The Lattice of Varieties of Completely Regular Semigroups

Polák’s theorem on the structure of the (lattice of) varieties of completely regular semigroups provides an isomorphic copy of the interval $[{cal S,CR}]$ of varieties which contain semilattices in terms of certain functions. We give a variant of this theorem for the lattice ${cal L(CR)}$ of all varieties of completely regular semigroups in terms of pairs with componentwise inclusion. The first entry of these pairs is a band variety and the second consists of a ?₀-tuple of members of ${cal K}_0$ . Here ${cal K}_0$ is the set of varieties which satisfy ${cal V}_K={cal V}$ where ${cal V}_K$ is the least element of the K-class containing ${cal V}$ . We have based the proof of our theorem on Polák’s theorem for the sake of expediency and comparison. It utilizes a set of varieties which we term canonical. Several corollaries treat various special cases.