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 typical designation for a few department of arithmetic. The collocation 'combinatorial crew conception' turns out to ap pear first because the name of the publication via A. Karras, W. Magnus, and D. Solitar [182] and, in a while, it served because the name of the booklet through R. C. Lyndon and P. Schupp [247]. these days, experts don't query the lifestyles of 'combinatorial algebra' as a distinct algebraic task. The task is unique not just via its gadgets of study (that are successfully given to a point) but additionally through its tools (ef fective to a few extent). To be extra precise, shall we nearly outline the time period 'combinatorial algebra' for the needs of this booklet, as follows: So we name part of algebra facing teams, semi teams , associative algebras, Lie algebras, and different algebraic structures that are given via turbines and defining kin {in the 1st and specific position, loose teams, semigroups, algebras, and so forth. )j an element within which we research common buildings, viz. unfastened items, lINN-extensions, and so forth. j and, eventually, a component the place particular equipment corresponding to the Composition technique (in different phrases, the Diamond Lemma, see [49]) are utilized. without doubt, the above rationalization is way from masking the complete scope of the time period (compare the prefaces to the books pointed out above).

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We proceed by induction on m. The case m = 1 is trivial, because t'l is representable in F = MI(F). Assume that for m - 1, m ::::: 2, the statement is valid. Thus, fm = t'l + ... + t;:, = fm-l + t;:', and fm-l is representable in A = Mq(F), q = nm-2. For a suitable linear form IE A[tl, ... ,tm ], we have fm-l = In. Since I and tm are representable, fm-l and t;:, are simultaneously representable in A. 11 is proved. 1. 7 More on Embeddings into Simple Algebras In this section, we shall prove various theorems on embedding algebras into simple ones which are generated by (or are the sums of) some algebras of a predetermined type (under some set-theoretical assumptions).

The degree of a word is the number of the occurrences of Xj, the degree of an element is the maximal degree of its the words. Agree that d(O) = -00. To prove the lemma, we need only to establish that d(fg) = d(f) + d(g) for f, g E Aa· First, verify this relation for f, g of degree 1. If 4 4 i=1 j=1 L xibi . L CjXj = g, where d(g) < 2, bi , Ci E A, Composition Method for Associative Algebras then either L Xibi = 0 or L CjX 15 j = o. (6) To obtain a contradiction, suppose both distinct from O. All the words XiajXk, except X4alx3, belong to the basis, hence, we get b3cj = 0 and b1cj = 0, 1 S; j S; 4, that is, b3 = bl = O.

Proof. If f is representable in F'-algebra A then, treating A as the F-algebra, we infer that f is representable over F. 7. 8. The finite sum of representable forms, of degree n, of F[t l , . , tmJ is representable. Proof. It is sufficient to consider the sum f + 9 of forms of degree n. 7, the form f + 9 is weakly representable. 5, it is representable, as needed. 9. Each form is representable. 10. The genemlised Clifford algebm CF(n, m, J) over a field F has a finite-dimensional representation.