On the regular part of varieties of algebras
An identity is said to be normal (or regular) if the same variables appear on each side of it. The variety N(V) defined by all normal identities satisfied in the variety V of algebras is called the normal closure of V. The reviewer has shown that the normal closure of a semigroup variety V consists of subalgebras of rigid commutative bands (or sums of direct systems) of semigroups from V [Izv. Vyssh. Uchebn. Zaved., Mat. 1969, No.5(84), 61-68 (1969; Zbl 0199.336), Teor. Polugrupp Prilozh. 2, 69-74 (1971; Zbl 0244.20077)]. It was stated by the reviewer [in Tr. Mol. Uchen. Mat. Mekh. 124-130 (1969; Zbl 0276.08001)] that the same is true for arbitrary algebras. The present authors refute this statement by an example. The second example shows that the join of infinitely many nonregular varieties can be regular thus disproving another erroneous claim from (op.cit.). \textit{I. I. Mel'nik} [Uporjad. Mnozestv. Resetki 1, 56-65 (1971; Zbl 0244.08003)] has proved that if N(V) has a finite basis then V is also finitely based. In the paper under review a finitely based variety V is exposed for which the normal closure N(V) has no finite basis.
scientific article; zbMATH DE number 3977072
scientific article