## Algorithmic and Combinatorial AlgebraEven 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

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 |

Survey of the Theory of Absolutely Free Algebras | 335 |

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A. I. Shirshov Algebra i logika arbitrary assume automorphism basic words basis belong calculus canonical form canonical words commutators complete under composition completes the proof conjugacy problem conjugate Consider constructed contain the letters Corollary defining relations Definition degree Denote element f embedding endomorphism epimorphism equality exists an algorithm expression field F finite number finite-dimensional finitely presented free algebra free associative algebra free group free Lie algebra free product fully regular G. P. Kukin group G homomorphism ideal identity implies induction integer irreducible set irreducible word isomorphic K(II leading term Lindon-Shirshov words linear combination linearly independent Math modulo monomials nilpotent nonzero obtain p-algebra p-ring polynomial product of Lie proof of Lemma recursive reduced regular words residually finite ring Russian satisfies Section semigroup solvable stable letters subalgebra subgroup subring subword summands Suppose Theorem unsolvable valid variety word problem words in form wreath product zero divisors