Graph algebra

Let ${\displaystyle D=(V,E)}$ be a directed graph, and ${\displaystyle 0}$ an element not in ${\displaystyle V}$. The graph algebra associated with ${\displaystyle D}$ has underlying set ${\displaystyle V\cup \{0\}}$, and is equipped with a multiplication defined by the rules

• ${\displaystyle xy=x}$ if ${\displaystyle x,y\in V}$ and ${\displaystyle (x,y)\in E}$,
• ${\displaystyle xy=0}$ if ${\displaystyle x,y\in V\cup \{0\}}$ and ${\displaystyle (x,y)\notin E}$.

This notion has made it possible to use the methods of graph theory in universal algebra and several other directions of discrete mathematics and computer science. Graph algebras have been used, for example, in constructions concerning dualities (Davey et al. 2000), equational theories (Pöschel 1989), flatness (Delić 2001), groupoid rings (Lee 1991), topologies (Lee 1988), varieties (Oates-Williams 1984), finite state automata (Kelarev, Miller & Sokratova 2005), finite state machines (Kelarev & Sokratova 2003) harv error: no target: CITEREFKelarev_&_Sokratova2003 (help), tree languages and tree automata (Kelarev & Sokratova 2001) etc.

• Group algebra (disambiguation)
• Incidence algebra
• Path algebra

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• Delić, Dejan (2001), "Finite bases for flat graph algebras", Journal of Algebra, 246 (1): 453–469, doi:10.1006/jabr.2001.8947, ISSN 0021-8693, MR 1872631
• McNulty, George F.; Shallon, Caroline R. (1983), "Inherently nonfinitely based finite algebras", Universal algebra and lattice theory (Puebla, 1982), Lecture Notes in Math., 1004, Berlin, New York: Springer-Verlag, pp. 206–231, doi:10.1007/BFb0063439, hdl:10338.dmlcz/102157, ISBN 978-3-540-12329-3, MR 0716184
• Kelarev, A.V. (2003), Graph Algebras and Automata, New York: Marcel Dekker, ISBN 0-8247-4708-9, MR 2064147
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• Kelarev, A.V.; Miller, M.; Sokratova, O.V. (2005), "Languages recognized by two-sided automata of graphs", Proc. Estonian Akademy of Science, 54 (1): 46–54, ISSN 1736-6046, MR 2126358
• Kelarev, A.V.; Sokratova, O.V. (2001), "Directed graphs and syntactic algebras of tree languages", J. Automata, Languages & Combinatorics, 6 (3): 305–311, ISSN 1430-189X, MR 1879773
• Kiss, E.W.; Pöschel, R.; Pröhle, P. (1990), "Subvarieties of varieties generated by graph algebras", Acta Sci. Math. (Szeged), 54 (1–2): 57–75, MR 1073419
• Lee, S.-M. (1988), "Graph algebras which admit only discrete topologies", Congr. Numer., 64: 147–156, ISSN 1736-6046, MR 0988675
• Lee, S.-M. (1991), "Simple graph algebras and simple rings", Southeast Asian Bull. Math., 15 (2): 117–121, ISSN 0129-2021, MR 1145431
• Oates-Williams, Sheila (1984), "On the variety generated by Murskiĭ's algebra", Algebra Universalis, 18 (2): 175–177, doi:10.1007/BF01198526, ISSN 0002-5240, MR 0743465
• Pöschel, R (1989), "The equational logic for graph algebras", Z. Math. Logik Grundlag. Math., 35 (3): 273–282, doi:10.1002/malq.19890350311, MR 1000970

• Raeburn, Iain (2005), Graph algebras, American Mathematical Society, ISBN 978-0-8218-3660-6