Algorithmic and Combinatorial Algebra

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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

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