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Gröbner-Shirshov Bases for Algebras, Groups, and Semigroups

In: Proceedings of the Third International Algebra Conference

Author

Listed:
  • L. Bokut

    (Institute of Mathematics)

  • Y. Fong

    (National Cheng Kung University, Department of Mathematics)

  • L.-S. Shiao

    (Chang Jung Christian University)

Abstract

What is now called by the Gröbner-Shirshov bases method had been invented in a highly original paper by A.I. Shirshov ([85], 1962) on Lie algebras or to be more precise on Lie polynomials (elements of a free Lie algebra). This very fact is widely accepted now (see below; see also a recent survey by V. Ufnarovsky [87]). As it happened, roughly speaking the same ideas had been discovered by B. Buchberger ([41], 1965) for ordinary polynomials in his Theses under supervision of W. Gröbner (published in ([42], 1970), see English translation in [45]). Let us have some parallels between papers [85] and [42]. The notion of composition of two Lie polynomials in [85] corresponds to the notion of s-polynomial in [42]. The process of constructing a set S* for any set S of Lie polynomials in [85] corresponds to the very Buchberger algorithm in [42]. A main lemma by Shirshov in [85] (that is called later by Composition lemma: If f ∈ S*, then % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGMbaaaiabg2da9iaadggadaqdaaqaaiaadohaaaGaamOyaaaa!3ACC! ]]

Suggested Citation

  • L. Bokut & Y. Fong & L.-S. Shiao, 2003. "Gröbner-Shirshov Bases for Algebras, Groups, and Semigroups," Springer Books, in: Proceedings of the Third International Algebra Conference, pages 17-32, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-0337-6_2
    DOI: 10.1007/978-94-017-0337-6_2
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