Advances In Algebra And Combinatorics: Proceedings of the by K. P. Shum, E. Zelmanov, Jiping Zhang, Li Shangzhi

By K. P. Shum, E. Zelmanov, Jiping Zhang, Li Shangzhi

This quantity is a compilation of lectures on algebras and combinatorics provided on the moment foreign Congress in Algebra and Combinatorics. It experiences on not just new effects, but in addition on open difficulties within the box. The complaints quantity turns out to be useful for graduate scholars and researchers in algebras and combinatorics. The participants contain eminent figures resembling E Bannai, P Hilton, M Jambu, I Kotsireas, B Schein and A Smoktunowicz.

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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”.

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