定向图的张量代数中的单元

Single Elements in Tensor Algebras of Directed Graphs

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

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 Gn be the graph consisting of a single vertex{p}and n loop edges{e1,e2,…en}i.e.,s(ei)=r(ei)=p,i=1,2…n.We show every element of the tensor algebra TGn+is a single element.Moreover,every element of the free semigroupoid algebra LGn is a single element.For a countable directed graph G,we show the linear span of the single elements of the tensor algebra TG+is dense inTG+.For a finite directed graph Gn,we show any element of TGn+is a linear span of n2 single elements of TCn+.