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Subvarieties of varieties generated by graph algebras. (English) Zbl 0713.08006
Graph algebras, which were first inroduced by C. R. Shallon [Ph. Dissertation, U.C.L.A. (1979)] have provided many examples in the theory of varieties generated by finite algebras. However, this interesting paper shows that we cannot settle the following two problems by seeking for example among graph algebras corresponding to undirected graphs.
Problem 1. Does there exist a finitely generated variety whose lattice of subvarieties satisfies the descending chain condition but not the ascending chain condition?
Problem 2. Is the intersection of two finitely generated varieties always finetely generated?
The theorem that achieves this result consists of a classification of those varieties satisfying the descending chain ondition which are generated by graph algebras corresponding to undirected graphs; there are precisely seventeen of these. One of the key lemmata used in proving this and several other interesting results, states that a critical algebra in a variety generated by graph algebras is itself a graph algebra. This result raises the problem of classifying those graphs whose graph algebras are critical.
Reviewer: S.Oates-Williams

08B15 Lattices of varieties