A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties

A finite combinatorial inverse semigroup Theta of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Theta possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Theta, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.