The relation ν defined on the
lattice ℒ(ℐ) of varieties of inverse semigroups by 𝒰ν𝒱 if and only if 𝒰∩𝒢 = 𝒱∩𝒢
and 𝒰∨𝒢 = 𝒱∨𝒢, where 𝒢 is the variety of groups, is a congruence. It
is known that varieties belonging to the first three layers of ℒ(ℐ) (those
varieties belonging to the lattice ℒ(𝒮ℐ) of varieties of strict inverse semigroups)
possess trivial ν-classes and that there exist non-trivial ν-classes in the next
layer of ℒ(ℐ). We show that there are infinitely many ν-classes in the fourth
layer of ℒ(ℐ), and also higher up in ℒ(ℐ), that in fact contain an infinite
descending chain of varieties. To find these chains we first construct a collection of
semigroups in ℬ1, 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 ℬ1, we obtain an infinite descending
chain in the ν-class of 𝒰∨ℬ1, for every non-trivial abelian group variety
𝒰.
