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羽 李. The ``Freedom Theorem" of Left-Commutative Algebras[J]. Journal of South China Normal University (Natural Science Edition), 2018, 50(1): 110-113.
Citation: 羽 李. The ``Freedom Theorem" of Left-Commutative Algebras[J]. Journal of South China Normal University (Natural Science Edition), 2018, 50(1): 110-113.

The ``Freedom Theorem" of Left-Commutative Algebras

Funds: 

; the Natural Science Foundation of Guangdong Province; the Natural Science Foundation of Guangdong Province

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  • Corresponding author:

    羽 李

  • Received Date: February 27, 2016
  • Revised Date: March 30, 2016
  • Let X be a finite set and LC(X)bethefreeleftcommutativealgebrainducedbyX.LetId(f) be the ideal of LC(X) induced by f where fLC(X). For any h, the problem is whether there is an algorithm to decide hId(f) or hId(f). This problem is studied by using the approach of Grobner-Shirshov bases theory. A well ordering on a linear basis of free left commutative algebra is defined. It is proved that the ordering is compatible with the product and that the element of the ideal of free left-commutative algebra induced by one polynomial is rewritten. The word problem for left-commutative algebras with a single defining relation is solved and the ``freedom theorem for left-commutative algebras is obtained.
  • [1] L.A. Bokut and G.P. Kukin. Algorithmic and combinatorial algebra., Mathematics and its Applications 255. Kluwer Academic Publishers Group, Dordrecht, 1994.
    [2] A. Dzhumadil'daev and C. L\"{o}fwall. Trees, free right-symmetric algebras, free Novikov algebras and identities. Homology, Homotopy and Applications. 4(2)(2002), 165–190.
    [3] D. Kozybaev, L. Makar-Limanov and U. Umirbaev. The freiheitssatz and automorphisms of free right-symmetric algebras. Asian-European Journal of Mathematics.
    1(2)(2008), 243-254.
    [4] J.-L. Loday. Cup product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77, Univ. Louis Pasteur, Strasbourg, 1995, 189-196.
    [5] W. Magnus. \"{U}ber discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssaz). J. Reine Angew. Math. 163(1930), 141-165.
    [6] L. Makar-Limanov. Algebraically closed skew fields. J. Algebra 93(1985), 117-135.
    [7] N.S. Romanovskii. A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees.(Russian) Mat. Sb. (N.S.) 89(131)(1972), 93-99.
    [8] A.I. Shirshov. Some algorithmic problems for Lie algebras. Sibirsk. Mat. Z. 3(1962), 292-296.

    [1] L.A. Bokut and G.P. Kukin. Algorithmic and combinatorial algebra., Mathematics and its Applications 255. Kluwer Academic Publishers Group, Dordrecht, 1994.
    [2] A. Dzhumadil'daev and C. L\"{o}fwall. Trees, free right-symmetric algebras, free Novikov algebras and identities. Homology, Homotopy and Applications. 4(2)(2002), 165–190.
    [3] D. Kozybaev, L. Makar-Limanov and U. Umirbaev. The freiheitssatz and automorphisms of free right-symmetric algebras. Asian-European Journal of Mathematics.
    1(2)(2008), 243-254.
    [4] J.-L. Loday. Cup product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77, Univ. Louis Pasteur, Strasbourg, 1995, 189-196.
    [5] W. Magnus. \"{U}ber discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssaz). J. Reine Angew. Math. 163(1930), 141-165.
    [6] L. Makar-Limanov. Algebraically closed skew fields. J. Algebra 93(1985), 117-135.
    [7] N.S. Romanovskii. A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees.(Russian) Mat. Sb. (N.S.) 89(131)(1972), 93-99.
    [8] A.I. Shirshov. Some algorithmic problems for Lie algebras. Sibirsk. Mat. Z. 3(1962), 292-296.

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