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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Relative Gröbner-Shirshov bases for algebras and groups


Authors: L. A. Bokut and K. P. Shum
Translated by: the authors
Original publication: Algebra i Analiz, tom 19 (2007), nomer 6.
Journal: St. Petersburg Math. J. 19 (2008), 867-881
MSC (2000): Primary 16S15, 16S34, 20C05, 20C07
Published electronically: August 21, 2008
MathSciNet review: 2411637
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Abstract: The notion of a relative Gröbner-Shirshov basis for algebras and groups is introduced. The relative composition lemma and relative (composition-)diamond lemma are established. In particular, it is shown that the relative normal forms of certain groups arising from Malcev's embedding problem are the irreducible normal forms of these groups with respect to their relative Gröbner-Shirshov bases. Other examples of such groups are given by showing that any group $ G$ in a Tits system $ (G, B, N,S)$ has a relative ($ B$-)Gröbner-Shirshov basis such that the irreducible words are the Bruhat words of $ G$.


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Additional Information

L. A. Bokut
Affiliation: Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, 630090, Russia
Email: bokut@math.nsc.ru

K. P. Shum
Affiliation: Department of Mathematics, the University of Hong Kong, Pokfulam Road, Hong Kong, China (SAR)
Email: kpshum@maths.hku.hk

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01025-X
PII: S 1061-0022(08)01025-X
Keywords: Relative Gr\"obner--Shirshov bases, irreducible normal form, rewriting system, Tits systems, Malcev's problem
Received by editor(s): August 6, 2007
Published electronically: August 21, 2008
Additional Notes: Supported by RFBR and SS, grant no. 344.2008.1, and by UGC (HK), grant no. 2060187, 2002/04
Article copyright: © Copyright 2008 American Mathematical Society