By S. B. Kizlik

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**Sample text**

We say that a monomial X u = Xyl . . Xgn, u E Z n , occurs in an element f E 3 if f # 0. Expand the valuation v t o f E 3 in the following way. If f E 3 then v( f) the least element from s u p p f . Put - 0 = {f E 3 I v ( f )2 O}, m = {f E 3 I v ( f )> 0). As in [3] it can be shown the set 3 is a division ring containing F . The skew field 3 is called a completion of F with respect to v. Moreover 0 is a subring in 3 and m is an ideal in 0. It is generated as an ideal in 0 by the elements XE' , ..

Conformal associative algebras (C, (n), n 2 0, D) ([12]). There are 6 types of compositions including inclusion, intersection, Dinclusion, D-intersection, left (right) multiplication by a generator. The condition (a) ((b)) in the CD-lemma is not equivalent to the condition that S is a Grobner-Shirshov basis. -Modules ([36], [29]). 2. CD-lemma for associative algebras In this section, we cite some concepts and results from the literature which are related to the Grobner-Shirshov bases for the associative algebras.

Some one-relator groups This section contains the results of Y. Q. Chen and C. Y. Zhong in [28]. It is well known the Magnus algorithm for a solution of the word problem for any one-relator group. Using the Magnus rewriting procedure (see R. C. Lyndon and P. E. Schupp [43)], one may embed any one-relator group into a tower of HNNextensions. For towers of HNN-extensions of groups, L. Bokut (see [13]) developed a method of groups with the standard normal forms in 1965. Actually, for these groups, Grobner-Shirshov bases are also “standard” in a sense, so we may speak about “groups with the standard Grobner-Shirshov bases”.