Ladders and canonical varieties of completely regular semigroups
The lattice \({\mathcal L} (\mathcal{CR})\) of varieties of completely regular semigroups contains as an ideal the well-studied lattice \({\mathcal{L}} (\mathcal{B})\) of bands. Two associated complete congruences are defined on \({\mathcal L} (\mathcal{CR})\): \(\mathcal{U} \mathrel{\mathbb{B}^{\wedge}} \mathcal{V}\) if \(\mathcal{U} \cap \mathcal{B} = \mathcal{V} \cap \mathcal{B}\); and \(\mathcal{U} \mathrel{\mathbb{B}^{\vee}} \mathcal{V}\) if \(\mathcal{U} \vee \mathcal{B} = \mathcal{V} \vee \mathcal{B}\). The meet-semilattice \(\Delta\) of upper ends \(\mathcal{V}^{\mathbb{B}^\wedge}\) of the \(\mathbb{B}^{\wedge}\)-classes can be expected to carry significant information. With a few exceptions, the members of \(\Delta\) are intersections of the `canonical' varieties the author introduced in [J. Aust. Math. Soc. 83, No. 1, 87--104 (2007; Zbl 1142.20035)] and so are quite well understood. The first main theorem computes the upper ends \(\mathcal{V}\vee \mathcal{B}\) of the \(\mathbb{B}^{\vee}\)-classes of the members of \(\Delta\). As a useful tool, the Polák ladders are calculated for most of these varieties. Various properties of the resulting meet-semilattice of upper ends are also found.
scientific article; zbMATH DE number 6909513
scientific article