Single Elements in Tensor Algebras of Directed Graphs

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^ +.