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Gröbner-Shirshov Bases for Algebras, Groups, and Semigroups

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Proceedings of the Third International Algebra Conference

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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 EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGMbaaaiabg2da9iaadggadaqdaaqaaiaadohaaaGaamOyaaaa!3ACC! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \overline f = a\overline s b $$ for some s ∈ S*; for notations see below) corresponds to a theorem in [42] that is called later Buchberger’s theorem.

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  1. Institute of Mathematics, Novosibirsk, Russia

    L. Bokut

  2. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

    Y. Fong

  3. Chang Jung Christian University, Kway Jen, Tainan, Taiwan

    L.-S. Shiao

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Bokut, L., Fong, Y., Shiao, LS. (2003). Gröbner-Shirshov Bases for Algebras, Groups, and Semigroups. In: Proceedings of the Third International Algebra Conference. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0337-6_2

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