左交换代数的“自由定理”

李羽

李羽. 左交换代数的“自由定理”[J]. 华南师范大学学报(自然科学版), 2018, 50(1): 110-113.
引用本文: 李羽. 左交换代数的“自由定理”[J]. 华南师范大学学报(自然科学版), 2018, 50(1): 110-113.
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.

左交换代数的“自由定理”

基金项目: 

国家自然科学基金-数学天元基金项目;国家自然科学基金青年科学基金项目;国家自然科学基金青年科学基金项目;广东省自然科学基金博士启动项目;广东省自然科学基金博士启动项目

详细信息
    通讯作者:

    李羽

  • 中图分类号: 中图分类号: O153.5

The ``Freedom Theorem" of Left-Commutative Algebras

Funds: 

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

More Information
    Corresponding author:

    羽 李

  • 摘要: 设X是一个有限集,LC(X)表示由X生成的自由左交换代数;fLC(X), Id(f)表示LC(X)的由f所生成的理想. 对于任意的h, 是否存在一个算法可以判断出hId(f)hId(f)?为了研究这个问题, 文中应用Gr\{o}bner-Shirshov 基理论的思想方法在自由左交换代数的线性基底上定义了一个良序,证明了这个良序保持运算,重写了由一个多项式所生成的自由左交换代数的理想的元素的表达式. 证明了一个定义关系的左交换代数具有可解的字问题并得到了左交换代数的自由定理.
    Abstract: 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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出版历程
  • 收稿日期:  2016-02-27
  • 修回日期:  2016-03-30
  • 刊出日期:  2018-02-24

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