Algorithmic and Combinatorial Algebra: (255 Mathematics and Its Applications)

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.- 1.1 Introduction.- 1.2 Free Semigroups and Free Groups.- 1.3 The Composition Lemma.- 1.4 The Composition Lemma for Semigroup Algebras.- 1.5 The Generalised Clifford Algebra and Some Other Examples.- 1.6 Finite-Dimensional Representation of a Generalised Clifford Algebra.- 1.7 More on Embeddings into Simple Algebras.- 2 Free Lie Algebras.- 2.1 Introduction.- 2.2 The Definition of Free Lie Algebras.- 2.3 Projective Algebras.- 2.4 Elementary Transformations and Automorphisms of Free Algebras.- 2.5 Lie Algebra Derivations.- 2.6 The Ideal of Codimension 1.- 2.7 Constructing Generators for an Arbitrary Subalgebra in a Free Lie Algebra.- 2.8 The Shirshov Theorem on Free Lie Algebra Subalgebras.- 2.9 Automorphisms of Free Lie Algebras of Finite Rank.- 2.10 A Criterion for a Lie Algebra to be Free.- 2.11 Bases of a Free Lie Algebra.- 2.12 Construction of Free Lie Algebras.- 2.13 Universal Enveloping Associative Algebra.- 2.14 On Subrings of Free Rings with Operators.- 2.15 Embedding Lie Rings into Associative Rings with Operators.- 2.16 Restricted Lie Algebras.- 2.17 Relatively Free Lie Algebras.- 2.18 Embedding Countable-Dimensional Lie Algebras into Lie Algebras with Two Generators.- 2.19 The Residual Finiteness of Associative and Lie Algebras.- 2.20 Residual Finiteness of Free Rings and Algebras.- 3 The Composition Method in the Theory of Lie Algebras.- 3.1 Introduction.- 3.2 The Composition Lemma.- 3.3 Formulation of Decision Problems. One-Relator Lie Algebras.- 3.4 Embedding Lie Algebras into Simple Lie Algebras.- 3.5 The Main Algorithmic Problems for Lie Algebras are Unsolvable.- 3.6 Unrecognizable Markov Properties for Finitely Presented Lie Algebras.- 3.7 Defining Relations of a Subalgebra.- 3.8 Residual Finiteness and Decision Problems.- 3.9 On Residual Finiteness of One-Relator Lie Algebras.- 3.10 Constructing Free Resolutions.- 3.11 Cohomological Dimension of Almost Free Lie Algebras.- 4 Amalgamated Products of Lie Algebras.- 4.1 Introduction.- 4.2 Definition of Amalgamated Products.- 4.3 Constructing the Free Product of Associative Algebras without 1.- 4.4 Constructing the Amalgamated Product of Lie Algebras.- 4.5 Subalgebras of the Free Product of Lie Algebras.- 4.6 Generators of a Subalgebra of the Free Product of Lie Algebras.- 4.7 Decomposition of a Free Product into the Sum of Two Subalgebras, One Free.- 4.8 Decomposition of a Subalgebra of a Free Product into the Sum of Two Subalgebras, One Free.- 4.9 The Theorem on Subalgebras of an Amalgamated Product of Lie Algebras.- 4.10 Free Subalgebras in a Free Product of Lie Algebras.- 4.11 The Case in Which the Kurosh Formula Almost Holds.- 4.12 Supplementary Facts on Free Products.- 4.13 Residual Finiteness of Free Products of Associative and Lie Algebras.- 4.14 Residual Finiteness of Free Soluble Lie Algebras with respect to Inclusion into Finitely Generated Subalgebras.- 4.15 On Residual Finiteness of Free Soluble Groups with respect to Inclusion into Finitely Generated Subgroups.- 4.16 On Residual Properties of Free Products of Lie Algebras. Central Systems in Free Products.- 5 The Problem of Endomorph Reducibility and Relatively Free Groups with the Word Problem Unsolvable.- 5.1 Introduction.- 5.2 When the Problem of Endomorph Reducibility for Relatively Free Rings is Unsolvable.- 5.3 When the Problem of Endomorph Reducibility is Solvable.- 5.4 The Problem of Endomorph Reducibility for Relatively Free Groups.- 5.5 The Variety R Included in NN.- 5.6 The Free Group T of the Variety $$ (\mathcal{A}_2^2 \cap \mathcal{N}_2 )\mathcal{A}_2 \mathcal{R} $$ and its Quotient Group S.- 5.7 The Main Construction.- 5.8 Application to Constructing Non-Finitely-Based Varieties.- 5.9 An Interpretation of Polynomials.- 5.10 Unsolvability of Some Algorithmic Problems in the Theory of Group.- 6 The Constructive Method in the Theory of HNN-extensions. Groups with Standard Normal Form.- 6.1 Introduction.- 6.2 Novikov-Boone Groups.- 6.3 The Novikov Lemma and the Britton Lemma.- 6.4 The Definition of Groups with Standard Normal Form.- 6.5 The Novikov Group AP1P2.- 6.6 The Boone Group.- 7 The Constructive Method for HNN-extensions and the Conjugacy Problem for Novikov-Boone Groups.- 7.1 Introduction.- 7.2 The Conjugacy Problem for the Group G1.- 7.3 The Group G2.- 7.4 Some Calculuses.- 7.5 The Conjugacy Problem for the Group AP1P2.- 7.6 Auxiliary facts.- Appendix 1 Calculations in Free Groups.- Appendix 2 Algorithmic Properties of the Wreath Products of Groups.- Appendix 3 Survey of the Theory of Absolutely Free Algebras.