Graph algebra: Difference between revisions

Revision as of 00:38, 1 March 2009

Revision as of 00:39, 1 March 2009

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The concept of a graph algebra was introduced by G.F. McNulty and C.R. Shallon in 1983. Let <math>D=(V,E)</math> be a directed graph (see [[Graph (data structure)]]), and let <math>0</math> be an element not in <math>V</math>. The ''graph algebra'' associated with <math>D</math> is the set <math>V \cup \{0\}</math> equipped with multiplication defined by the rules <math>xy = x</math> if <math>x,y\in V,(x,y)\in E</math>, and <math>xy = 0</math> if <math>x,y\in V\cup \{0\},(x,y)\notin E</math>.

This notion has made it possible to use the methods of [[graph theory]] in several other directions of [[universal algebra]], and several other directions of [[mathematics]] and [[computer science]]. Graph algebras have been used, for example, in constructions concerning dualities, equational theories (see [[equational theory]]), flatness, groupoid rings (see [[groupoid (algebra)]] and [[ring (mathematics)]]), [[topologies]], varieties (see [[variety (universal algebra)]]), [[finite state automata]] or [[finite state machines]], etc.

References