Kernel classes of varieties of completely regular semigroups I

Abstract

Several complete congruences on the lattice ${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$ of varieties of completely regular semigroups have been fundamental to studies of the structure of ${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$. These are the kernel relation K, the left trace relation $T_{\ell }$, the right trace relation $T_r$ and their intersections $K\cap T_{\ell }, K\cap T_r$. However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K classes beyond the fact that they are intervals in ${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$ (with the notable exception of the remarkable results in the very recent article by Kad’ourek (Int J Algebra Comput, https://doi.org/10.1142/S0218196719500541, 2019)). Here we present a number of general results that are pertinent to the study of K classes. This includes a variation of the renowned Polák Theorem and its relationship to the complete retraction ${\mathcal {V}} \longrightarrow {\mathcal {V}}\cap {\mathcal {B}}$, where ${\mathcal {B}}$ denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K classes. The paper concludes with results hinting at the complexity of K classes in general, such as that the classes of relation $K/(K\cap T_{\ell })$ may have the cardinality of the continuum.