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 workforce idea' turns out to ap pear first because the identify of the e-book by way of A. Karras, W. Magnus, and D. Solitar [182] and, in a while, it served because the name of the publication by way of R. C. Lyndon and P. Schupp [247]. these days, experts don't query the life of 'combinatorial algebra' as a unique algebraic job. The job is wonderful not just by way of its items of study (that are successfully given to a point) but in addition by way of its equipment (ef fective to a couple extent). To be extra distinct, shall we 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 via turbines and defining family members {in the 1st and specific position, loose teams, semigroups, algebras, and so on. )j an element within which we research common buildings, viz. loose items, lINN-extensions, and so on. j and, eventually, a component the place particular equipment similar to the Composition procedure (in different phrases, the Diamond Lemma, see [49]) are utilized. absolutely, the above clarification is much from protecting the total scope of the time period (compare the prefaces to the books pointed out above).

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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). All algebras are considered over a fixed field F. Let X be a totally ordered set. Introduce the following order on a free semigroup X*. , d(Xi) > d(xj) implies Xi > Xj for Xi, Xj E X. Extend this degree on X* as usual: d(Xil ... Xi n ) = d(XiJ + ...

19, akybm =I- o. Hence, 0' =I- 0 and ;1 =I- o. 14 and due to an = bn = 0). Since x = y and (x, y) is the independent pair of homogeneous elements, O'yX - O'xY is a homogeneous nonzero element. Designate it as z. It is clear that z < y and (z, y) E Dc. 19, we obtain akzbm =I- 0 and thus, akzbm = akzbm • Then O'akzbm = (O'y;1- O'O'x)akybm (it follows from O'akx = ;1a k y) contradicts z < y. This completes the proof. 25. An algebra R over a commutative ring F is called filtered if Un>oRn for some F-submodules Rn, n E N, such that 1 E Ro C Rl C ...

Hence, U = 0 or v = O. We arrive at a contradiction, because all the words UiXlbiCjX2Vj are of the basis and elements UiX2aidjXIVj can be transformed into the canonical form as follows: aidj = aijal + Ek#l afjak, Uj = u~( x2aJ)k;, kj ~ 0, ujx2ajdjXIVj = (-1)k;+lui x I(alx2)k;+lVj + Ek#l arjUi(x2ai)k;x2akXIVj + 0:, d{c) < d(u) + d(v). 12 is complete. 13. Every algebra A without zero divisors is embeddable into an algebra A lacking zero divisors and such that every equation of the form xay+yax = b, o -:f.