Cardinality bounds for subdirectly irreducible algebras
This is a well-written and nicely presented paper that everyone should read who is interested in the trends of modern universal algebra. The author tells the history of the topic, which enlightens the difficulty of the subject and the depth of its proof. Previous attempts to prove that a residually small variety generated by a finite algebra is residually bounded by a finite integer, failed. The reason was undecidability, as was shown by \textit{R. McKenzie} [``The residual bound of a finite algebra is not computable'', Int. J. Algebra Comput. 6, No. 1, 29-48 (1996; Zbl 0844.08010)]. The positive result of this paper was proved by the author in 1991, but remained unpublished, hoping for a complete solution. The result is the following: In a finitely generated residually small variety there is a finite cardinal bound which holds for all subdirectly irreducible algebras, which omits type \textbf{5} and whose monolith is of type \textbf{2}, \textbf{3}, or \textbf{4}. In fact, if the generating algebra has exactly \(n\) elements, then the bound is \(n^{n^{n^{n-2}}}\). As the result shows, the author uses tame congruence theory.
scientific article; zbMATH DE number 937164
scientific article