Graph algebra

This article is about the mathematical concept of Graph Algebras. For "Graph Algebra" as used in the social sciences, see Graph algebra (social sciences).

In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon,[1] and has seen many uses in the field of universal algebra since then.

## Contents

* 1 Definition
* 2 Applications
* 3 See also
* 4 Citations
* 5 Works cited
* 6 Further reading

## Definition

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

* xy = x if x,y∈V and (x,y)∈E,
* xy = 0 if x,y∈V∪{0} and (x,y)∉E.

## Applications

This notion has made it possible to use the methods of graph theory in universal algebra and several other areas of discrete mathematics and computer science. Graph algebras have been used, for example, in constructions concerning dualities,[2] equational theories,[3] flatness,[4] groupoid rings,[5] topologies,[6] varieties,[7] finite-state machines,[8][9] tree languages and tree automata,[10] etc.

## See also

* Group algebra (disambiguation)
* Incidence algebra
* Path algebra

## Citations

1. ↑ McNulty & Shallon 1983, pp. 206–231.
2. ↑ Davey et al. 2000, pp. 145–172.
3. ↑ Pöschel 1989, pp. 273–282.
4. ↑ Delić 2001, pp. 453–469.
5. ↑ Lee 1991, pp. 117–121.
6. ↑ Lee 1988, pp. 147–156.
7. ↑ Oates-Williams 1984, pp. 175–177.
8. ↑ Kelarev, Miller & Sokratova 2005, pp. 46–54.
9. ↑ Kelarev & Sokratova 2003, pp. 31–43.
10. ↑ Kelarev & Sokratova 2001, pp. 305–311.

## Works cited

* Davey, Brian A.; Idziak, Pawel M.; Lampe, William A.; McNulty, George F. (2000). "Dualizability and graph algebras". Discrete Mathematics 214 (1): 145–172. doi:10.1016/S0012-365X(99)00225-3. ISSN 0012-365X.
* 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.
* 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.
* 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.
* Kelarev, A.V.; Sokratova, O.V. (2003). "On congruences of automata defined by directed graphs". Theoretical Computer Science 301 (1–3): 31–43. doi:10.1016/S0304-3975(02)00544-3. ISSN 0304-3975. https://eprints.utas.edu.au/157/1/congruences.pdf.
* Lee, S.-M. (1988). "Graph algebras which admit only discrete topologies". Congr. Numer 64: 147–156. ISSN 1736-6046.
* Lee, S.-M. (1991). "Simple graph algebras and simple rings". Southeast Asian Bull. Math 15 (2): 117–121. ISSN 0129-2021.
* McNulty, George F.; Shallon, Caroline R. (1983). "Inherently nonfinitely based finite algebras". in Freese, Ralph S.; Garcia, Octavio C.. Universal algebra and lattice theory (Puebla, 1982). Lecture Notes in Math.. 1004. Berlin, New York City: Springer-Verlag. pp. 206–231. doi:10.1007/BFb0063439. ISBN 978-354012329-3. https://archive.org/details/universalalgebra0000unse.
* 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.
* Pöschel, R. (1989). "The equational logic for graph algebras". Z. Math. Logik Grundlag. Math. 35 (3): 273–282. doi:10.1002/malq.19890350311.

## Further reading

* Kelarev, A.V. (2003). Graph Algebras and Automata. New York City: Marcel Dekker. ISBN 0-8247-4708-9. https://archive.org/details/graphalgebrasaut0000kela.
* Kiss, E.W.; Pöschel, R.; Pröhle, P. (1990). "Subvarieties of varieties generated by graph algebras". Acta Sci. Math. 54 (1–2): 57–75.
* Raeburn, Iain (2005). Graph algebras. American Mathematical Society. ISBN 978-082183660-6.

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