Graph algebra

The concept of a graph algebra was introduced by G.F. McNulty and C.R. Shallon in [1]. Let $D=(V,E)$ be a directed graph, and let $0$ by an element not in $V$. The graph algebra associated with $D$ is the set $V\cup \{0\}$ equipped with multiplication defined by the rules $xy=x$ if $x,y\in V,(x,y)\in E$, and $xy=0$ if $x,y\in V\cup \{0\},(x,y)\notin E$.

Graph algebras have been used in several directions of mathematical research as a convenient source of examples required for the proofs of various theorems.

References

[1] B.A. Davey, P.M. Idziak, W.A. Lampe and G. F. McNulty, ``Dualizability and graph algebras, Discrete Math. {\bf 214} (1-3) (2000), 145-172.

[2] G.F. McNulty and C.R. Shallon, Inherently nonfinitely based finite algebras, Universal Algebra and Lattice Theory (Puebla, 1982), Springer, Berlin, 1983, 206-231.

[3] A.V. Kelarev, "Graph Algebras and Automata", Marcel Dekker, New York, 2003. ISBN: 0-8247-4708-9.

[4] A.V. Kelarev and O.V. Sokratova, On congruences of automata defined by directed graphs, Theoretical Computer Science 301 (2003), 31-43.

[5] E.W. Kiss, R. P"oschel, and P. Pr"ohle, Subvarieties of varieties generated by graph algebras, Acta Sci. Math. (Szeged)} 54(1-2) (1990), 57-75.