Subdirect decompositions of the lattice of varieties of completely regular semigroups

Abstract

It is shown that if V is an element of the lattice of the title then the map given by U → (V ∧ U, V ∨ U) is a complete lattice embedding of into (V] × [V) if and only if V is a join-infinitely distributive element. In this case the image of the map is a subdirect product of the principal ideal (V] by the principal filter [V) generated by V. Some important varieties in are shown to be join-infinitely distributive.