Bigroups and a Limit Variety of Groups
by
Alexei Krasilnikov
Department of Mathematics, University of Brasilia, Brasilia-DF, 70904-970, Brazil

It follows easily from Zorn's lemma that if a variety of groups V is not finitely based then it contains a subvariety V^* such that all proper subvarieties of V^* are finitely based, but V^* itself is not. Any variety with these properties is called a limit or a just non-finitely based variety. In this sense limit varieties of groups form a "border" between those which are finitely based and those which are not. It is known that infinitely many such varieties exist (Newman, 1971) although no explicit examples are known.

In 2001 the first example of a limit variety V of bigroups was constructed by C.K.Gupta and the speaker. A bigroup is a pair (H, p) consisting of a group H and an idempotent endomorphism (projection) p of H. One can consider p as a unary operation on H so a bigroup is a universal algebra.

Let U be the variety of groups defined by

U = var { H | H = (H, p) ∈ V for some projection p: H → H } .

In other words, the variety of groups U is generated by all the bigroups H ∈ V if we consider them as groups and "forget" about the additional operation p on H. C.K.Gupta and the speaker conjectured that U is a limit variety of groups, that is, (i) U is a non-finitely based variety and (ii) each proper subvariety of U is finitely based. Our main result confirms the item (i) of the conjecture.

Theorem. The variety of groups U described above is non-finitely based.

Date received: February 22, 2006