Nonfinitely based pseudovarieties and inherently nonfinitely based varieties
An algebra is a set together with a finite number of finitary operations. The pseudovariety generated by an algebra \(A\) is the smallest class containing \(A\) and closed under homomorphisms, subalgebras, and finite products. The pseudovariety generated by \(A\) consists of all finite algebras in the variety generated by \(A\). If \(A\) is finitely based then it generates a finitely based pseudovariety. The general question: if \(A\) is nonfinitely based, does it necessarily generate a nonfinitely based pseudovariety? is open. The author shows that if \(A\) has a property called \(\omega\)-bounded, then it generates a nonfinitely based pseudovariety. In this paper he gives two examples of \(\omega\)-bounded algebras, Perkins' semigroup and Lyndon's groupoid and shows that every nonfinitely based finite algebra is either inherently nonfinitely based or \(\omega\)-bounded. Perkins' semigroup falls in the intersection of these classes while Lyndon's groupoid must be \(\omega\)-bounded since it is not inherently nonfinitely based.
scientific article; zbMATH DE number 436441