Algebras with single defining relation (Q6597860)

The paper under review surveys classical and recent results concerning Schreier varieties. Recall that a Schreier variety of algebras is one where the subalgebras of every free algebra in the variety are once again free. The name comes fom Group Theory: the well known theorem of Nielsen and Schreier gives us that the variety of all groups satisfies this property. Theorems of Witt and of Shirshov state that the variety of all Lie algebras is Schreier one as well. On the other hand, the variety of all associative algebras is not Schreier. Clearly one can extend the above notion to any class of algebraic systems that can be defined in terms of defining relations, and where it is meaningful to define a free object. The author gives an extensive list of theorems concerning Schreier varieties of algebras. He also recalls results concerning the well known Freiheitssatz. The relation of these two notions to varieties defined by one relation is considered also in great detail. The readers of the paper will benefit from the huge list of references, some 114 titles in all. These include most of the major contributions to the area.