On finite bases for varieties and pseudovarieties
Let P be a partition of an algebra A, and let S be a set of distinct representatives for P. Then the (S,P)-reduction of A is the algebra on S of the same type as A with operations defined via representatives modulo P. Such a reduction may or may not preserve composite terms; if it does preserve all terms of length at most n, it is called an n-reduction. The author proves that a variety of finite type is finitely based if and only if it is closed under n-reductions for some finite n. By related means, he shows that certain classes of finite algebras generating locally finite varieties generate nonfinitely based pseudovarieties; as examples, he mentions \textit{P. Perkins}' 6-element semigroup [J. Algebra 11, 298-314 (1969; Zbl 0186.03401)] and \textit{R. C. Lyndon}'s 7-element groupoid [Proc. Am. Math. Soc. 5, 8-9 (1954; Zbl 0055.027)].
scientific article; zbMATH DE number 4068958
scientific article