Some infinite chains in the lattice of varieties of inverse semigroups

The relation v defined on the lattice £?{J) of varieties of inverse semigroups by W υ Ψ* if and only if % Π & = T Π & and ^ V & = ^V-f, where S? is the variety of groups, is a congruence. It is known that varieties belonging to the first three layers of &(*?) (those varieties belonging to the lattice Sfψί?) of varieties of strict inverse semigroups) possess trivial v-classes and that there exist non-trivial z/-classes in the next layer of Jΐf^). We show that there are infinitely many v -classes in the fourth layer of S?(*f), and also higher up in J?(J r ), that in fact contain an infinite descending chain of varieties. To find these chains we first construct a collection of semigroups in 38ι, the variety generated by the five element combinatorial Brandt semigroup with an identity adjoined. By considering wreath products of abelian groups and these semigroups from J^we obtain an infinite descending chain in the v -class of ^ V 38X, for every non-trivial abelian group variety %.