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^ +.