A just non-finitely based variety of bigroups

A bigroup is a pair (H, pi) consisting of a group H and an idempotent endom orphism pi of H. One can consider pi as a unary operation on H so a bigroup is a universal algebra. The aim of our paper is to construct the first exa mple of a just non-finitely based variety of bigroups i.e. a variety which is non-finitely based but all whose proper subvarieties are finitely based. There is a close similarity between varieties of bigroups and varieties of groups so we hope that our result could help to construct a just non-finit ely based variety of groups.