Kernel Classes of Varieties of Completely Regular Semigroups I

TL;DR

This paper investigates the internal structure of kernel classes within the lattice of varieties of completely regular semigroups, providing new theoretical results and highlighting their potential complexity.

Contribution

It introduces general results on K-classes, including a variation of Polak's Theorem, and explores their structure and complexity in the context of completely regular semigroups.

Findings

K-classes are intervals in L(CR)

Variation of Polak's Theorem related to retraction V -> V ∩ B

K/Kl classes may have continuum cardinality

Abstract

Several complete congruences on the lattice L(CR) of varieties of completely regular semi- groups have been fundamental to studies of the structure of L(CR). These are the kernel relation K , the left trace relation Tl , the right trace relation Tr and their intersections K\capTl,K\cap Tr . 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 L(CR). Here we present a number of general results that are pertinent to the study of K -classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction V -> V \cap B , where 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/Kl may have the cardinality of the continuum.