# Algorithmic and Combinatorial Algebra

Springer Science & Business Media, Dec 6, 2012 - Mathematics - 384 pages
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).

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### Contents

 Composition Method for Associative Algebras 1 Free Lie Algebras 53 Amalgamated Products of Lie Algebras 141 The Problem of Endomorph Reducibility and Relatively Free Groups 177 The Constructive Method in the Theory of HNNextensions Groups 209 The Constructive Method for HNNextensions and the Conjugacy 237 Calculations in Free Groups 315 Algorithmic Properties of the Wreath Products of Groups 325
 A3 3 Constructing a Free Algebra 337 Properties of Subalgebras of a Free Algebra 338 The Word Problem for 8algebras A3 7 Properties of Ideals of a Free Algebra 343 Elementary Theories of Free Algebras 346 Endomorphisms and Automorphisms of Free Algebras 347 Bibliography 349 Index 380 Copyright

 Survey of the Theory of Absolutely Free Algebras 335