such as the Composition Method (in other words, the Diamond Lemma, see 49]) are applied. Surely, the above explanation is far from covering the full scope of the term (compare the prefaces to the books mentioned above)."
Table of Contents
1. Composition Method for Associative Algebras. 2. Free Lie Algebras. 3. The Composition Method in the Theory of Lie Algebras. 4. Amalgamated Products of Lie Algebras. 5. Decision Problems and Embedding Theorems in the Theory of Varieties of Lie Algebras. 6. The Word Problem and Embedding Theorems in the Theory of the Varieties of Groups. 7. The Problem of Endomorph Reducibility and Relatively Free Groups with the Word Problem Undecidable. 8. The Constructive Method in the Theory of HNN-Extensions. Groups with Standard Normal Form. 9. The Constructive Method for HNN-Extensions and the Conjugacy Problem for Novikov-Boone Groups. A1: Calculations in Free Groups. A2: Algorithmic Properties of Wreath Products of Groups. A3: Survey of the Theory of Absolutely Free Algebras. Bibliography. Index.