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    徐一丹, 李建奎, 李姗. 定向图的张量代数中的单元[J]. 华东理工大学学报(自然科学版), 2020, 46(6): 833-837. DOI: 10.14135/j.cnki.1006-3080.20191029003
    引用本文: 徐一丹, 李建奎, 李姗. 定向图的张量代数中的单元[J]. 华东理工大学学报(自然科学版), 2020, 46(6): 833-837. DOI: 10.14135/j.cnki.1006-3080.20191029003
    XU Yidan, LI Jiankui, LI Shan. Single Elements in Tensor Algebras of Directed Graphs[J]. Journal of East China University of Science and Technology, 2020, 46(6): 833-837. DOI: 10.14135/j.cnki.1006-3080.20191029003
    Citation: XU Yidan, LI Jiankui, LI Shan. Single Elements in Tensor Algebras of Directed Graphs[J]. Journal of East China University of Science and Technology, 2020, 46(6): 833-837. DOI: 10.14135/j.cnki.1006-3080.20191029003

    定向图的张量代数中的单元

    Single Elements in Tensor Algebras of Directed Graphs

    • 摘要: 对于任何可数的定向图 G,证明了张量代数 T_ G^ +中的单元生成的线性子空间在 T_ G^ +中是稠密的。对于有限的定向图 C_n,证明了 T_ C_n^ +中的每个元素可以写成n^2个单元的线性组合。

       

      Abstract: The purpose of this article is to study the linear span of the single elements in the tensor algebras of directed graphs. The notion of ‘single element’ may prove to be useful in other fields. Let G_n be the graph consisting of a single vertex \ p\ and n loop edges \ e_1,e_2, \cdots e_n\ i.e., s(e_i) = r(e_i) = p, i = 1,2, \cdots n. We show every element of the tensor algebra T_ G_n^ + is a single element. Moreover, every element of the free semigroupoid algebra L_ G_n^ is a single element. For a countable directed graph G, we show the linear span of the single elements of the tensor algebra T_ G^ + is dense in T_ G^ +. For a finite directed graph C_n, we show any element of T_ C_n^ + is a linear span of n^2 single elements of T_ C_n^ +.

       

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