By L.A. Bokut', G.P.. Kukin

Even 3 many years in the past, the phrases 'combinatorial algebra' contrasting, for in stance, the phrases 'combinatorial topology,' weren't a standard designation for a few department of arithmetic. The collocation 'combinatorial staff thought' turns out to ap pear first because the name of the e-book via A. Karras, W. Magnus, and D. Solitar [182] and, in a while, it served because the identify of the publication by way of R. C. Lyndon and P. Schupp [247]. these days, experts don't query the lifestyles of 'combinatorial algebra' as a unique algebraic job. The task is exotic not just through its gadgets of study (that are successfully given to a point) but in addition through its equipment (ef fective to a couple extent). To be extra specific, lets nearly outline the time period 'combinatorial algebra' for the needs of this publication, as follows: So we name part of algebra facing teams, semi teams , associative algebras, Lie algebras, and different algebraic platforms that are given by means of turbines and defining kin {in the 1st and specific position, unfastened teams, semigroups, algebras, and so forth. )j an element during which we examine common structures, viz. unfastened items, lINN-extensions, and so on. j and, eventually, a component the place particular equipment resembling the Composition technique (in different phrases, the Diamond Lemma, see [49]) are utilized. absolutely, the above rationalization is much from protecting the whole scope of the time period (compare the prefaces to the books pointed out above).

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8. 4 it is convenient to use G. Bergman's notion of reduction. sb and fixes the other elements of G. A reduction r is called nontrivial on 1 E A if r(J) f. 1, that is, if If I 3 asb. The sequence r(1), ... ,r(t) of reductions is called nontrivial on 1 if each r(i) is nontrivial on li-l, where fj is the image of Ij-l under r· U) and f = 10. Such sequence cannot be infinite. It suffices to prove this claim for v E G. In the case, r(I)(v) is an F-linear combination of elements in G that are less than v.

It is clear that fg = Jg. As was noted seen, the Composition Lemma holds for every linear order on X* meeting the minimality condition, so the lemms is valid in our case. In this section, we will prove the following theorems by essentially using the Composition Lemma. 1. Let F be a field that is at most countable, and A, I

Before we continue our proving, we agree to use the following simplifications in relations (9) and (13). These relations are homogeneous, and any term of each relation (such as u)n)ujm), ...