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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).
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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