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 $D = (V, E)$ 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
 * $xy = x$ if $$x,y \in V$$ and $$(x,y) \in E$$,
 * $xy = 0$ 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.