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

Even 3 a long time 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 team conception' turns out to ap­ pear first because the name of the booklet by means of A. Karras, W. Magnus, and D. Solitar [182] and, in a while, it served because the identify of the e-book by means of R. C. Lyndon and P. Schupp [247]. these days, experts don't query the lifestyles of 'combinatorial algebra' as a different algebraic job. The task is unusual not just through its items of analysis (that are successfully given to a point) but additionally through its tools (ef­ fective to a couple extent). To be extra detailed, shall we nearly outline the time period 'combinatorial algebra' for the needs of this ebook, as follows: So we name part of algebra facing teams, semi teams , associative algebras, Lie algebras, and different algebraic platforms that are given through turbines and defining family members {in the 1st and specific position, unfastened teams, semigroups, algebras, and so on. )j a component within which we learn common structures, viz. unfastened items, lINN-extensions, and so on. j and, ultimately, a component the place particular tools resembling the Composition approach (in different phrases, the Diamond Lemma, see [49]) are utilized. definitely, the above clarification is way from overlaying the complete scope of the time period (compare the prefaces to the books pointed out above).

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Conditions (2) and (3) just state the completeness of Sunder composition. 8 implies the next G. 10 (Diamond Lemma). Let S ~ F(X) \ {OJ. Assume (X) to be a semigroup partially ordered by ~, meeting the minimality condition, and such that for any s E S there exists a leading word s in s, s having the coefficient 1. Then the following are equivalent: (a") The algebm R = F(X; s = 0 (s E S)) is a free F-module and the image (under the injective map) of the set M consisting of the words of < X > without any occurrences of subwords s, s E S, forms the basis of this F-module.

35. For n > 3 or m > 2, C = CF(n, m, 0) is not weakly noethe- nan. Proof. Consider the following cases. Case 1. Suppose m > 2, then a < c < b. Put Xt = b(ac)tba n- 1 and verify that the ideal generated by Xk'S, 1 ~ k < t, does not contain Xt. This will prove the theorem. By way of contradiction, suppose that t-l Xt = L L /iUiXkVi, (4) k=1 iEh where Ii E F, Ui, and Vi are words and h's, 1 ~ k < t, are finite sets. Since the defining relations of C are homogeneous, Ui and Vi can be considered as words in a and c.

B - a51ds~b E Iv (v = a51ds2b). b. It follows that uS2b, aSlw < v, and so the difference is contained in Iv. , v = a51ds 2b, al = a, bl = ds 2b, a2 = a51d and b2 = b. The following lemma is clear. 6. Let v E G and a, b E G#. Then aIvb ~ I avb . 7. This lemma shows that if condition (1) is met by some Si, ai, and bi (i = 1,2), then it is also satisfied by Si, aai, and bib (i = 1,2 and a,b E G#). s2. In this case, (1) expresses the completeness of 5 under composition. 4 is exactly our Composition Lemma for the semigroup algebra FG.

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