Varieties of algebras whose critical algebras are basic (Q799777)

A finite algebra R is critical if the proper subalgebras and factor- algebras of R satisfy a polynomial identity which does not vanish for R. By analogy with group theory a critical algebra R is basic if the variety generated by R cannot be expressed as a union of proper subvarieties. The author considers algebras over a field with p elements, p being odd and prime. He obtains a characterization of the varieties \({\mathfrak M}\) such that every critical algebra from \({\mathfrak M}\) is basic. Two algebras A and B are found (A consists of 3\(\times 3\) upper triangular matrices with a zero diagonal and B is commutative) such that \({\mathfrak M}\) satisfies the desired property if and only if neither A nor B belongs to \({\mathfrak M}\). A description of \({\mathfrak M}\) in the language of polynomial identities is given as well.