Some Varieties of Groups Closed to Being Just Non-Finitely Based
by
Alexei Krasilnikov
Department of Algebra, Moscow Pedagogical State University, Russia, visiting Department of Mathematics, University of Brasilia, Brazil

A variety of groups V is just non-finitely based (or limit ) if all proper subvarieties of V are finitely based but V itself is not. It follows easily from Zorn's lemma that if a variety is not finitely based then it contains a just non-finitely based subvariety. In this sense varieties which are just non-finitely based form a ``border'' between those which are finitely based and those which are not. It is known that infinitely many such varieties exist (Newman [1]) although no explicit examples are known. The problem of the construction of such examples is one of the most important open problems in the theory of varieties of groups.

Three years ago C.K.Gupta and the present speaker [2] constructed a non-finitely based variety of groups which is closed, in a certain sense, to being just non-finitely based. In my talk I am going to give new examples of non-finitely based varieties which are smaller (and so closer to being just non-finitely based) than one constructed in [2]. I will also discuss some conjecture related to the problem above.

1. Newman, M.F. Just non-finitely based varieties of groups. Bull. Austral. Math. Soc. 4 (1971), 343-348.

2. Gupta, C.K. and Krasilnikov, A.N. A solution of a problem of Plotkin and Vovsi and an application to varieties of groups. J. Austral. Math. Soc. (Series A) 67 (1999), 329-355.

Date received: December 31, 2001