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Graph algebras and graph varieties. (English) Zbl 0725.08002
Let $$G=(V,E)$$ be a directed graph without multiple edges, V denotes the set of vertices, E is the set of edges. Let $$\infty$$ be an adjoined element. The operation $$a,b=a$$ if (a,b)$$\in E$$ and $$a,b=\infty$$ otherwise defines the graph algebra on $$V\cup \{\infty \}$$. The author proves a “Birkhoff-type” theorem: a class of finite directed graphs is a graph variety iff it is closed with respect to finite restricted pointed subproducts and isomorphic copies. Several applications are given.

##### MSC:
 08B05 Equational logic, Mal’tsev conditions 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
directed graph; graph algebra; graph variety
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##### References:
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