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Graph algebra

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, and has seen many uses in the field of universal algebra since then.

Definition

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

  • if x,y \in V and (x,y) \in E,
  • if x,y \in V \cup {0} and (x,y)\notin 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, equational theories, flatness, groupoid rings, topologies, varieties, finite-state machines, tree languages and tree automata, etc.

Citations

Works cited

  • {{Cite journal | title = Dualizability and graph algebras | doi-access = free }}
  • {{Cite journal | title = Finite bases for flat graph algebras | doi-access = free
  • {{Cite journal | title = Languages recognized by two-sided automata of graphs
  • {{Cite journal | title = Directed graphs and syntactic algebras of tree languages
  • {{Cite journal | title = On congruences of automata defined by directed graphs
  • {{Cite journal | title = Graph algebras which admit only discrete topologies
  • {{Cite journal | title = Simple graph algebras and simple rings
  • {{Cite book| chapter = Inherently nonfinitely based finite algebras | editor1-last = Freese | editor1-first = Ralph S. | editor2-last = Garcia | editor2-first = Octavio C. | hdl-access = free
  • {{Cite journal | title = On the variety generated by Murskiĭ's algebra | author-link = Sheila Oates Williams
  • {{Cite journal | title = The equational logic for graph algebras
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

graph-theory universal-algebra
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