Varieties of structurally trivial semigroups II

Melnik [5] determined completely the lattice of all 2-nilpotent extensions of rectangular band varieties; and Koselev [4] determined a distributive sublattice formed by certain varieties of n-nilpotent extensions of left zero bands. In [2] the author described the skeleton of the lattice of all 3-nilpotent extensions of rectangular bands. We generalize these results by proving that a certain family of semigroup varieties which includes all the varieties mentioned above, and referred to here as planar varieties, consisting of certain n-nilpotent extensions of rectangular bands forms a distributive sublattice that looks somewhat like an inverted pyramid. Our proof makes use of a countably infinite family of injective endomorphisms on the lattice of all semigroup varieties that was introduced by the author in [1]. Although we do not determine completely the lattice of all n-nilpotent extensions of rectangular band varieties, our result unifies certain previously known results and provides a framework for further research.